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PRAISE FOR AMIR D. ACZEL'S
FERMAT'S LAST THEOREM
"This is a captivating volume. . . . The brilliant backdoor method
used by Mr. Wiles as he reached his solution, along with the debt he
owed to many other contemporary mathematicians, is graspable in
Mr. Aczel's lucid prose. Equally important is the sense of awe that
Mr. Aczel imparts for the hidden, mystical harmonies of numbers,
and for that sense of awe alone, his slender volume is well worth the
effort."
—The New York Times
"For more than three centuries, Fermat's Last Theorem was the
most famous unsolved problem in mathematics, here's the story of
how it was solved. . . . An excellent short history of mathematics,
viewed through the lens of one of its great problems— and
achievements."
—Kirkus Reviews
"This exciting recreation of a landmark discovery reveals the great
extent to which modern mathematics is a collaborative
enterprise. . . . While avoiding technical details, Aczel maps the
strange, beautiful byways of modern mathematical thought in ways
the layperson can grasp."
—Publishers Weekly
"Briefly chronicles the history of the famous problem of the title,
which was recently solved by a mathematician named Andrew Wiles
after he had devoted seven years to the task. . . . Aczel does a
superb job of creating in the nonmathematical reader the illusion of
comprehension."
—The New Yorker
Pierre de Fermat (1601-1665).
FERMAT'S LAST THEOREMUnlocking the Secret ojan Ancient
Mathematical Problem
By Amir D. Aczel
D e l t a
A Delta Book
Published by Dell
Publishing a
division of
Bantam Doubleday Dell Publishing Group, Inc.
1540 Broadway
New York, New York 10036
Copyright © 1996 by Amir D Aczel
All rights reserved. No part of this book may be reproduced or transmitted in any
form or by any means, electronic or mechanical, including photocopying,
recording, or by any information storage and retrieval system, without the written
permission of the Publisher, except where permitted by law. For information
address: Four Walls Eight Windows, New York, NY, 10011.
The trademark Delta® is registered in the U.S. Patent and Trademark Office and
in other countries.
ISBN: 0-385-31946-0
Reprinted by arrangement with Four Walls Eight Windows
Manufactured in the United States of America Published
simultaneously in Canada
October 1997
10 9 8 7 6 5 4 3 2 1
BVG
Preface
In June 1993, my old friend Tom Schulte was visiting me in
Boston from California. We were sitting at a sunny sidewalk cafe on
Newbury Street, tall, icy drinks in front of us. Tom had just gotten
divorced, and he was ruminative. He half turned toward me. "By the
way," he said, "Fermat's Last Theorem has just been proved." This
must be a new joke, I thought, as Tom's attention was back on the
sidewalk. Twenty years earlier, Tom and I were roommates, both of
us undergraduate students in mathematics at the University of
California at Berkeley. Fermat's Last Theorem was something we
often talked about. We also discussed functions, and sets, and
number fields, and topology. None of the math students slept much
at night, since our assignments were so difficult. That's what
distinguished us from students in most other areas. Sometimes
we'd have math nightmares . . . trying to prove some theorem or
another before it was due in the morning. But Fermat's Last
Theorem? No one ever believed it would be proven in our lifetime.
The theorem was so difficult, and so many people had tried to
prove it for over three hundred years. We were well aware that
entire branches of mathematics had been developed as the result
of attempts to prove the theorem. But the attempts failed, one by
one. Fermat's Last Theorem had come to symbolize the unat-
tainable. I once even used the theorem's perceived impossibility to
my advantage. It was a few years later, also at Berkeley, when I
had already graduated in math and was getting my Master's in
operations research. An arrogant graduate student
in mathematics, unaware of my own background in math, offered
me his help when we met at the International House where we both
lived. "I'm in pure mathematics," he said. "If you ever have a math
problem you can't solve, feel free to ask me." He was about to
leave, when I said "Urn, yes. There is something you can help me
with . . . " He turned back, "Why, sure, let me see what it is." I
pulled open a napkin—we were at the dining room. I slowly wrote
on it:
xn + yn = zn has no whole number solution when n is greater
than 2.
"I've been trying to prove this one since last night," I said,
handing him the napkin. I could see the blood draining from his
face. "Fermat's Last Theorem," he grunted. "Yes," I said, "You're in
pure math. Could you help me?" I never saw that person from up
close again.
"I am serious," Tom said, finishing his drink. "Andrew Wiles. He
proved Fermat's Last Theorem in Cambridge last month. Remember
that name. You'll hear it a lot." That night, Tom was in the air, flying
back to California. In the next months, I realized Tom didn't play a
joke on me, and I followed the sequence of events where Wiles was
at first applauded, then the hole in his proof was found, then he
withdrew for a year, and finally reemerged with a corrected proof.
But following the continuing saga, I learned that Tom was wrong. It
wasn't Andrew Wiles's name I should have paid attention to, or at
least not his alone. I, and the world, should have recognized that
the proof of Fermat's Last Theorem was far from the work of one
mathematician. While Wiles got much of the praise,
the accolades belong to others as much: Ken Ribet, Barry Mazur,
Goro Shimura, Yutaka Taniyama, Gerhard Frey and others. This
book tells the entire story, including what happened behind the
scenes and out of the view of the media's cameras and floodlights.
It is also a story of deception, intrigue, and betrayal.
IN
"Perhaps I could best describe my experience of doing
mathematics in terms oj entering a dark mansion. You go into the
first room and it's dark, completely dark. You stumble around,
bumping into the furniture. Gradually, you learn where each piece
of furniture is. And finally, after six months or so, you find the light
switch and tum it on. Suddenly, it's all illuminated and you can see
exactly where you were. Then you enter the next dark room . . . "
This is how Professor Andrew Wiles described bis seven-year
Quest for the mathematicians' Holy Grail.
Just before dawn on June 23, 1993, Professor John Conway
approached the darkened mathematics building on the Princeton
University campus. He unlocked the front door and quickly walked
up to his office. For weeks preceding his colleague Andrew Wiles'
departure for England, persistent but unspecific rumors had been
circulating in the world's mathematical community. Conway was
expecting something important to happen. Exactly what it was, he
had no idea. He turned on his computer and sat down to stare at
the screen. At 5:53 AM, a terse e-mail message flashed from
across the Atlantic: "Wiles proves FL.T."
Cambridge, England, June 1993
Late in June of 1993, Professor Andrew Wiles flew to England. He
was returning to Cambridge University, where he had been a
graduate student twenty years earlier. Wiles' former doctoral thesis
adviser at Cambridge, Professor John Coates, was organizing a
conference on Iwasawa Theory—the particular area within number
theory in which Andrew Wiles did his dissertation and about which
he knew a great deal. Coates had asked his former student if he
would mind giving a short, one-hour talk at the conference on a
topic of his choice. To his great surprise and that of the other
conference organizers, the shy Wiles—previously reluctant to speak
in public—responded by asking if he could be given three hours of
presentation.
The 40-year-old Wiles looked the typical mathematician when he
arrived in Cambridge: white dress shirt with sleeves rolled up
carelessly, thick horn-rimmed glasses, unruly strands
of thinning light hair. Born in Cambridge, his return was a very
special kind of homecoming—it was the realization of a childhood
dream. In pursuit of this dream, Andrew Wiles had spent the last
seven years of his life a virtual prisoner in his own attic. But he
hoped that soon the sacrifice, the years of struggle and the long
hours of solitude would end. Soon he might be able to spend more
time with his wife and daughters, of whom he had seen so little for
seven years. He had often failed to show up for lunch with his
family, missed afternoon tea, barely made it to dinner. But now the
accolades would be his alone.
The Sir Isaac Newton Institute for Mathematical Sciences at
Cambridge had only recently opened by the time Professor Wiles
arrived to deliver his three hour-long lectures. The Institute is
spacious, set in scenic surroundings at some distance from the
University of Cambridge. Wide areas outside the lecture halls are
furnished with plush, comfortable chairs, designed to help facilitate
the informal exchange of ideas among scholars and scientists, and
to promote learning and knowledge.
Although he knew most of the other mathematicians who came
to the specialized conference from around the world, Wiles kept to
himself. When colleagues became curious about the length of his
scheduled presentation, Wiles would only say they should come to
his lectures and find out for themselves. Such secretiveness was
unusual, even for a mathematician. While they often work alone
trying to prove theorems and are generally not known to be the
world's most gregarious people, mathematicians usually share
research results with each other. Mathematical results are freely
circulated by their authors in the form of research preprints. These
preprints bring their authors outside
comments that help them improve the papers before they are
published. But Wiles didn't hand out preprints and didn't discuss his
work. The title of Wiles' talks was "Modular Forms, Elliptic Curves,
and Galois Representations," but the name gave no hint where the
lectures would lead, and even experts in his field could not guess.
The rumors intensified as time went on.
On the first day, Wiles rewarded the 20 or so mathematicians
who came to his lecture with a powerful and unexpected
mathematical result—and there were still two more lectures to go.
What was coming? It became clear to everyone that Wiles' lectures
were the place to be, and the suspense grew as expectant
mathematicians flocked to the lectures.
On the second day, Wiles' presentation intensified. He had
brought with him over 200 pages of formulas and derivations,
original thoughts stated as new theorems with their lengthy,
abstract proofs. The room was now filled to capacity. Everyone
listened intently. Where would it lead? Wiles gave no hint. He
dispassionately continued writing on the blackboard and when he
was done for the day, he quickly disappeared.
The next day, Wednesday, June 23, 1993, was his last talk.
As he neared the lecture hall, Wiles found it necessary to push his
way in. People stood outside blocking the entrance and the room
was overflowing. Many carried cameras. As Wiles again wrote
seemingly endless formulas and theorems on the board, the
tension increased. "There was only one possible climax, only one
possible end to Wiles' presentation," Professor Ken Ribet of the
University of California at Berkeley later told me. Wiles was
finishing the last few lines of his proof of an enigmatic and
complicated conjecture in mathematics, the
Shimura-Taniyama Conjecture. Then suddenly he added one final
line, a restatement of a centuries-old equation, one which Ken Ribet
had proved seven years earlier would be a consequence of the
conjecture. "And this proves Fermat's Last Theorem," he said,
almost offhandedly. "I think I'll stop here."
There was a moment of stunned silence in the room. Then the
audience (erupted in spontaneous applause. Cameras flashed as
everyone stood up to congratulate a beaming Wiles. Within
minutes, electronic mail flashed and faxes rolled out of machines
around the world. The most celebrated mathematical problem of all
time appeared to have been solved.
"What was so unexpected was that the next day we were
deluged by the world press," recalled Professor John Coates, who
organized the conference without having the slightest idea that it
would become the launching ground for one of the greatest
mathematical achievements. Headlines in the world's newspapers
hailed the unexpected breakthrough. "At Last, Shout of 'Eureka!' In
Age-Old Math Mystery" announced the front page of the New York
Times on June 24, 1993. The Washington Post called Wiles in a
major article "The Math Dragon-Slayer," and news stories
everywhere described the person who apparently solved the most
persistent problem in all of mathematics, one that had defied
resolution for over 350 years. Overnight, the quiet and very
private Andrew Wiles became a household name.
Pierre de Fermat
Pierre de Fermat was a seventeenth-century French jurist who was
also an amateur mathematician. But while he was technically
an "amateur" since he had a day job as a jurist, the leading histo-
rian of mathematics E. T. Bell, writing in the early part of the
twentieth century, aptly called Fermat the "Prince of Amateurs."
Bell believed Fermat to have achieved more important mathe-
matical results than most "professional" mathematicians of his day.
Bell argued that Fermat was the most prolific mathematician of the
seventeenth century, a century that witnessed the work of some of
the greatest mathematical brains of all time.1
One of Fermat's most stunning achievements was to develop the
main ideas of calculus, which he did thirteen years before the birth
of Sir Isaac Newton. Newton and his contemporary Gottfried
Wilhelm von Leibniz are jointly credited in the popular tradition with
having conceived the mathematical theory of motion, acceleration,
forces, orbits, and other applied mathematical concepts of
continuous change we call calculus.
Fermat was fascinated with the mathematical works of ancient
Greece. Possibly he was led to his conception of calculus ideas by
the work of the classical Greek mathematicians Archimedes and
Eudoxus, who lived in the third and fourth centuries B.C.,
respectively. Fermat studied the works of the ancients—which were
translated into Latin in his day—in every spare moment. He had a
full-time job as an important jurist, but his hobby—his passion—was
to try to generalize the work of the ancients and to find new beauty
in their long-buried discoveries. "I have found a great number of
exceedingly beautiful theorems," he once said. These theorems he
would jot down in the margins of the translated copies of ancient
books he possessed.
Fermat was the son of a leather merchant, Dominique Fer-
Pierre de Fermat's "Last Theorem" as reproduced in an edition of Diophantus'
Arithmetica published by Fermat's son Samuel. The original copy of Diophantus with
Fermat's handwritten note has never been found.
mat, who was Second Consul in the town of Beaumont-de-Lomagne,
and of Claire de Long, the daughter of a family of parliamentary
judges. The young Fermat was born in August, 1601 (baptized
August 20 in Beaumont-de-Lomagne), and was raised by his
parents to be a magistrate. He went to school in Toulouse, and was
installed in the same city as Commissioner of Requests at the age
of thirty. He married Louise Long, his mother's cousin, that same
year, 1631. Pierre and Louise had three sons and two daughters.
One of their sons, Clement Samuel, became his father's scientific
executor and published his father's works after his death. In fact, it
is the book containing Fermat's work, published by his son, that has
come down to us and from which we know his famous Last
Theorem. Clement Samuel de Fermat recognized the importance of
the theorem scribbled in the margin and added it to the translation
of the ancient work he republished.
Fermat's life is often described as quiet, stable, and uneventful.
He did his work with dignity and honesty, and in 1648 was
promoted to the important position of the King's Councillor-ship in
the local Parliament of Toulouse, a title he held for seventeen years
until his death in 1665. Considering the great work Fermat did for
the Crown, a lifetime of devoted, able, and conscientious service,
many historians are puzzled that he had the time and the mental
energy to do first-rate mathematics—and volumes of it. One French
expert suggested that Fermat's official work was actually an asset
to his mathematical studies, since French parliamentary councilors
were expected to minimize their unofficial contacts in order to
avoid the temptations of bribery and other corruption. Since Fermat
certainly required a
diversion from his hard work, and since he had to limit his social
life, mathematics probably offered a much-needed break. And the
ideas of calculus were far from Fermat's only achievement. Fermat
brought us number theory. An important element in number theory
is the concept of a prime number.
Prime Numbers
The numbers two and three are prime numbers. The number four is
not prime because it is the product of two and two: 2 x 2 = 4.
The number five is prime. The number six is not prime since, like
four, it is the product of two numbers: 2 x 3 = 6. Seven is
prime, eight is not (2 x 2 x 2 = 8), nine is not (3 x 3=9), and
ten is not (2 x 5=10). But eleven again is a prime number since
there are no integers (other than eleven itself and one), which can
be multiplied together to give us 1 1 . And we can continue this
way: 12 is not prime, 13 is, 14 is not, 15 is not, 16 is not, 17
is prime, and so on. There is no apparent structure here, such as
every fourth number is not a prime, or even any more complicated
pattern. The concept has mystified human beings since early
antiquity. Prime numbers are the essential elements in number
theory, and the lack of easily-seen structure tends to make number
theory seem un-unified as a field, and its problems isolated, difficult
to solve, and without clear implications to other fields of
mathematics. In the words of Barry Mazur: "Number theory
produces, without effort, innumerable problems which have a
sweet, innocent air about them, tempting flowers,- and yet . . .
number theory swarms with bugs, waiting to bite the tempted
flower-lovers who, once bitten, are inspired to excesses of effort!"2
A Famous Note on the Margin
Fermat was smitten by the charm of numbers. In them he found
beauty and meaning. He came up with a number of theorems in
number theory, one of which was that every number of the form
22An+l (two raised to the power two raised to the power n, plus one)
is a prime number. Later, it was discovered that the theorem was
false when a number of this form was found not to be a prime.
Among Fermat's cherished Latin translations of ancient texts was
a book called the Arithmetka, written by the Greek mathematician
Diophantus, who lived in Alexandria in the third century A.D. Around
1637, Fermat wrote in Latin in the margin of his Diophantus, next
to a problem on breaking down a squared number into two squares:
On the other hand, it is impossible to separate a cube into two
cubes, or a biquadrate into two biquadrates, or generally any
power except a square into two powers with the same exponent.
I have discovered a truly marvelous proof of this, which,
however, the margin is not large enough to contain.
This mysterious statement kept generations of mathematicians
busy trying to supply the "truly marvelous proof" Fermat claimed to
have possessed. The statement itself, that while a square of a
whole number could be broken down into two other squares of
whole numbers (for example, five squared, which is twenty-five,
equals the sum of four squared (sixteen) and three squared [nine]),
but that the same cannot be done with cubes or any higher powers,
looked deceptively simple.
All of Fermat's other theorems were either proved or disproved by
the early 1800s. This seemingly simple statement remained
unsettled, and therefore was given the name "Fermat's Last The-
orem." Was it indeed true? Even in our own century, computers
were stymied in attempts to verify that the theorem was true.
Computers could verify the theorem for very large numbers, but
they couldn't help for all numbers. The theorem could be tried on
billions of numbers, and there still would be infinitely many—and
infinitely many exponents—to check. To establish Fermat's Last
Theorem, a mathematical proof was required. Awards were offered
in the 1800s by the French and German scientific academies to
anyone who would come up with a proof, and every year thousands
of mathematicians and amateurs, along with cranks, sent "proofs"
to mathematical journals and judging committees—always coming
up empty-handed.
July-August, 1993—A Fatal Flaw is Discovered Mathematicians were
cautiously optimistic when Wiles stepped down from the podium
that Wednesday in June. Finally, the 350 -year-old mystery seemed
to have been solved. Wiles' lengthy proof, using complicated
mathematical notions and theories which were not known during
the time of Fermat or indeed until the twentieth century, needed to
be validated by independent experts. The proof was sent to a
number of leading mathematicians. Perhaps seven years of working
alone in the seclusion of his attic had finally paid off for Wiles. But
the optimism was short-lived. Within weeks, a hole was discovered
in Wiles' logic. He tried to patch it, but the gap would simply not go
away. Princeton mathematician Peter Sarnak, a
close friend of Andrew Wiles, watched him agonize every day over
the proof he had told the entire world he possessed only two
months earlier in Cambridge. "It's as if Andrew was trying to lay an
over-sized carpet on the floor of a room," Sarnak explained. "He'd
pull it out, and the carpet would fit perfectly in one side of the
room, but across the room it would be up against the wall, so he
would go there and pull it down . . . and then it would pop up in
another place. Whether or not the carpet had the right size for the
room was not something he was able to determine." Wiles withdrew
into his attic. The reporters from the New York Times and the rest
of the media left him to his lonely task. As time went by without a
proof, mathematicians and the public in general began to wonder
whether Fermat's theorem was at all true. The marvelous proof
Professor Wiles convinced the world he possessed became no more
real than Fermat's own "truly marvelous proof which the margin is
unfortunately too small to contain."
Between the Tigris and Euphrates Rivers, Circa 2 0 0 0 B.C. The story
of Fermat's Last Theorem is much, much older than Fermat himself.
It's even older than Diophantus, whose work Fermat was trying to
generalize. The origins of this simple-looking yet profound theorem
are as old as human civilization itself. They are rooted in the Bronze
Age culture that developed in the Fertile Crescent between the
Tigris and Euphrates rivers of ancient Babylon (an area within
today's Iraq). And while Fermat's Last Theorem is an abstract
statement with no applications in science, engineering,
mathematics—not even in number theory, its own niche within
mathematics—the
roots of this theorem are grounded in the everyday life of the
people of Mesopotamia of 2000 B.C. .
The era from 2000 B.C. to 600 B.C. in the Mesopotamian
valley is considered the Babylonian Era. This time saw remarkable
cultural developments, including writing, the use of the wheel, and
metal works. A system of canals was used for irrigating large tracts
of land between the two rivers. As civilization flourished in the
fertile valley of Babylon, the ancient people who inhabited these
planes learned to trade and to build prosperous cities such as
Babylon and Llr (where Abraham was born). Even earlier, by the
end of the fourth millennium B.C., a primitive form of writing had
already developed in both the Mesopotamian and the Nile river
valleys. In Mesopotamia, clay was abundant and wedge-shaped
marks were impressed with a stylus on soft clay tablets. These
tablets were then baked in ovens or left to harden in the sun. This
form of writing is called cuneiform, a word derived from the Latin
word cuneus, meaning wedge. The cuneiform constitutes the first
known writing the world has ever seen. Commerce and
construction in Babylon and in ancient Egypt brought the need for
accurate measurements. The early scientists of these Bronze Age
societies learned to estimate the ratio between the circumference
and the diameter of a circle, which gave them a number close to
what we call today pi. The people who built the giant Ziggurat, the
biblical Tower of Babel, and the Hanging Gardens, one of the Seven
Wonder of the Ancient World, needed a way to compute areas and
volumes.
Wealth Is a Squared Quantity
A sophisticated number system was developed using base sixty and
Babylonian engineers and builders were able to compute the
quantities required in their everyday professional lives. Squares of
numbers appear naturally in life, although it doesn't seem so at first
glance. Squares of numbers can be viewed as representing wealth.
A farmer's prosperity is dependent on the amount of crops he is
able to produce. These crops, in turn, depend on the area that is
available to the farmer. The area is a product of the length and the
width of the field, and this is where squares come in. A field that
has length and width equal to a has area equal to a-squared. In this
sense, therefore, wealth is a squared quantity.
The Babylonians wanted to know when such squares of whole
numbers could be partitioned into other squares of whole numbers.
A farmer who owned one field of twenty-five square units of land
could swap it for two square fields: one measuring sixteen squared
units and the other nine squared units. So a field five units by five
units was equivalent to two fields, one four by four and the other
three by three. This was important information for the solution of a
practical problem. Today we would write this relationship in the
form of an equation: 52 = 3 2 + 4 2 . And triples of such integers,
here 3, 4, and 5, whose squares satisfy this relation, are called
Pythagorean triples—even though they were known to the
Babylonians over one thousand years before the time of the famous
Greek mathematician, Pythagoras, after whom they are named. We
know all this from an unusual clay tablet dated to around 1900
B.C.
F E R M A T ' S L A S T T H E O R E M
"Plimpton 322"
The Babylonians were obsessed with tables. And the abundance of
clay and the cuneiform writing technology they possessed allowed
them to create many of them. Because of the clay tablets'
durability, many of them survive today. From one location alone,
the site of ancient Nippur, over 50,000 tablets were recovered
and are now in the collections at the museums of Yale, Columbia,
and the University of Pennsylvania, among others. Many of these
tablets are in the basements of the muse-' urns, gathering dust,
lying there unread and undeciphered.
One tablet that was deciphered is remarkable. This tablet, in the
museum of Columbia University, is called Plimpton 322. All it
contains are 15 triples of numbers. Each one of the triples has the
property that the first number is a square and is the sum of the
other two, each being itself a squared number—the table contains
fifteen Pythagorean triples.3 The numbers 25 = 16 + 9, given
earlier, form a Pythagorean triple. Another Pythagorean triple on
Plimpton 322 is 169 = 144 + 25 (13 2 = 122 + 5 2). Not
all scholars agree on the reason for the ancient Babylonians'
interest in these numbers. One theory is that the interest was solely
for practical purposes, and the fact that they used a number system
with base sixty and therefore preferred integers to fractions
supports this need to solve practical problems with nice, whole
square numbers. But other experts think that an inherent interest in
numbers themselves may also have been a motivator for the
Babylonians' interest in square numbers. It seems that, whatever
the motive, Plimpton 322 may have served as a tool for teaching
students to solve problems where the numbers are perfect squares.
The Babylonians' approach was not to develop a general theory
for solving such problems, but rather to provide tables listing triples
of numbers and—apparently—to teach pupils how to read and use
these tables.
An Ancient Society of Number-Worshippers Sworn to Secrecy
Pythagoras was born on the Greek island of Samos around 580 B.C.
He travelled extensively throughout the ancient world and visited
Babylon, Egypt, and possibly even India. In his travels, especially in
Babylon, Pythagoras came in contact with mathematicians and likely
became aware of their studies of numbers now named after him—
the Pythagorean triples,
which Babylonian scientists and mathematicians had known about
for over 1500 years. Pythagoras came in contact with the
builders of magnificent works of art and architecture, and the
mathematical aspects of these wonders could not escape him.
Pythagoras was also exposed in his travels to religious and
philosophical ideas of the East.
When Pythagoras returned to Greece he left the island of Samos
and moved to Crotona, on the Italian "boot." It is interesting to note
that Pythagoras certainly saw most of the Seven Wonders of the
Ancient World. One of these wonders, the Temple of Hera, is right
where Pythagoras was born on Samos. Today, the ruins of the
magnificent temple—only one standing column remains from
among hundreds—are a short walk away from the modern town of
Pythagorion, named in honor of the island's illustrious son. Just
across the strait and a few miles to the north, in modern-day
Turkey, lie another of the Wonders in the remains of ancient
Ephesus. The Colossus of Rhodes is nearby, to the south of Samos;
the Pyramids and the Sphynx are in Egypt and Pythagoras saw
those,- and in Babylon he must have seen the Hanging Gardens.
The Italian boot, including Crotona where Pythagoras settled, as
well as much of the rest of southern Italy, were at that time part
of the Greek world—Magna Graecia. This "greater Greece" included
settlements al l over the eastern Mediterranean, including
Alexandria in Egypt with its large ethnic Greek population—
descendants of which remained there through the early 1900s.
Not far from Crotona were caves for oracles just like the Oracle of
Delphi, who was believed to foretell fortunes and futures of people
and nations.
A M I R D . A C Z E L
"Number Is
Everything"
In the barren, stark surroundings of the tip of Italy, Pythagoras
founded a secret society dedicated to the study of numbers. The
society, whose members became known collectively as the
Pythagoreans, is believed to have developed a substantial body of
mathematical knowledge—all in complete secrecy. The
Pythagoreans are believed to have followed a philosophy sum-
marized by their motto that "number is everything." They wor-
shipped numbers and believed them to have magical properties. An
object of interest to them was a "perfect" number. One of the
definitions of a perfect number—a concept that continued to be
pursued in the Middle Ages, and appears in mystical systems such
as the Jewish Kabbalah—is a number that is the sum of its
multiplicative factors. The best and simplest example of a perfect
number is the number six. Six is the product of three and two and
one. These are the multiplicative factors of this number, and we
have: 6 = 3 x 2 x 1 . But note also that if you add the same
factors you will again get the number six: 6 = 3 + 2 + 1 . In
that sense, six is "perfect." Another perfect number is 28, since
the numbers that can divide 28 (without remainder) are 1, 2, 4,
7, and 14, and we note that also: 1 + 2 + 4 + 7 + 14 =
28.
The Pythagoreans followed an ascetic lifestyle, and were strict
vegetarians. But they did not eat beans, thinking they resembled
testicles. Their preoccupations with number were very much in the
spirit of a religion, and their strict vegetarianism originated in
religious beliefs. While no documents survive dating to the time of
Pythagoras, there is a large body of later literature about the
master and his followers, and Pythagoras himself is considered one
of the greatest mathematicians of
antiquity. To him is attributed the discovery of the Pythagorean
Theorem concerning the squares of the sides of a right triangle,
which has strong bearing on Pythagorean triples and, ultimately, on
Fermat's Last Theorem two thousand years later.
The Square oj the Hypotenuse Is Equal to the
Sum oj the Squares oj the Other Two Sides. . .
The theorem itself originated in Babylon, since the Babylonians
clearly understood "Pythagorean" triples. The Pythagoreans,
however, are credited with setting the problem in geometric
terms, and thus generalizing them away from strictly the natural
numbers (positive integers without zero). The Pythagorean
Theorem says that the square of the hypotenuse of a right-triangle
is equal to the sum of the squares of the two remaining sides of the
triangle, as shown above.
When the hypotenuse is an integer (such as 5, whose square is
25), the general Pythagorean solution in terms of the sum of two
squares will be the integers four (whose square is sixteen) and
three (whose square is nine). So the Pythagorean Theorem, when
applied to integers (whole numbers such as 1, 2, 3, . . . )
gives us the Pythagorean triples which were known a millennium
earlier in Babylon.
Incidentally, the Pythagoreans also knew that squared numbers
are sums of sequences of odd numbers. For example, 4 = 1 +
3,- 9 = 1 + 3 + 5,- 16= 1 + 3 + 5 + 7, and so on. This
property they represented by a visual array of numbers in a square
pattern. When the odd number of dots along two adjacent sides is
added to the previous square, a new square is formed:
Whole Numbers, Fractions, and What Else?
But the Pythagoreans knew a lot more than whole numbers and
fractions (numbers such as 1/2, 1/3, 5/8, 147/1769, etc.),
which were known in antiquity both in Babylon and in Egypt. The
Pythagoreans are the ones who discovered the irrational numbers—
that is, numbers that cannot be written as fractions but have to be
written as unending, non-repeating decimals. One such example is
the number pi (3.141592654 . . . ), the ratio of the
circumference of a circle to its diameter. The number pi is
unending,- it would take forever to write it down completely since it
has infinitely many (that is, neverending) distinct digits. To write it
down, we simply say: pi ( K ) . Or, we can write it to any finite
number of decimals, such as 3.14, 3.1415, or so. Computers
were used in our century to compute and write down pi to a million
or more digits, but this is rarely necessary. Pi was known to within
various approximations to the Babylonians and Egyptians of the
second millennium B.C. They took it as roughly three, and it arose
naturally as a consequence of the discovery of the wheel. Pi also
arose in various measurements of a pyramid. Pi is even alluded to
in the Old Testament: in Kings I, 7:23, we read about a circular
wall being constructed. From the given number of units for the
circumference and the diameter, we can conclude that the ancient
Israelites took pi to be roughly three.
The Pythagoreans discovered that the square root of two was an
irrational number. From an application of the Pythagorean Theorem
to a right triangle with two sides both equal to one unit, the
Pythagoreans obtained that the hypotenuse was a strange number:
the square root of two. They could tell that this
number was not an integer, nor even a fraction, a ratio of two
integers. This was a number with an unending decimal represen-
tation which did not repeat itself. As is the case with pi, to write
down the exact number that is the square root of two
(1.414213562 . . . ) would take forever, since there are infi-
nitely many digits, forming a unique sequence (rather than a
repeating one such as
1.857142857142857142857142857 . . . , etc., which
one could describe without having to actually write down every
digit). Any number with a repeating decimal representation (here,
the sequence 857142 repeats itself over and over again in the
decimal part of this number), is a rational number, that is, a
number that can also be written in the form alb, that is, the ratio of
two integers. In this example, the two integers are 13 and 7. The
ratio 13/7 is equal to 1.857142857142857142857142 . .
. , the pattern 857142 repeating itself forever.
The discovery of the irrationality of the square root of two
surprised and shocked these diligent number admirers. They swore
never to tell anyone outside their society. But word got out. And
legend has it that Pythagoras himself killed by drowning the
member who divulged to the world the secret of the existence of
the strange, irrational numbers.
The numbers on the number line are of two distinct kinds:
rational and irrational. Looked at together, they fill the entire line
with no holes. The numbers are very, very close (inflnites-imally
close) to each other. The rational numbers are said to be
everywhere dense within the real numbers. Any neighborhood, any
tiny interval, around a rational number contains infinitely many of
these irrational numbers. And vice versa,
around every irrational number there are infinitely many rational ones.
Both sets, the rational and the irrational numbers, are infinite. But the
irrationals are so numerous that there are more of them than there are
rational numbers. Their order of infinity is higher. This fact was shown
in the 1800s by the mathematician Georg Cantor (1845-1918). At
the time, few people believed Cantor. His arch-enemy Leopold
Kronecker (1823-1891) taunted and ridiculed Cantor for his
theories about how many rational and irrational numbers- there are.
Kronecker is known for his statement, "God made the integers, and all
the rest is the work of man," meaning that he did not even believe that
the irrational numbers, such as the square root of two, existed! (This,
over two millennia after the Pythagoreans.) His antagonism is blamed
for having prevented Cantor from obtaining a professorship at the
prestigious University of Berlin, and ultimately for Cantor's frequent
nervous breakdowns and his ending up at an asylum for the mentally
infirm. Today, all mathematicians know that Cantor was right and that
there are infinitely many more irrational numbers than rational ones,
even though both sets are infinite. But did the ancient Greeks know
that much?4
F E R M A T ' S L A S T T H E O R E M The
Pythagorean Legacy
An important aspect of Pythagorean life, with its dietary rules and
number worship and secret meetings and rituals, was the pursuit of
philosophical and mathematical studies as a moral basis. It is
believed that Pythagoras himself coined the words philosophy: love
of wisdom,- and mathematics-, that which is learned. Pythagoras
transformed the science of mathematics into a liberal form of
education.
Pythagoras died around 500 B.C. and left no written records of
his work. His center at Crotona was destroyed when a rival political
group, the Sybaritics, surprised the members and murdered most
of them. The rest dispersed about the Greek world around the
Mediterranean, carrying with them their philosophy and number
mysticism. Among those who learned the philosophy of
mathematics from these refugees was Philolaos of Tarentum, who
studied at the new center the Pythagoreans established in that city.
Philolaos is the first Greek philosopher to have written down the
history and theories of the Pythagorean order. It is from the book
written by Philolaos that Plato learned of the Pythagorean
philosophy of number, cosmology, and mysticism, about which he
later wrote himself. The special symbol of the Pythagorean order
was the five-pointed star embedded in a pentagon. The diagonals
which form the five-pointed star intersect in such a way that they
form another, smaller pentagon, in a reversed direction. If the
diagonals inside this smaller pentagon are drawn, they form yet
another pentagon, and so on ad infinitum. This pentagon and five-
pointed star made up of its diagonals have some fascinating
properties, which the Pythagoreans believed
were mystical. A diagonal point divides a diagonal into two unequal
parts. The ratio of the entire diagonal to the larger segment is
exactly the same as the ratio of the larger segment to the smaller
one. This same ratio exists in all smaller and smaller diagonals. This
ratio is called the Golden Section. It is an irrational number equal to
1.618 . . . If you divide 1 by this number, you get exactly the
decimal part without the 1 . That is, you get 0.618. . . . As we
will see later, the Golden Section appears in natural phenomena as
well as proportions that the human eye perceives as beautiful. It
appears as the limit of the ratio of the famous Fibonacci numbers
we will soon encounter.
You can find the Golden Section by an interesting sequence of
operations on a calculator. Do 1 + 1 =, then press 1 /x, then +1
= , then 1 /x, then +1 = , then l/x and keep going. The number on
your display should become 1.618.. . and 0.618.. . inter-
changeably, once you have done the repetitive set of operations
enough times. This is the Golden Section. It is equal to the square
root of five, minus 1, divided by 2. This is the way it is obtained
geometrically from the Pythagorean Pentagon. Since this ratio
never becomes a ratio of two integers, hence never a rational
number, it proves that the square root of five is also an irrational
number. We will see more of the Golden Section later.
The Pythagoreans discovered that harmony in music corre-
sponded to simple ratios of numbers. According to Aristotle, the
Pythagoreans believed that all of heaven was musical scale and
numbers. Musical harmony and geometrical designs are what
brought the Pythagoreans to their belief that "All is number." The
Pythagoreans thought that the basic ratios in music involved only
the numbers 1, 2, 3, and 4, whose sum is 10. And 10, in
turn, is the base of our number system. The Pythagoreans
represented the number 10 as a triangle, which they called
tetraktys-.5
The Pythagoreans considered the tetraktys holy and even swore
oaths by it. Incidentally, according to Aristotle as well as Ovid and
other classical writers, the number ten was chosen as the base for
the number system because people have ten fingers. Recall that
the Babylonians, on the other hand, used a number system based
on sixty. There are some remnants of other number systems even
today. The French word for eighty (quatre-vingt, meaning "four
twenties") is a relic of an archaic number system based on twenty.
The Ropes, The Nile, and the Birth oj Geometry Much of what we
know about ancient Greek mathematics comes from the Elements
of Euclid of Alexandria, who lived around 300 B.C. It is believed
that the first two volumes of the Elements are all on the work of
Pythagoras and his secret society. The mathematics of the ancient
Greeks was done for its beauty and concerned abstract geometrical
figures. The Greeks developed an entire theory of geometry and it
is this theory, mostly unchanged, that is taught in schools today. In
fact, the Elements, or what remains of it today, is considered the
greatest textbook of all time.
Herodotus, the great Greek historian of antiquity, believed that
geometry was developed in ancient Egypt of 3,000 B.C., long
before the Greeks of Alexandria and elsewhere. He tells how the
overflow of the Nile would destroy boundaries between fields in the
river's fertile delta, and how this necessitated complicated
surveying techniques. For the purpose of this work, the surveyors
had to develop geometrical concepts and ideas. In his Histories,
Herodotus writes:
If the river carried away any portion of a man's lot, the king sent persons to
examine and determine by measurement the exact extent of the loss. From
this practice, I think, geometry first came to be known in Egypt, whence it
passed into Greece.6
Geometry is the study of shapes and figures made of circles and
straight lines and arcs and triangles and their intersections forming
various angles. It stands to reason that such science would be
essential for good surveying work. Egyptian geometers were indeed
called "rope stretchers," since ropes were used for outlining
straight lines necessary both in building temples and pyramids and
in realigning boundaries between fields. But it is possible that the
origins of geometry are even more ancient. Neolithic finds show
examples of congruence and symmetry of design, and these may
have been the precursors of Egyptian geometry, inherited centuries
later by the ancient Greeks. The same concerns that the
Babylonians had with areas of fields, leading to their need to
understand square numbers and their relations, may have been
shared by the ancient Egyptians, who were faced with the same
agrarian quandaries as well as construction problems with their
own pyramids. It is possible, therefore, that the ancient Egyptians
also had a knowledge of Pythagorean triples. What the Greeks did
with geometry, however, was to establish it as a pure mathematical
endeavor. They postulated and proved theorems.
What is a Theorem?
The Greeks brought us the concept of a theorem. A theorem is a
mathematical statement whose proof is given. The proof of a
theorem is a rigorous justification of the veracity of the theorem in
such a way that it cannot be disputed by anyone who follows the
rules of logic, and who accepts a set of axioms put forth as the
basis for the logic system. Euclid's axioms include the definition of
a point, a line, and the statement that two parallel lines never
meet. Following axioms and logical progressions, such as if A
implies B and B implies C then A implies C, the ancient Greeks were
able to prove many beautiful theorems about the geometry of
triangles and circles and squares and octagons and hexagons and
pentagons.
"Eureka! Eureka!"
The great Greek mathematicians Eudoxus (fifth century B.C.) and
Archimedes (third century B.C.) extended such work on geometrical
figures to the finding of areas using infinitesimal (meaning infinitely
small) quantities. Eudoxus of Cnidus (408-355 B.C.) was a friend
and student of Plato. He was too poor to live in the Academy in
Athens, and so he lived in the cheaper harbor town of Piraeus, from
which he commuted daily to Plato's Academy. While Plato himself
was not a mathematician, he encouraged mathematical work,
especially that of gifted students such as Eudoxus. Eudoxus
traveled to Egypt and there, as well as in Greece, learned much
geometry. He invented a "method of exhaustion," which he used to
find areas of geometric figures by using infinitesimal quantities.
Eudoxus would, for example, approximate the area of a circle by
the sum of the areas of many small rectangles—whose areas are
easy to calculate as the base times the height. This is essentially
the method used today in integral calculus, and the mod
em limit arguments are not different from Eudoxus' "exhaustion"
method.
But the most brilliant mathematician of antiquity was
undoubtedly Archimedes (287-212 B.C.), who lived in the city of
Syracuse on the island of Sicily. Archimedes was the son of the
astronomer Pheidias, and was related to Hieron II, the king of
Syracuse. Like Eudoxus, Archimedes developed methods for finding
areas and volumes, which were the forerunners of calculus. His
work anticipated both integral calculus and differential calculus
(there are two parts to calculus—Archimedes understood both of
them). But while he was mostly interested in pure mathematics:
numbers, geometry, areas of geometric figures, and so on, he is
also known for his achievements in the applications of
mathematics. A well-known story is the one about Archimedes'
discovery of what we call today the first law of hydrostatics: the law
that a submerged body loses from its weight the weight of the
liquid it displaces. There was at that time a dishonest goldsmith in
Syracuse, and King Hieron asked
his mathematician friend to find a way to prove this. Archimedes
started by studying the loss of weight of submerged bodies, using
his own body in the experiments. He took a bath and made some
measurements. When he discovered the law, he jumped out of the
bath and ran naked through the streets of Syracuse shouting
"Eureka, eureka!" ("I found it, I found it!").
Archimedes is also credited with the discovery of archimedes'
screw, a device to raise water by turning a hand-operated crank. It
is still in use by farmers around the world.
When the Roman General Marcellus attacked Syracuse in 214-
212 B.C., Hieron again asked for the help of his illustrious relative.
As the Roman fleet was approaching, Archimedes devised great
catapults based on his study of levers, and the Syracusans were
able to defend themselves well. But Marcellus regrouped his forces
and some time later attacked from the rear and was able to take
Syracuse by surprise. This time Archimedes was not even aware of
the attack and was sitting quietly on the ground above the city
drawing geometrical figures in the sand. A Roman soldier
approached and stepped on the figures. Archimedes jumped,
exclaiming: "Don't disturb my circles!" at which the soldier drew his
sword and killed the 75-year-old mathematician. In his will,
Archimedes had apparently requested that his gravestone bear a
carving of the particular geometric figure he admired—a sphere
inside a cylinder. The neglected grave was covered and the site
lost, but the Roman orator Cicero found it many years later and
restored it, and then the sands of time covered it again. In 1963,
ground was broken for a new hotel near Syracuse, and there
workers rediscovered Archimedes' tomb.
Archimedes' favorite theorem had to do with the sphere inside
the cylinder and he wrote the theorem in a book called The Method.
As with most ancient texts, it was assumed lost. In 1906, the
Danish scholar J. L. Heiberg heard that in Constantinople there was
a faded parchment manuscript with writings of a mathematical
nature. He traveled to Constantinople and found the manuscript,
consisting of 185 leaves of parchment. Scientific studies proved it
was a tenth-century copy of Archimedes' book, over which Eastern
Orthodox prayers had been added in the thirteenth century.
Alexandria, Greek Egypt, Circa A.D. 250
Around A.D. 250 a mathematician by the name of Diophantus
lived in Alexandria. All that we know about the life of Diophantus is
what is given in the following problem in a collection called the
Palatine Anthology, written roughly a century after Diophantus'
death.7
Here you see the tomb containing the remains of Diophantus, it is
remarkable: artfully it tells the measures of his life. The sixth part of his life
God granted him for his youth. After a twelfth more his cheeks were
bearded. After an additional seventh he kindled the light of marriage, and in
the fifth year he accepted a son. Alas, a dear but unfortunate child, half of
his father he was when chill Fate took him. He consoled his grief in the
remaining four years of his life. By this devise of numbers, tell us the extent
of his life.
(If you solve the implied equation, you will find that the answer
is 84.)
It is not certain when Diophantus lived. We can date his period
based only on two interesting facts. First, he quotes in his writings
Hypsicles, who we know lived around 150 B.C. And second,
Diophantus himself is quoted by Theon of Alexandria. Theon's time
is dated well by the solar eclipse which occurred on June 16, A.D.
364. So Diophantus certainly lived before A.D. 364 but after
150 B.C. Scholars, somewhat arbitrarily, place him at about A.D.
250.
Diophantus wrote the Arithmetica, which developed algebraic
concepts and gave rise to a certain type of equation. These are the
Diophantine equations, used in mathematics today. He wrote
fifteen volumes, only six of which came down to us. The rest were
lost in the fire that destroyed the great library of Alexandria, the
most monumental collection of books in antiquity. The volumes that
survived were among the last Greek texts to be translated. The first
known Latin translation was published in 1575. But the copy
Fermat had was the one translated by Claude Bachet in 1621 . It
was Diophantus' Problem 8 in Volume II, asking for a way of
dividing a given square number into the sum of two squares—the
Pythagorean problem whose solution was known to the Babylonians
two thousand years earlier—that inspired Fermat to write his
famous Last Theorem in the margin. The mathematical
achievements of Diophantus and his contemporaries were the final
glory of the ancient Greeks.
Arabian Nights
While Europe was busy fighting little feudal wars of the vassals of
one king or prince against another, surviving the Great Plague, and
going on costly and often deadly expeditions
called the Crusades, the Arabs ruled a flourishing empire from the
Middle East to the Iberian Peninsula. Among their great
achievements in medicine, astronomy, and the arts, the Arabs
developed algebra. In A.D. 632, the prophet Mohammed
established an Islamic state centered at Mecca, which remains the
religious center of Islam. Shortly afterwards, his forces attacked the
Byzantine Empire, an offensive which continued after Mohammed's
death in Medina that same year. Within a few years, Damascus,
Jerusalem, and much of Mesopotamia fell to the forces of Islam, and
by A.D. 641 so did Alexandria—the mathematical center of the
world. By A.D. 750, these wars as well as the ones among the
Moslems themselves subsided and the Arabs of Morocco and the
west were reconciled with the eastern Arabs centered in Baghdad.
Baghdad became a center of mathematics. The Arabs absorbed
mathematical ideas as well as discoveries in astronomy and other
sciences from the inhabitants of the areas they overcame. Scholars
from Iran, Syria, and Alexandria were called to Baghdad. During the
reign of the caliph Al Mamun in the early 800s, the Arabian Nights
was written and many Greek works, including Euclid's Elements,
were translated into Arabic. The caliph established a House of
Wisdom in Baghdad, and one of its members was Mohammed Ibn
Musa Al-Khowarizmi. Like Euclid, Al-Khowarizmi was to become
world-renowned. Borrowing Hindu ideas and symbols for numerals,
as well as Mesopotamian concepts and Euclid's geometrical
thought, Al-Khowarizmi wrote books on arithmetic and algebra. The
word "algorithm" is derived from Al-Khowarizmi. And the word
"algebra" is derived from the first words in the title of Al-
Khowarizmi's most well-known book: Al Jabr Wa'l Mucjabalah. It was
from this book that Europe was later to learn the branch of
mathematics called algebra. While algebraic ideas are in the root of
Diophantus' Aritbmetica, the Al Jabr is more closely related to the
algebra of today. The book is concerned with straightforward
solutions of equations of first and second degree. In Arabic, the
name of the book means "restoration by transposing terms from
one side of an equation to the other"— the way first-order
equations are solved today.
Algebra and geometry are related, as are all branches of
mathematics. One field that links these two together is algebraic
geometry, developed in our century. It is the linkage of branches of
mathematics, and areas that lie within different branches and
connect them, that would pave the way to Wiles' work on Fermat's
problem centuries later.
The Medieval Merchant and the Golden Section
The Arabs were interested in a problem that was very closely
related to the Diophantine question of finding Pythagorean triples.
The problem was to find Pythagorean triples giving an area of a
right-triangle that is also an integer. Hundreds of years later, this
problem found itself as the basis for the book Liber Quadratorum,
written in 1225 by Leonardo of Pisa (1180-1250). Leonardo
was better known as Fibonacci (which means "son of Bonaccio").
Fibonacci was an international merchant born in Pisa. He also lived
in North Africa and Constantinople, and he traveled extensively
throughout his life and visited Provence, Sicily, Syria, Egypt, and
many other areas in the Mediterranean. His travels and his relations
with the elite of Mediterranean soci
ety of the time brought him into contact with Arab mathematical
ideas, as well as Greek and Roman culture. When the emperor
Frederick II came to Pisa, Fibonacci was introduced to the emperor's
court and became a member of the imperial entourage.
In addition to Liber Quadratorum, Fibonacci is known for another
book he wrote at that time, Liber Abaci. A problem about
Pythagorean triangles in Fibonacci's book also appears in a
Byzantine manuscript of the eleventh century now in the Old Palace
library in Istanbul. It could be a coincidence,- on the other hand,
Fibonacci might have seen that same book in Constantinople during
his travels.
Fibonacci is best known for the sequence of numbers named
after him, the Fibonacci Numbers. These numbers originate in the
following problem in the Liber Abaci-.
How many pairs of rabbits will be produced in a year, beginning with a
single pair, if in every month each pair bears a new pair which becomes
productive from the second month on?
The Fibonacci sequence, which is derived from this problem, is
one where each term after the first is obtained by adding together
the two numbers that precede it. The sequence is= 1, 1 ,2, 3, 5,
8, 13, 21, 34, 55, 89, 144, . . .
This sequence (which is taken to continue beyond the 12
months of the problem) has unexpectedly significant properties.
Amazingly, the ratio of two successive numbers in the sequence
tends to the Golden Section. The ratios are: 1/1, 1 /2, 2/3, 3/5,
5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144, etc.
Note that these numbers get closer and closer to (sqr5 - l)/2.
This is the Golden Section. It can also be obtained using a cal-
culator by repeating the operation l/l + l/l + 1/. . . . as described
earlier. Recall that the reciprocal (l/x) of the Golden Section gives
the same number, less 1 . The Fibonacci sequence appears
everywhere in nature. Leaves on a branch grow at distances from
one another that correspond to the Fibonacci sequence. The
Fibonacci numbers occur in flowers. In most flowers, the number of
petals is one of: 3, 5, 8, 13, 2 1 , 34, 55, or 89. Lilies have
three petals, buttercups five, delphiniums often eight, marigolds
thirteen, asters twenty-one, daisies usually thirty-four or fifty-five or
eighty-nine.
The Fibonacci numbers appear in sunflowers too. The little
florets that become seeds in the head of the sunflower are
arranged in two sets of spirals: one winding in a clockwise direction
and the other counter-clockwise. The number of spirals in the
clockwise orientation is often thirty-four and the counter-clockwise
fifty-five. Sometimes the numbers are fifty-five and eighty-nine, and
sometimes even eighty-nine and a hundred and forty-four. All are
consecutive Fibonacci numbers (whose ratio approaches the golden
section). Ian Stewart argues in Nature's Numbers that when spirals
are developed, the angles between them are 137.5 degrees,
which is 360 degrees multiplied by one minus the golden ratio,
and they also give rise to two successive Fibonacci numbers for the
number of clockwise and counter-clockwise spirals, as shown
below.8
If a rectangle is drawn with sides in the Golden Section ratio to
each other, then the rectangle can be divided into a square and
another rectangle. This second rectangle is similar to the large one
in that it, too, has ratio of sides equal to the Golden Section. The
smaller rectangle can now be divided into a square and a remaining
rectangle, also in the Golden ratio . . . and so on. A spiral through
successive vertices of the sequence of rectangles that can be
drawn is one that appears often in shells, in the arrangement of
sunflower florets as mentioned, and in the arrangement of leaves
on a branch.
The rectangle has appealing proportions. The Golden Section
appears not only in nature but also in art as the classic ideal of
beauty. There is something divine about the sequence, and in fact
the Fibonacci Society, which is active today, is headed by a priest
and centered at St. Mary's College in California. The Society is
dedicated to the pursuit of examples of the Golden Section and
Fibonacci Numbers in nature, art, and architecture, with the belief
that the ratio is a gift of God to the world. As the ideal of beauty,
the Golden Section appears in such places as the Athenian
Parthenon. The ratio of the height of the Parthenon to its length is
the Golden Section.
The great Pyramid at Giza, built many hundreds of years before
the Greek Parthenon, has ratio of height of a face to half the side of
the base also in the Golden Section. The Egyptian Rhind Papyrus
refers to a "sacred ratio." Ancient statues as well as renaissance
paintings display the proportions equal to the Golden Section, the
Divine Ratio.
The Golden Section has been searched for as the ideal of beauty
beyond flowers or architecture. In a letter to the Fibonacci Society
some years ago, a member described how someone looking to find
the Golden Section asked several couples to perform an
experiment. The husband was asked to measure the height of his
wife's navel and divide it by his wife's height. The writer claimed
that in all couples the ratio was close to 0.618.
The Cossists
Mathematics entered Medieval Europe through Fibonacci's works
and from Spain, then part of the Arab world, with the work of Al
Khowarizmi. The main idea of algebra in those days was to solve
equations for an unknown quantity. Today, we call the unknown
quantity "x," and try to solve an equation for whatever value "x"
may have. An example of the simplest equation is: x - 5 = 0.
Here, we use simple math operations to find the value of "x." If we
add 5 to both sides of the equation we get on the left side x - 5 +
5, and on the right side we get 0 + 5. So the left side is "x" and
the right side is 5. That is, x = 5. The Arabs of Al Khowarizmi's
day called the unknown quantity "thing." The word "thing" in Arabic
is shai. And so they solved equations for the unknown shai, as done
above with "x."
When these ideas were imported to Europe, the Arabic shai was
translated into Latin. In Latin, "thing" is res, and in Italian it is cosa.
Since the first European algebraists were Italian, the word cosa was
attached to them. Since they were concerned with solving
equations for an unknown cosa, they became known as the
Cossists.
As in Babylon three and a half millennia earlier, mathematics in
the Middle Ages and the early Renaissance was mainly used as an
aid in commerce. Mercantile society of that time was increasingly
concerned with problems of trade, exchange rates, profits, costs,
and these could sometimes be cast as mathematical problems
requiring the solution of some equation. The cossists were people
such as Luca Pacioli (1445-1514), Geronimo Cardano (1501-
1576), Niccolo Tartaglia (1500-1557), and others who
competed with each other as problem-solvers in the service of
merchants and traders. These mathematicians used the solution of
more abstract problems as a form of advertising. Since they had to
compete for clients, they would also spend time and effort solving
these more difficult problems, such as cubic equations (equations
where the unknown quantity "cosa," or our "x," is in the third
power, x3), so that they could publish the results and become ever
more sought-after for the solution of applied problems.
In the early 1500s, Tartaglia found a way to solve cubic
equations and kept the method secret, so he could retain an edge
over his competitors in the lucrative problem-solving market. After
Tartaglia won a problem-solving competition against another
mathematician, Cardano pressed him to reveal his secret of how he
could solve these cubic equations.
Tartaglia revealed his method, on the condition that Cardano keep
it a secret from the rest of the world. When Cardano later learned
the same methods from another cossist, Scippi-one del Ferro
(1456-1526), he immediately assumed that Tartaglia got his
system from this person, and felt free to reveal the secret. Cardano
then published the methodology for solving cubic equations in his
1545 book Ars Magna. Tartaglia felt betrayed and became furious
with Cardano. In his last years, he spent much of his time vilifying
his former friend, and he succeeded in diminishing Cardano's
reputation.
The cossists were considered mathematicians of a lower level
than the ancient Greeks. Their preoccupation with applied problems
in pursuit of financial success, and their unconstructive fights
among themselves, kept them from looking for beauty in
mathematics and the pursuit of knowledge for its own sake. They
did not develop an abstract, general theory of mathematics. For
that, one needed to go back to the ancient Greeks. That is exactly
what happened a century later.
Renaissance Quest for Ancient Knowledge
Thirteen hundred years had passed since Diophantus. The medieval
world gave way to the Renaissance and the beginning of the
modern age. Out of the darkness of the Middle Ages, Europe awoke
with a thirst for knowledge. Many people turned their interest to
classical works of the ancients. Whatever ancient books existed
were translated into Latin—the language of the educated—in this
revival of the search for knowledge and enlightenment. Claude
Bachet, a French nobleman, was a translator with a great interest in
mathematics. He
obtained a copy of the Greek Arithmetica of Diophantus, translated
it and published it as Diophanti Alexandrini Arithmeticorum Libri Sex
in Paris in 1621. It was a copy of this book that found its way to
Fermat.
Fermat's theorem says that there are no possible Pythagorean
triples for anything beyond squares. There are no triples of
numbers, two adding up to the third, where the three numbers are
perfect cubes of integers, or fourth-powers of integers, fifth, sixth, or
any other powers. How could Fermat come up with such a theorem?
Squares, Cubes, and Higher Dimensions
A theorem is a statement with a proof. Fermat claimed to have had
a "marvelous proof," but without seeing and validating the proof no
one could call his statement a theorem. A statement may be very
deep, very meaningful and important, but without the proof that it is
indeed true, it must be called a conjecture, or sometimes a
hypothesis. Once a conjecture is proven it then can be called a
theorem, or a lemma if it is a preliminary proven statement which
then leads to a more profound theorem. Proven results that follow a
theorem are called corollaries. And Fermat had a number of such
statements. One such statement was that the number 22 A | 1+1 was
always a prime number. This conjecture was not only not proven,
hence not a theorem, it was actually proven to be wrong. This was
done by the great Swiss mathematician Leonhard Euler (1707-
1783) in the following century. So there was no reason to believe
that the "last theorem" was true. It could be true, or it could be
false. To prove that Fermat's Last Theorem was false all some-
one would have to do is to find a triple of integers, a , b , and c, and
a power n , greater than 2, that satisfy the relation a" + b" = c".
No one has ever found such a set of integers. (However, assuming
that a solution exists was to be a key element in attempts to prove
the theorem later.) And by the 1990s it was shown that no such
integers exist for any n less than four million. But that did not
mean that such numbers could not be found some day. The
theorem had to be proven for all integers and all possible powers.
Fermat himself was able to prove his Last Theorem for «=4. He
used an ingenious method he called the method of "infinite
descent," to prove that no integers a , b , and c exist satisfying a 4
+ b4 = c4. He also recognized that i f a solution existed for any
power n , then it would also exist for any multiple of n. One would
therefore only have to consider prime numbers (greater than 2) as
the exponents, that is , numbers that cannot be divided by any
number other than 1 or themselves in integers. The first few prime
numbers are 1, 2, 3, 5, 7, 1 1 , 1 3 , 17. .. None of these
numbers can be divided by any number other than 1 or themselves
and an integer result. An example of a number that is not a prime
is 6, since 6 divided by 3 gives 2—an integer. Fermat was also
able to prove his theorem for n = 3. Leon-hard Euler proved the
case n = 3 and n = 4 independently of Fermat, and Peter G. L.
Dirichlet in 1828 was able to prove the case n = 5. The same
case was proved by Adrien-Marie Legendre in 1830. Gabriel
Lame, and Henri Lebesgue who corrected him in 1840, were able
to establish the case n=7. Thus, two hundred years after Fermat
wrote his famous note in the margin of his Diophantus, his
theorem was only proven
correct for the exponents 3, 4, 5, 6, and 7. It would be a long
way to infinity, which is what one had to do to prove the theorem
for any exponent n. Clearly, what was needed was a general proof
that would work for all exponents, however large they might be.
Mathematicians were all looking for the elusive general proof, but
unfortunately what they were finding were proofs for particular
exponents only.
The Algorist
An algorist is a person who devises computing systems, or
algorithms. One such person was the prolific Swiss mathematician
Leonhard Euler, who was said to be able to calculate as naturally as
other people breathed. But Euler was much more than a walking
calculator. He was the most productive Swiss scientist of all times,
and a mathematician who wrote so many volumes of work that the
Swiss government established a special fund to collect all of his
works. He is said to have produced mathematical papers between
two calls for dinner at his large household.
Leonhard Euler was born in Basel on April 15, 1707. The fol-
lowing year, the family moved to the village of Riechen, where the
father became the Calvinist pastor. When young Leonhard went to
school, his father encouraged him to pursue the study of theology
so that he would eventually take his place as the village pastor. But
Euler showed much promise in mathematics and was tutored by
Johannes Bernoulli, a well-known Swiss mathematician of the day.
Daniel and Nicolaus Bernoulli, two younger members of the large
Bernoulli family of mathematicians, became his good friends. These
two convinced Leon-
hard's parents to allow him to pursue mathematics, since he would
become a great mathematician. Leonhard, however, continued with
the theology in addition to mathematics, and religious feelings and
customs would be a part of his entire life.
Mathematicial and scientific research in Europe in those days did
not take place primarily at universities, as it does today. The
universities were more devoted to teaching and did not allow much
time for other activities. Research in the eighteenth century was
primarily done at royal academies. There, the monarch would
support the leading scientists of the day in their pursuit of
knowledge. Some of the knowledge was applied, and would help
the government improve the standing of the entire nation. Other
research was more "pure," that is, research for its own sake—for
the advancement of human knowledge. The royals supported such
research generously and the scientists working at the academies
were able to lead comfortable lives.
When he finished his studies of mathematics, as well as theology
and Hebrew, at the University of Basel, Euler applied for a
professorship. Despite the great achievements he had already
made, he was turned down. In the meantime, his two friends Daniel
and Nicolaus were appointed as research mathematicians at the
royal academy in St. Petersburg, Russia. The two stayed in touch
with Leonhard and promised that, somehow, they would get him
there too. One day, the two Bernoullis wrote an urgent letter to
Euler telling him that there was an opening in the medical section
of the St. Petersburg academy. Euler immediately put himself to
work studying physiology and medicine at Basel. Medicine was not
something that interested him, but he was desperate to have a job
and hoped that
this way he could join his two friends who had such excellent
positions doing nothing but their own research in Russia.
Euler found mathematics in whatever he studied, medicine
included. Studying ear physiology led him to a mathematical
analysis of the propagation of waves. At any rate, soon an invitation
came from St. Petersburg, and in 1727 he joined his two friends.
However, on the death of Peter's wife Catherine, there was chaos at
the academy since she had been the great supporter of research. In
the confusion, Leonhard Euler slipped out of the medical section
and somehow got his name listed with the mathematical section,
where he would rightfully belong. For six years he kept his head low
to prevent the detection of his move, and he avoided all social
interactions lest the deception be discovered. All through this
period he worked continuously, producing volumes of top-rated
mathematical work. In 1733 he was promoted to the leading
mathematical position at the academy. Apparently Euler was a
person who could work anywhere, and as his family was growing,
he would often do his mathematics while holding a baby in one
arm.
When Anna Ivanova, Peter the Great's niece, became empress of
Russia, a period of terror began and Euler again hid himself in his
work for ten years. During this period he was working on a difficult
problem in astronomy for which a prize was offered in Paris. A
number of mathematicians requested several months' leave from
the academy to work on the problem. Euler solved it in three days.
But the concentrated exertion took its toll and he became blind in
his right eye.
Euler moved to Germany to be at the royal academy there, but
did not get along with the Germans, who enjoyed long
philosophical discussions not to his taste. Catherine the Great of
Russia invited Euler back to the St. Petersburg academy and he was
more than happy to come back. At that time, the philosopher Denis
Diderot, an atheist, was visiting Catherine's court. The empress
asked Euler to argue with Diderot about the existence of God.
Diderot, in the meantime, was told that the famous mathematician
had a proof of God's existence. Euler approached Diderot and said
gravely: "Sir, a + bin = x, hence God exists,- reply!" Diderot, who
knew nothing about mathematics, gave up and immediately
returned to France.
During his second stay in Russia, Euler went blind in his second
eye. He continued, however, to do mathematics with the help of his
sons, who did the writing for him. Blindness increased his mental
ability to do complicated calculations in his head. Euler continued to
do mathematics for seventeen years and died while playing with his
grandson in 1783. Much of the mathematical notation we use
today is due to Euler. This includes the use of the letter i for the
basic imaginary number, the square root of - 1 . Euler loved one
mathematical formula, which he considered the most beautiful and
put it above the gates of the Academy. The formula is:
e* + 1 = 0
This formula has 1 and 0, basic to our number system,- it has the
three mathematical operations: addition, multiplication, and
exponentiation,- and it has the two natural numbers pi and e,- and
it has i, the basis for the imaginary numbers. It is also visually
appealing.
The Seven Bridges oj Konigsberg
Euler was such an incredible visionary in mathematics that his
pioneering work on imaginary numbers (and what today is called
complex analysis), was not his only innovation. He did pioneering
work in a field which, in our century, would become indispensable
in the work of mathematicians—and in attempts to solve the
Fermat mystery. The field is topology, a visual theory of spatial
configurations that can remain unchanged when transformed by
continuous functions. It is the study of shapes and forms, some with
intricate, unexpected geometry, which is extended to four, five, or
higher dimensions beyond our normal three-dimensional world. We
will visit this fascinating area again when we get to the modern
approach to Fermat's problem, since topology—much as it seems
unrelated to the Fermat equation—has great importance for
understanding it.
Predating the development of topology, Euler's contribution to
the field is the famous problem of the Seven Bridges of Konigsberg.
This is the puzzle that started the whole interest in topology. In
Euler's time, seven bridges crossed the Pregel River in Konigsberg.
These are shown in the diagram below.
Euler asked whether or not it was possible to cross all seven
bridges without passing twice on any bridge. It is impossible. Other
problems, which were studied in modern times and were posed
because of interest in the seven bridges problem, are the various
map-coloring problems. A cartographer draws a map of the world.
In this map, each country is colored differently, to distinguish it
from its immediate neighbors. Any two countries or states that are
separated completely from each other may be colored in the same
exact color. The question is, what is the minimum number of colors
required so that no two states that touch each other are in the
same color? Of course, this is a general problem, not bounded by
how the map of the world looks today. The question is really, given
all possible configurations of maps on a plane, what is the minimum
number of colors that can be used? Given boundaries between
states in the former Yugoslavia or in the Middle East, with very
unusual lines between political entities, this general problem
becomes relevant in applications.
Mathematically, this is a topological problem. In October 1852,
Francis Guthrie was coloring a map of England. He wondered what
would be the minimum number of colors to be used for the
counties. It occurred to him that the number should be four. In
1879 a proof was given that the number was indeed four, but
later the proof was found to be false. Almost a century later, in
1976, two mathematicians, Haken and Appel, proved what had
become known as the Four Color Map problem. To this day,
however, their proof is considered controversial since it made use
of computer work, rather than pure mathematical logic.
Gauss, The Great German Genius
An alleged error in Euler's proof for n = 3 (that is, for cubes) was
corrected by Carl Friedrich Gauss (1777-1855). While most of
the renowned mathematicians of this time were French, Gauss,
undoubtedly the greatest mathematician of the time—and arguably
of all times—was unmistakably German. In fact, he never left
Germany, even for a visit. Gauss was the grandson of a very poor
peasant, and the son of a laborer in Brunswick. His father was harsh
with him, but his mother protected and encouraged her son. Young
Carl was also taken care of by his uncle Friedrich, the brother of
Gauss' mother Dorothea. This uncle was wealthier than Carl's par-
ents and made a reputation for himself in the field of weaving.
When Carl was three years old, he once watched his uncle add up
his accounts in a ledger. "Uncle Friedrich," he interrupted, "this
calculation is wrong." The uncle was shocked. From that day on, the
uncle did everything possible to contribute to the education and
care of the young genius. Although Gauss showed incredible
promise in school, his behavior sometimes left something to be
desired. One day, the teacher punished young Gauss by telling him
to stay in the classroom until he added up all the numbers from 1
to 100 while everyone else went to play outside. Two minutes
later, the 10-year-old Gauss was outside playing with the rest of
the class. The teacher came out furious. "Carl Friedrich!" the
teacher called, "do you want a more severe punishment? I told you
to stay inside until you have finished adding up all the numbers!"
"But I have," he said, "here is the answer." Gauss handed the
teacher a piece of paper with the right answer,
5,050, written on it. Apparently, Gauss figured out that he could
write two rows of 101 numbers:
0 1 2 3..... 97 98 99 100
100 99 98 97..... 3 2 1 0
He noted that the sum of each column was 100, so there was
nothing long to add up. Since there were 101 columns, the sum of
all the numbers was 101 x 100=10,100. Now, either of the two
rows had the sum he needed—all the numbers from 1 to 100.
Since he needed only one of the two rows, the answer was half of
10,100, or 5,050. Very simple, he thought. The teacher,
however, learned a lesson and never again assigned the young
Gauss a math problem as punishment.
When he was fifteen, Gauss attended college at Brunswick,
through the help of the Duke of Brunswick. The Duke later also
supported the young mathematician in enrolling at the renowned
university at Gottingen. There, on March 30, 1796, Gauss wrote
the first page in his famous diary. The diary had only nineteen
pages, but in these pages Gauss recorded 146 brief statements of
important and powerful mathematical results he had derived. It was
later discovered that almost every important mathematical idea
published by any mathematician in the late eighteenth and in the
nineteenth centuries had been preceded by one of the entries in
Gauss' unpublished diary. The diary remained hidden until it was
found in the possession of Gauss' grandson in Hamlin in 1898.
Gauss' results in number theory, which were shared with the
mathematicians of his day by regular correspondence, were of
great importance in all attempts by mathematicians to prove
Fermat's Last Theorem. Many of these results were contained in a
book on number theory which Gauss published in Latin in 1801,
when he was 24. The book, Distfuisitiones Arithmeticae, was
translated into French and published in Paris in 1807 and received
much attention. It was recognized as the work of genius. Gauss
dedicated it to his patron, the Duke of Brunswick.
Gauss was a distinguished scholar of classical languages as well.
On entering college he was already a master of Latin, and his
interest in philology precipitated a crisis in his career. Should he
follow the study of languages or of mathematics? The turning point
was March 30, 1796. From his diary, we know that on that day
the young Gauss decided definitely to specialize in mathematics. In
mathematics and statistics (where he is credited with the ingenious
method of least squares for finding a line to fit a data set) he
contributed to many areas, but he believed that number theory was
the heart of all of mathematics.
But why did the world's greatest mathematical genius never try
to prove Fermat's Last Theorem? Gauss' friend H. W. M. Olbers
wrote him a letter from Bremen on March 7, 1816, in which he
told Gauss that the Paris Academy offered a big prize for anyone
who would present a proof or a disproof of Fermat's Last Theorem.
Gauss surely could use the money, his friend suggested. At that
time, as he did throughout his mathematical career, Gauss received
financial support from the Duke of Brunswick, which allowed him to
do his mathematical work without the need for additional
employment. But he was far from rich. And, as Olbers suggested, no
other mathematician had nearly his expertise or ability. "It seems
right to me, dear Gauss, that you should get busy about this," he
concluded.
But Gauss would not be tempted. Possibly he knew how
deceptive Fermat's Last Theorem really was. The great genius in
number theory may have been the only mathematician in Europe to
realize just how difficult it would be to prove. Two weeks later,
Gauss wrote to Olbers his opinion of Fermat's Last Theorem: "I am
very much obliged for your news concerning the Paris prize. But I
confess that Fermat's Theorem as an isolated proposition has very
little interest for me, because 1 could easily lay down a multitude
of such propositions, which one could neither prove nor dispose of."
Ironically, Gauss made great contributions to the branch of
mathematics known as complex analysis—an area incorporating the
imaginary numbers worked on by Euler. Imaginary numbers would
have a decisive role in twentieth-century understanding of the con-
text of Fermat's Last Theorem.
Imaginary Numbers
The complex number field is a field of numbers based on the usual
real numbers, and what are called imaginary numbers, which were
known to Euler. These numbers arose when mathematicians were
looking to define as a number the solution of an equation such as x2
+ 1 = 0. There is no "real" solution to this simple equation,
because there is no real number which, when squared, gives -1—
the number that when added to 1 will give the answer zero. But if
we could somehow define the square root of negative one as a
number, then—while not a real number—it would be the solution to
the equation.
The number line was therefore extended to include imaginary
numbers. These numbers are multiples of the square root
of - 1 , denoted by i. They were put on their own number line,
perpendicular to the real number line. Together, these two axes
give us the complex plane. The complex plane is shown below. It
has many surprising properties, such as rotation being
multiplication by i.
Multiplication by i rotates counterclockwise.
The complex plane is the smallest field of numbers that contains
the solutions of all quadratic equations. It was found to be very
useful, even in applications in engineering, fluid mechanics, and
other areas. In 1 8 1 1 , decades ahead of his time, Gauss was
studying the behavior of functions on the complex plane. He
discovered some amazing properties of these functions, known as
analytic functions. Gauss found that analytic functions had special
smoothness, and they allowed
for particularly neat calculations. Analytic functions preserved
angles between lines and arcs on the plane—an aspect that would
become significant in the twentieth century. Some analytic
functions, called modular forms, would prove crucial in new
approaches to the Fermat problem.
In his modesty, Gauss did not publish these impressive results.
He wrote of them in a letter to his friend Friedrich Wilhelm Bessel
(1784-1846). When the theory reemerged without Gauss' name
attached to it years later, other mathematicians were given credit
for work on the same analytic functions Gauss understood so well.
Sophie Germain
One day Gauss received a letter from a certain "Monsieur Leblanc."
Leblanc was fascinated with Gauss' book, Discjuisitiones
Arithmeticae, and sent Gauss some new results in arithmetic the-
ory. Through the ensuing correspondence on mathematical
matters, Gauss gained much respect for Mr. Leblanc and his work.
This appreciation did not diminish when Gauss discovered that
neither was his correspondent's real name Leblanc, nor was the
writer of the letters a "Mr." The mathematician writing so eloquently
to Gauss was one of very few women active in the profession at
that time, Sophie Germain (1776-1831). In fact, upon
discovering the deception, Gauss wrote her:
But how to describe to you my admiration and
astonishment at seeing my esteemed
correspondent Mr. Leblanc metamorphose himself
into this illustrious personage who gives such a
brilliant example of what 1 would find difficult to
believe...
(A letter from Gauss to Sophie Germain, written from Brunswick on
Gauss' birthday, as stated in French at the end of his letter:
"Bronsvic ce 30 avril 1807 jour de ma naissance.")
Sophie Germain assumed a man's name to avoid the prejudice
against women scientists prevalent in those days and to gain
Gauss' serious attention. She was one of the most important
mathematicians to attempt a proof of Fermat's Last Theorem, and
to make considerable headway on the problem. Sophie Germain's
Theorem, which gained her much recognition, states that if a
solution of Fermat's equation with «=5 existed, then all three
numbers had to be divisible by 5. /The theorem separated
Fermat's Last Theorem into two cases: Case I for numbers not
divisible by 5, and Case II for numbers that are. The theorem was
generalized to other powers, and Sophie Germain gave a general
theorem which allowed for a proof of Fermat's Last Theorem for all
prime numbers, n, less than 100 in Case I. This was an important
result, which reduced the possible cases where Fermat's Last
Theorem might fail for primes less than 100 to only Case II.10
Sophie Germain had to drop the disguise she was using when
Gauss asked his friend "Leblanc" for a favor. In 1807, Napoleon
was occupying Germany. The French were imposing war fines on
the Germans, and were determining the amounts owed by each
resident based on what they perceived each person was worth. As
a prominent professor and astronomer at Gottingen, Gauss was
determined to owe 2,000 francs—far beyond his means. A number
of French mathematicians who were friends of the great Gauss
offered their help, but he refused to take their money. Gauss
wanted some
one to intercede on his behalf with the French General Per-nety in
Hanover.
He wrote his friend, Monsieur Leblanc, to ask if Leblanc might
contact the French General on Gauss' behalf. When Sophie Germain
gladly complied, it became clear who she was. But Gauss was
thrilled, as seen from his letter, and their correspondence continued
and developed further on many mathematical topics. Unfortunately,
the two never met. Sophie Germain died in Paris in 1831, before
the University of Gottingen could award her the honorary doctorate
which Gauss had recommended she receive.
Sophie Germain had many other achievements to her credit in
addition to her contributions to the solution of Fermat's Last
Theorem. She was active in the mathematical theories of acoustics
and elasticity, and other areas in applied and pure mathematics. In
number theory, she also proved theorems on which prime numbers
can lead to solvable equations.
The Blazing Comet of i 8 a
Gauss did much important work in astronomy, determining the
orbits of planets. On August 22, 1811, he first observed a comet
barely visible in the night sky. He was able to predict the comet's
precise trajectory toward the sun. When the comet became clearly
visible and blazed across the sky the superstitious and oppressed
peoples of Europe saw in it a sign from heaven signifying the
coming demise of Napoleon. Gauss saw in the comet a realization
of the orbit he had predicted for it to exact numerical accuracy. But
the unscientific masses were also right—the next year Napoleon
was defeated and
retreated from Russia. Gauss was amused. He was not unhappy to
see the Emperor defeated after French forces had extorted such
high amounts from him and his countrymen.
The Disciple
The Norwegian mathematician Niels Henrik Abel visited Paris in
October of 1826. There, he tried to meet other mathematicians—
Paris was at that time a mecca for mathematics. One of the people
who impressed Abel the most was Peter Gustav Lejeune Dirichlet
(1805-1859), a Prussian who was also visiting Paris and
gravitated toward the young Norwegian, at first thinking he was a
fellow Prussian. Abel was much impressed by the fact that Dirichlet
had proved Fermat's Last Theorem for M=5. He wrote about it in a
letter to a friend, mentioning that this was also done by Adrien-
Marie Legendre (1752-1833). Abel described Legendre as
extremely polite but very old. Legendre had proved the Fermat
result for n = 5 independently of Dirichlet, two years after him.
Unfortunately, this was always happening to Legendre—much of his
work was superseded by that of younger mathematicians.
Dirichlet was a friend and disciple of Gauss. When Gauss' great
book, the Discfuisitiones Arithmeticae, was published, it quickly
went out of print. Even mathematicians whose work was related to
that of Gauss could not obtain a copy. And many who did, did not
understand the depth of Gauss' work. Dirichlet had his own copy.
He carried this copy of Gauss' book with him on his many travels, to
Paris, Rome, and elsewhere on the Continent. Everywhere he went,
Dirichlet slept with the book under his pillow. Gauss' book became
known as
the book of the seven seals: The gifted Dirichlet is known as the
person who broke the seven seals. Dirichlet did more than anyone
else to explain and interpret the book of his great master to the rest
of the world.
Besides amplifying and explaining the DisQuisitiones, as well as
proving Fermat's Last Theorem for the power five, Dirichlet did
other great mathematics. One interesting result that Dirichlet
proved was connected with the following progression of numbers:
a, a + b, a + lb, a + 3b, a + 4b,... and so on, where the
numbers a and b are integers that have no common divisor other
than 1 (that is, they are numbers such as 1 and 3, or 3 and 5,-
rather than numbers such as 1 and 4, which have the common
divisor 1, or the numbers 6 and 9, which have the common
divisor 3). Dirichlet proved that this progression of numbers
contains infinitely many prime numbers. The amazing element of
Dirichlet's proof was that he had crafted it using an area of
mathematics which seemed in those days to be very far from
number theory, where this problem rightfully belongs. In his proof,
Dirichlet used the field called analysis, an important area of
mathematics which contains the calculus. Analysis deals with
continuous things: functions on a continuum of numbers on the line,
which seems very far from the discrete world of integers and prime
numbers—the realm of number theory.
It would be a similar bridge across seemingly different branches
of mathematics that would usher in the modern philosophy solving
the Fermat mystery in our own century. Dirichlet was a bold pioneer
in this area of unifying disparate branches of mathematics. The
student later inherited the mas
ter's position. When Gauss died in 1855, Dirichlet left his
prestigious job in Berlin to accept the honor of replacing Gauss at
Gottingen.
Napoleon's Mathematicians
The Emperor of the French loved mathematicians, even if he wasn't
one himself. Two who were especially close to him were
GaspardMonge (1746-1818), and Joseph Fourier (1768-1830).
In 1798, Napoleon took the two mathematicians with him to Egypt,
to help him "civilize" that ancient country.
Fourier was born in Auxerre, France, on March 21, 1768, but at
the age of eight he was orphaned and was helped by the local
Bishop to enter the military school. Even at the age of twelve,
Fourier showed great promise and was writing sermons for church
dignitaries in Paris, which they passed off as their own. The French
Revolution of 1789 saved the young Fourier from life as a priest.
Instead he became a professor of mathematics, and an enthusiastic
supporter of the revolution. When the revolution gave way to the
Terror, Fourier was repulsed by its brutality. He used his eloquence,
developed over years of writing sermons for others, to preach
against the excesses. Fourier also used his great public speaking
skills in teaching mathematics at the best schools in Paris.
Fourier was interested in engineering, applied mathematics, and
physics. At the Ecole Polytechnique, he did serious research in
these areas, and many of his papers were presented to the
Academy. His reputation brought him to the attention of Napoleon,
and in 1798 the Emperor asked Fourier to accompany him, aboard
his flagship, along with the French fleet of five hun-
dred ships headed for Egypt. Fourier was to be part of the Legion of
Culture. The Legion's charge was to "endow the people of Egypt
with all the benefits of European civilization." Culture was to be
brought to these people while they were being conquered by the
invading armada. In Egypt, the two mathematicians founded the
Egyptian Institute, and Fourier stayed there until 1802, when he
returned to France and was made a prefect of the region around
Grenoble. There he was responsible for many good public works
such as draining marshlands and stamping out malaria. With all that
work, Fourier, the mathematician-turned-administrator, managed to
find time to do his best mathematical thinking. Fourier's
masterpiece was the mathematical theory of heat, answering the
important question: How is heat conducted? This work won him a
Grand Prize from the Academy in 1812. Part of his work was based
on experiments he had done in the deserts of Egypt during his
years there. Some of his friends believed that these experiments,
including putting himself through the intense heat that developed in
closed rooms, contributed to his early death at the age of 62.
Fourier spent the last years of his life telling tales about
Napoleon and his close association with him, both in Egypt and after
Napoleon's escape from Elba. Fourier's research on heat, however,
was what immortalized him, for he developed an important theory
of periodic functions. A series of such periodic functions, when used
in a particular way to estimate another function, is called a Fourier
Series.
AMIR D . ACZEL
Periodic Functions
The clearest example of a periodic function is your watch. Minute by
minute, the large hand moves around a circle, and after sixty
minutes it comes back to the exact same place where it started.
Then it continues and in exactly sixty minutes it again returns to the
same spot. (Of course the little hand will have changed positions as
the hours go by.) The minutes hand on a watch is a periodic
function. Its period is exactly sixty minutes. In a sense, the space of
all minutes of eternity—the set of infinitely many minutes from now
until forever—can be wrapped by the big hand of a watch on the
outer edge of the face of the watch:
To take another example, on a locomotive speeding on the railroad
track the arm transferring power from the engine to the wheel
moves up and down along that wheel as it rotates. Every full circle
of the wheel, the arm comes back to its original position—this arm,
too, is periodical. The vertical height of the arm, when the radius of
the train wheel is one unit, is
defined as the sine function. This is the elementary
trigonometric function taught at school. The cosine
function is the horizontal measure of the arm. Both
sine and cosine are functions of the angle the arm
makes with a horizontal line through the center of
the wheel. This is shown below.
As the train moves forward, the vertical height of the arm traces
a wave-pattern as seen above. This pattern is periodic. Its period is
360 degrees. First the arm's height is zero, then it goes up in a
wave-like fashion until it reaches one, then it decreases, then it
goes to zero again, then to negative values until one, then its
negative magnitude decreases to zero. And then the cycles starts
all over again.
What Fourier discovered was that most functions can be
estimated to any degree of accuracy by the sum of many (the-
oretically infinitely many, for near-perfect accuracy) sine and cosine
functions. This is the famous result of the Fourier series. This
expansion of any function in terms of the sum of many sine and
cosine functions is useful in many applications of mathematics
when the actual mathematical expression of interest is difficult to
study but the sum of the sines and cosines, all multiplied by
different factors, can be easily
manipulated and evaluated—and this is especially
practical on a computer. The area of mathematics
known as numerical analysis is concerned with
computer methods for evaluating functions and other
numerical quantities. Fourier analysis is a substantial
part of numerical analysis, and consists of techniques
for studying difficult problems, many with no closed-
form (that is, given by a simple mathematical
expression) solution, using the Fourier series of
periodic function. After Fourier's pioneering work,
expansions using other simple functions, mostly
polynomials (that is, increasing powers of a variable:
squares, cubes, and so on) were also developed.
When your calculator computes the square root of a
number, it is doing so by an approximation based on
such a method. The Fourier series of sines and
cosines is especially useful in estimating phenomena
that are naturally the sum of periodic elements—for
example, music. A musical piece may be decomposed
into its harmonics. The tides, the phases of the
moon, and sun spots are examples of simple periodic
phenomena.
While the applications of Fourier's periodic
functions to natural effects and computational
methods are of great importance, the surprising fact
is that Fourier series and analysis found some useful
applications in pure mathematics, a field that was
never one of Fourier's main interests. In the
twentieth century, Fourier series found a role in
number theory as the tool for transforming
mathematical elements from one area to another in
the work of Goro Shimura. (The proof of Shimura's
conjecture was at the crux of proving Fermat's Last
Theorem.) Extensions of the periodic functions of
Fourier to the complex plane, linking together these
two areas of mathematics, would
lead to the discovery of automorphic functions and
modular forms—which also had a crucial impact on
Fermat's Last Theorem through the early twentieth
century work of another French mathematician, Henri
Poincare.
A Lame Proof
At the March 1, 1847 meeting of the Paris Academy,
the mathematician Gabriel Lame (1795-1870)
announced very excitedly that he had obtained a
general proof of Fermat's Last Theorem. Until then,
only specific powers, n, were attacked, and a proof
had been given of the theorem in the cases n = 3, 4,
5, 7. Lame suggested that he had a general approach
to the problem, which would work for any power n.
Lame's method was to factor the left side of Fermat's
equation, xn + yn, into linear factors using complex
numbers. Lame then went on to state modestly that
the glory should not all go to him, since the method
he suggested was introduced to him by Joseph
Liouville (1809-1882). But Liouville took the podium
after Lame and brushed aside any praise. Lame had
not proved Fermat's Last Theorem, he quietly stated,
because the factorization he suggested was not
unique (that is, there were many ways to carry out
the factorization, so there was no solution). It was a
gallant attempt, one of many, but it did not bear
fruit. However, the idea of factorization, that is,
breaking down the equation to a product of factors,
was to be tried again.
Ideal Numbers
The person to again try factorization was Ernst
Eduard Kum-mer (1810-1893)—the man who got
closer than anyone else in
his time to a general solution of Fermat's problem.
Kummer, in fact, invented an entire theory in
mathematics, the theory of ideal numbers, in
attempting to prove Fermat's Last Theorem.
Kummer's mother, a widow when her son was only
three, worked hard to assure her boy a good
education. At the age of eighteen he entered the
University of Halle, in Germany, to study theology
and prepare himself for life in the church. An
insightful professor of mathematics with enthusiasm
for algebra and number theory got the young
Kummer interested in these areas, and he soon
abandoned theology for mathematics. In his third
year as a student, the young Kummer solved a dif-
ficult problem in mathematics for which a prize had
been offered. Following this success, he was awarded
a doctorate in mathematics at the age of twenty-one.
But Kummer was unable to find a university
position and therefore had to take a teaching job at
his old gymnasium (high school). He remained a
schoolteacher for another ten years. Throughout this
period, Kummer did much research, which he
published and wrote in letters to several leading
mathematicians of his time. His friends realized how
sad it was to have such a gifted mathematician spend
his life teaching high school math. With the help of
some eminent mathematicians, Kummer was given
the position of professor at the University of Breslau.
A year later, in 1855, Gauss died. Dirichlet had taken
Gauss' place at Gottingen, leaving behind his old job
at the prestigious University of Berlin. Kummer was
chosen to replace Dirichlet in Berlin. He remained in
that position until his retirement.
Kummer worked on a wide range of problems in
mathemat-
fi7
ics, ranging from very abstract to very applied—even
applications of mathematics in warfare. But he made
his name for his extensive work on Fermat's Last
Theorem. The French mathematician Augustin-Louis
Cauchy (1789-1857), thought on a number of
occasions that he had found a general solution to the
Fermat problem. But the restless, careless Cauchy
realized in every attempt that the problem was much
bigger than he had assumed. The fields of numbers
he was working with always failed to have the
property he required. Cauchy left the problem and
went to work on other things.
Kummer became obsessed with Fermat's Last
Theorem, and his attempts at a solution led him
down the same futile track taken by Cauchy. But
instead of giving up hope when he recognized that
the number fields involved failed to have some prop-
erty, he instead invented new numbers that would
have the property he needed. These numbers he
called "ideal numbers." Kummer thus developed an
entirely new theory from scratch, which he used in
his attempts to prove Fermat's Last Theorem. At one
point, Kummer thought he finally had a general
proof, but this unfortunately fell short of what was
needed.
Nonetheless, Kummer did achieve tremendous
gains in his attack on the Fermat problem. His work
with ideal numbers enabled him to prove Fermat's
Last Theorem for a very extensive class of prime
numbers as the exponent n. Thus, he was able to
prove that Fermat's Last Theorem was true for an
infinite number of exponents, namely those that are
divisible by "regular" primes. The "irregular" prime
numbers eluded him. The only irregular primes less
than 100 are 37, 59, and 67. Kummer then worked
separately on the problem of these
irregular prime numbers and was eventually able to
prove Fermat's theorem for these numbers. By the
1850s, using Kum-mer's incredible breakthroughs,
Fermat's Last Theorem was known to be true for all
exponents less than n = 100, as well as infinitely
many multiples of prime numbers in this range. This
was quite an achievement, even if it was not a
general proof and still left out infinitely many
numbers for which it was not known whether the
theorem held true.
In 1816, the French Academy of Sciences offered a
prize to anyone who would prove Fermat's Last
Theorem. In 1850 the Academy again offered a gold
medal and 3,000 francs to the mathematician who
would prove Fermat's Last Theorem. In 1856, the
Academy decided to withdraw the award, since it
didn't seem like a solution to Fermat's problem was
imminent. Instead, the Academy decided to give the
award to Ernst Eduard Kummer, "for his beautiful
researches on the complex numbers composed of
roots of unity and integers." Thus Kummer was
awarded a prize for which he never even applied.
Kummer continued his tireless efforts on Fermat's
Last Theorem, stopping his research only in 1874.
Kummer also did pioneering work on the geometry of
four-dimensional space. Some of his results are
useful in the field of modern physics known as
quantum mechanics. Kummer died of the flu in his
eighties, in 1893.
Kummer's success with ideal numbers is even
more highly praised by mathematicians than the
actual advances he made in the solution of Fermat's
problem using these numbers. The fact that this
notable theory was inspired by attempts to solve Fer-
mat's Last Theorem shows how new theories can be
developed
by attempts to solve particular problems. In fact,
Kummer's theory of ideal numbers led to what are
now known as "ideals," which have had an impact on
the work of Wiles and other mathematicians on
Fermat's theorem in the twentieth century.
Another Prize
In 1908, the Wolfskehl Prize of one hundred thousand
marks was offered in Germany for anyone who would
come up with a general proof of Fermat's Last
Theorem. In the first year of the prize, 621
"solutions" were submitted. All of them were found to
be false. In the following years, thousands more
"solutions" were submitted, with the same effect. In
the 1920s, German hyperinflation reduced the real
value of the 100,000 marks to nothing. But false
proofs of Fermat's Last Theorem continued to pour
in.
Geometry Without Euclid
New developments started taking place in
mathematics in the nineteenth century. Janos Bolyai
(1802-1860) and Nicolas Ivanovitch Lobachevsky
(1793-1856), a Hungarian and a Russian respectively,
changed the face of geometry. By doing away with
Euclid's axiom that two parallel lines never meet,
these two were able independently to formulate a
geometrical universe which maintained many of
Euclid's properties but allowed two parallel lines to
meet at a point at infinity. A different geometry can
be seen, for example, on a sphere such as the globe.
Locally, two longitude lines are parallel. But in
reality, as they are followed to the North Pole, the
two lines meet there. New geome
tries solve many problems and explain situations
which, until that time, seemed mysterious and
without a solution.
Beauty and Tragedy —
Abstract algebra, a field derived from the familiar
algebra taught in school as a system for solving
equations, was developed in the nineteenth century.
In this area, the beautiful theory of Galois stands out.
Evariste Galois was born in 1811 in the small
village of Bourg-la-Reine, outside Paris. His father
was the mayor of the town and a staunch republican.
Young Evariste grew up on the ideals of democracy
and freedom. Unfortunately, most of France was at
that time heading in the opposite direction. The
French Revolution had come and gone, and so had
Napoleon. But the dreams of freedom, equality, and
fraternity had not yet been achieved. And the
royalists were enjoying a comeback in France, with a
Bourbon once again crowned King of the French—now
to rule together with the people's representatives.
Evariste's life was steeped in the lofty ideals of the
revolution. He was a good ideologue, and he gave
some moving speeches to the republicans. As a
mathematician, on the other hand, he was a genius
of unparalleled ability. As a teenager, Galois
absorbed the entire theories of algebra and
equations known to accomplished mathematicians of
his day, and—while still a schoolboy—developed his
own complete system, known today as Galois Theory.
Unfortunately, he was not to enjoy any recognition in
his tragically short life. Galois stayed up nights at his
boarding school, while everyone else slept, and
wrote down his theory. He sent it to the head of the
French Academy of Science—the mathematician
Cauchy—hoping that Cauchy would help him publish
the theory. But Cauchy was not only very busy, he
was also arrogant and careless. And Galois' brilliant
manuscript ended up in a trash can, unread.
Galois tried again, with the same result. In the
meantime, he failed to pass the entry exams to the
Lxole Polytechnique, where most of the celebrated
mathematicians of France got their training. Galois
had a habit of always working things out in his head.
He never took notes or wrote things down until he
had actual results. This method concentrated on ideas
rather than detail. Young Evariste had little patience
for, or interest in, the details. It was the great idea,
the beauty of the larger theory, that interested him.
Consequently, Galois was not at his best when taking
an examination in front of a blackboard. And this is
what caused him to fail twice in his attempts to gain
entrance to the school of his dreams. Twice in front
of the blackboard he did not perform well writing
things down, and he got irritated when asked for
details he just didn't consider
important. It was a tragedy that an
incrediblyintelligent young person would be
questioned by much less able examiners who could
not understand his deep ideas and took his
reluctance to give trivial details for ignorance. When
he realized he was going to fail the second and last
permissible attempt, and that the gates of the school
would be closed to him forever, Galois threw the
blackboard eraser in the face of his examiner.
Galois had to make do with a second-best choice,
the Ecole Normale. But even there he did not fare
well. Galois' father, the mayor of Bourg-la-Reine, was
the target of clerical intrigues in the town. An
unscrupulous priest circulated pornographic verses,
signing them with the mayor's name. Months of such
persecution caused Galois' father to lose his
confidence and he became convinced that the world
was out to get him. Slowly losing touch with reality,
he went to Paris. There, in an apartment not far from
where his son was studying, the father killed himself.
The young Galois never recovered from this tragedy.
Obsessed with the lost cause of the 1830 revolution,
and frustrated with the school director, whom he
considered an apologist for the royalists and clerics,
Galois wrote a scathing letter criticizing the director.
He was inspired to do so after three days of rioting in
the streets, when students all over Paris were
revolting against the regime. Galois and his
classmates were kept locked in their school, unable
to scale the tall fence. The angry Galois sent his
blistering letter criticizing the school director to the
Gazette des ficoles. As a result, he was expelled from the
school. But Galois was undaunted—he wrote a second
letter to the Gazette, and addressed the students at
the school to speak up for honor and conscience. He
got no response.
Out of school, Galois started offering private
lessons in mathematics. He wanted to teach his own
mathematical theories, outside the French schools,
when he was all of nineteen years old. But Galois
could not find students to teach—his theories were
too advanced; he was far ahead of his time.
Facing an uncertain future and doomed not to be
able to pursue a decent education, in desperation
Galois joined the artillery branch of the French
National Guard. Among the National Guard, headed
in its past by Lafayette, there were many liberal
elements close to young Galois' political philosophy.
While in the Guard, Galois made one last attempt to
publish his mathematical work. He wrote a paper on
the general solution of equations—today recognized
as the beautiful Galois Theory—and sent it to Simeon-
Denis Poisson (1781-1840) at the French Academy of
Sciences. Poisson read the paper but determined
that it was "incomprehensible." Once again, the
nineteen-year-old was so far ahead of any of the
older French mathematicians of his day that his
elegant new theories went way over their heads. At
that moment, Galois decided to abandon
mathematics and to become a full-time revolutionary.
He said that if a body was needed to get the people
involved with the revolution, he would donate his.
On May 9, 1831, two hundred young republicans
held a banquet in which they protested against the
royal order disbanding the artillery of the National
Guard. Toasts were offered to the French Revolution
and its heroes, as well as to the new revolution of
1830. Galois stood up and proposed a toast. He said
"To Louis Philippe," the Duke of Orleans, who was
now King of the French. While saying this, holding up
his
glass, Galois was also holding up an open pocket-
knife in his other hand. This was interpreted as a
threat on the life of the King, and caused a riot. The
next day Galois was arrested.
In his trial for threatening the life of the King,
Galois' attorney claimed that Galois had actually
said, while holding his knife, "To Louis Philippe, if he
becomes a traitor." Some of Galois' artillery friends
who were present testified to this, and the jury found
him not guilty. Galois retrieved his knife from the
evidence table, closed it and put it in his pocket, and
left the courtroom a free man. But he was not free
for very long. A month later he was arrested as a
dangerous republican, and kept in jail without a
charge while the authorities looked for a charge
against him that would stick. They finally found one—
wearing the uniform of the disbanded artillery. Galois
was tried for this/:harge and sentenced to six months
in jail. The royalists wercpleased to finally put away a
twenty-year-old they considered to be a dangerous
enemy of the regime. Galois was paroled after some
time and was moved to a halfway facility. What
happened next is open to question. While on parole,
Galois met a young woman and fell in love. Some
believe he was set up by his royalist enemies who
wanted to put an end to his revolutionary activities
once and for all. At any rate, the woman with whom
he got involved was of questionable virtue ("une
cocjuette de has e'tage"). As soon as the two became
lovers, a royalist came to "save her honor" and
challenged Galois to a duel. The young
mathematician was left no way out of the mess. He
tried everything he could to talk the man out of the
duel, but to no avail.
The night before the duel, Galois wrote several
letters.
These letters to his friends lend support to the
theory that he was framed by the royalists. He wrote
that he was challenged by two of the royalists and
that they had put him on his honor not to tell his
republican friends about the duel. "I die the victim of
an infamous coquette. It is in a miserable brawl that
my life is extinguished. Oh, why die for so trivial a
thing, why die for something so despicable!" But
most of that last night before the duel, Galois
carefully put down on paper his entire mathematical
theory, and sent it to his friend Auguste Chevalier. At
dawn on May 30, 1832, Galois faced his challenger on
a deserted field. He was shot in the stomach and left
lying in agony alone in the field. No one bothered to
call a doctor. Finally a peasant found him and
brought him to the hospital, where he died the next
morning. He was twenty years old. In 1846, the
mathematician Joseph Liouville edited and published
Galois' elegant mathematical theory in a journal.
Galois' theory would supply the crucial step in the
method used a century and a half later in attacking
Fermat's Last Theorem.
Another Victim
Cauchy's carelessness and arrogance ruined the life
of at least one other brilliant mathematician. Niels
Henrik Abel (1802-1829) was the son of the pastor of
the village of Findo in Norway. When he was sixteen,
a teacher encouraged Abel to read Gauss' famous
book, the Discjuisitiones. Abel was even successful in
filling in some gaps in the proofs of some of the
theorems. But two years later, his father died and
young Abel had to postpone his study of mathematics
and concentrate his efforts on supporting the family.
In spite of the great difficulties he
faced, Abel managed to continue some study of
mathematics and made a remarkable mathematical
discovery at the age of nineteen. He published a
paper in 1824 in which he proved that no solution
was possible for equations of the fifth degree. Abel
had thus solved one of the most celebrated problems
of his day. Yet the gifted young mathematician had
still not been offered an academic position, which he
badly needed to support his family and so he sent his
work to Cauchy for evaluation and possible
publication and recognition. The .paper Abel sent
Cauchy was of extraordinary power and generality.
But Cauchy lost it. When it appeared in print, years
later, it was much too late to help Abel. In 1829 Abel
died from tuberculosis, brought on by poverty and
the strain of supporting his family in dire
circumstances. Two days after his death, a letter
arrived addressed to Abel informing him he had been
appointed professor at the University of Berlin.
The concept of an Abelian Group (now considered a
word and written with a small "a," abelian) is very
important in modern algebra and is a crucial element
in the modern treatment of the Fermat problem. An
abelian group is one where the order of mathematical
operations can be reversed without affecting the
outcome. An abelian variety is an even more abstract
algebraic entity, and its use was also important in
modern approaches to the solution of Fermat's Last
Theorem.
D ed ek in d 's Idea ls
The legacy of Carl Friedrich Gauss continued through
the centuries. One of Gauss' most notable
mathematical successors was Richard Dedekind
(1831-1916), born in the same
town as the great master, in Brunswick, Germany.
Unlike Gauss, however, as a child Dedekind showed
no great interest in or capacity for mathematics. He
was more interested in physics and chemistry, and
saw mathematics as a servant to the sciences. But at
the age of seventeen, Dedekind entered the same
school where the great Gauss got his mathematical
training—Caroline College—and there his future
changed. Dedekind became interested in
mathematics and he pursued that interest in
Gottingen, where Gauss was teaching. In 1852, at the
age of 21, Richard Dedekind received his doctorate
from Gauss. The master found his pupil's dissertation
on the calculus "completely satisfying." This was not
such a great compliment, and in fact Dedekind's
genius had not yet begun to manifest itself.
In 1854, Dedekind was appointed a lecturer in
Gottingen. When Gauss died in 1855 and Dirichlet
moved from Berlin to take his position, Dedekind
attended all of Dirichlet's lectures at Gottingen and
edited the latter's pioneering treatise on number
theory, adding a supplement based on his own work.
This supplement contained an outline of the theory
Dedekind developed for algebraic numbers, which
are defined as solutions of algebraic equations. They
contain multiples of square roots of numbers along
with rational numbers. Algebraic number fields are
very important in the study of Fermat's equation, as
they arise from the solution of various kinds of
equations. Dedekind thus developed a significant
area within number theory.
Dedekind's greatest contribution to the modern
approach to Fermat's last Theorem was his
development of the theory of ideals, abstractions of
Kummer's ideal numbers. A century
after their development by Dedekind, ideals would
inspire Barry Mazur, and Mazur's own work would be
exploited by Andrew Wiles.
In the 1857-8 academic year, Richard Dedekind
gave the first mathematics course on Galois theory.
Dedekind's understanding of mathematics was very
abstract, and he elevated the theory of groups to the
modern level at which it is understood and taught
today. Abstraction made the twentieth-century
approach to Fermat's problem possible. Dedekind's
groundbreaking course on the theories developed by
Galois was a great step in this direction. The course
was attended by two students.
Then Dedekind's career took a strange turn. He left
Gottingen for a position in Zurich, and after five
years, in 1862, he returned to Brunswick, where he
taught at a high school for fifty years. No dne has
been able to explain why a brilliant mathematician
who brought algebra to an incredibly high level of
abstraction and generality suddenly left one of the
most prestigious positions at any European
university to teach at an unknown high school.
Dedekind never married and lived for many years
with his sister. He died in 1916, and maintained a
sharp, active mind to his last day.
Fin de Siecle
At the turn of the nineteenth century, there lived in
France a mathematician of great ability in a
surprisingly wide variety of areas. The breadth of
knowledge of Henri Poincare (1854-1912) extended
beyond mathematics. In 1902 and later, when he was
already a renowned mathematician, Poincare wrote
popular
books on mathematics. These paperbacks, read by
people of all ages, were a common sight in the cafes
and the parks of Paris.
Poincare was born to a family of great achievers.
His cousin, Raymond Poincare, rose to be the
president of France during the First World War.
Other family members held government and public
service positions in France as well.
From a young age, Henri displayed a powerful
memory. He could recite from any page of a book he
read. His absentmind-edness, however, was
legendary. A Finnish mathematician once came all
the way to Paris to meet with Poincare and discuss
some mathematical problems. The visitor was kept
waiting for three hours outside Poincare's study
while the absentminded mathematician paced back
and forth—as was his habit throughout his working
life. Finally, Poincare popped his head into the
waiting room and exclaimed: "Sir, you are disturbing
me!" upon which the visitor summarily left, never to
be seen in Paris again.
Poincare's brilliance was recognized when he was
in elementary school. But since he was such a
universalist—a renaissance man in the making—his
special aptitude for mathematics did not yet
manifest itself. He distinguished himself with his
excellence in writing at an early age. A teacher who
recognized and encouraged his ability treasured his
school papers. At some point, however, the
concerned teacher had to caution the young genius:
"Don't do so well, please...try to be more ordinary."
The teacher had good reason to make this sugges-
tion. Apparently, French educators had learned
something from the misfortunes of Galois half a
century earlier—teachers found that gifted students
often failed at the hands of uninspired examiners.
His teacher was genuinely worried that
Poincare was so brilliant that he might fail those
exams. As a child, Poincare was already
absentminded. He often skipped meals because he
forgot whether or not he had eaten.
Young Poincare was interested in tjie classics and
he learned to write well. As a teenager he became
interested in mathematics and immediately excelled
in it. He would work out problems entirely in his head
while pacing in a room—then sit down and write
everything very hastily. In this he resembled Galois
and Euler. When Poincare finally took his exams, he
almost failed in math, as his elementary school
teacher had feared years earlier. But he did pass,
only because—at seventeen—his renown as a
mathematician was so great that the examiners
didn't dare fail him. "Any student other than Poincare
would have been given a failing grade," the chief
examiner declared as he passed the student who
went on to the Ecole Polytechnique and became the
greatest French mathematician of his time.
Poincare wrote scores of books on mathematics,
mathematical physics, astronomy, and popular
science. He wrote research papers of over five
hundred pages on new mathematical topics he
developed. He made major contributions to topology,
the area started by Euler. However, Poincare's
results were so important that this branch of
mathematics is considered to have truly been
launched only in 1895, with the publication of
Poincare's Analysis Situs. Topology—the study of shapes
and surfaces and continuous functions—was
important in understanding Fermat's problem in the
late twentieth century. But even more essential to
the modern approach to Fermat's Last Theorem was
another area pioneered by Henri Poincare.
FERMAT'S LAST THEOREM
Modular Forms
Poincare studied periodic functions, such as the sines
and cosines of Fourier—not on the number line as
Fourier had done, but in the complex plane. The sine
function, sin x, is the vertical height on a circle with
radius 1 when the angle is x. This function is periodic:
it repeats itself over and over every time the angle
completes a multiple of its period, 360 degrees. This
periodicity is a symmetry. Poincare examined the
complex plane, which contains real numbers on the
horizontal axis and imaginary ones on the vertical one
as shown below.
Here, a periodic function could be conceived as having a periodicity
both along the real axis and along the imaginary axis. Poincare went
even further and posited the existence of functions with a wider
array of symmetries. These were functions that remained
unchanged when the complex variable z was changed
according to/(z) >j(az+b/cz+d). Here the elements a, b, c,
d, arranged as a matrix, formed an algebraic group. This means
t h a t t h e r e a r e i n f i n i t e l y m a n y p o s s i b l e v a r i a t i o n s . T h e y a l l c o m m u t e w i t h e a c h o t h e r a n d t h e f u n c t i o n / i s invariant u n d e r t h i s g r o u p o f t r a n s f o r m a t i o n s . P o i n c a r e c a l l e d s u c h w e i r d f u n c t i o n s a u t o m o r p h i c f o r m s .
T h e a u t o m o r p h i c f o r m s w e r e v e r y , v e r y s t r a n g e c r e a t u r e s s i n c e t h e y s a t i s f i e d m a n y i n t e r n a l s y m m e t r i e s . P o i n c a r e w a s n ' t q u i t e s u r e t h e y e x i s t e d . A c t u a l l y , P o i n c a r e d e s c r i b e d h i s r e s e a r c h a s f o l l o w s . H e s a i d t h a t f o r f i f t e e n d a y s h e w o u l d w a k e u p i n t h e m o r n i n g a n d s i t a t h i s d e s k f o r a c o u p l e o f h o u r s t r y i n g t o c o n v i n c e h i m s e l f t h a t t h e a u t o m o r p h i c f o r m s h e i n v e n t e d c o u l d n ' t p o s s i b l y e x i s t . O n t h e f i f t e e n t h d a y , h e r e a l i z e d t h a t h e w a s w r o n g . T h e s e s t r a n g e f u n c t i o n s , h a r d t o i m a g i n e v i s u a l l y , d i d e x i s t . P o i n c a r e e x t e n d e d t h e m t o e v e n m o r e c o m p l i c a t e d f u n c t i o n s , c a l l e d m o d u l a r f o r m s . T h e m o d u l a r f o r m s l i v e o n t h e u p p e r h a l f o f t h e c o m p l e x p l a n e , a n d t h e y h a v e a h y p e r b o l i c g e o m e t r y . T h a t i s , t h e y l i v e i n a s t r a n g e s p a c e w h e r e t h e n o n -E u c l i d e a n g e o m e t r y o f B o l y a i a n d L o b a c h e v s k y r u l e s . T h r o u g h a n y p o i n t i n t h i s h a l f - p l a n e , v m a n y " l i n e s " p a r a l l e l t o a n y g i v e n l i n e e x i s t .
P o i n c a r e l e f t b e h i n d t h e s y m m e t r i c a u t o m o r p h i c f o r m s a n d t h e i r e v e n m o r e i n t r i c a t e m o d u l a r f o r m s a n d w e n t o n t o d o o t h e r m a t h e m a t i c s . H e w a s b u s y i n s o m a n y f i e l d s , o f t e n i n a f e w o f t h e m a t o n c e , t h a t h e h a d n o t i m e t o s i t a n d p o n d e r t h e b e a u t y o f h a r d l y i m a g i n a b l e , i n f i n i t e l y s y m m e t r i c e n t i t i e s . B u t u n b e k n o w n s t t o h i m , h e h a d j u s t s o w e d a n o t h e r s e e d i n t h e g a r d e n t h a t w o u l d e v e n t u a l l y b r i n g a b o u t t h e F e r m a t s o l u t i o n .
An Unexpected Connection with a Doughnut
I n 1922, t h e A m e r i c a n m a t h e m a t i c i a n L o u i s J.
M o r d e l l d i s c o v e r e d w h a t h e t h o u g h t w a s a v e r y s t r a n g e c o n n e c t i o n b e t w e e n
t h e s o l u t i o n s o f a l g e b r a i c e q u a t i o n s a n d t o p o l o g y . T h e e l e m e n t s o f t o p o l o g y a r e s u r f a c e s a n d s p a c e s . T h e s e s u r f a c e s c o u l d b e i n a n y d i m e n s i o n : t w o d i m e n s i o n s , s u c h a s t h e f i g u r e s i n a n c i e n t G r e e k g e o m e t r y , o r t h e y c o u l d b e i n t h r e e -d i m e n s i o n a l s p a c e , o r m o r e . T o p o l o g y i s a s t u d y o f c o n t i n u o u s f u n c t i o n s t h a t a c t o n t h e s e s p a c e s , a n d t h e p r o p e r t i e s o f t h e s p a c e s t h e m s e l v e s . T h e p a r t o f t o p o l o g y w h i c h c o n c e r n e d M o r d e l l w a s t h e o n e o f s u r f a c e s i n t h r e e -d i m e n s i o n a l s p a c e . A s i m p l e e x a m p l e o f s u c h a s u r f a c e i s a s p h e r e : i t i s t h e s u r f a c e o f a b a l l , s u c h a s a b a s k e t b a l l . T h e b a l l i s t h r e e -d i m e n s i o n a l , b u t i t s s u r f a c e ( a s s u m i n g n o d e p t h ) i s a t w o - d i m e n s i o n a l o b j e c t . T h e s u r f a c e o f t h e e a r t h i s a n o t h e r e x a m p l e . T h e e a r t h i t s e l f i s t h r e e d i m e n s i o n a l : a n y p l a c e o n t h e e a r t h o r i n s i d e i t c a n b e g i v e n b y i t s l o n g i t u d e ( o n e d i m e n s i o n ) , i t s l a t i t u d e ( a s e c o n d d i m e n s i o n ) , a n d i t s d e p t h ( t h e t h i r d d i m e n s i o n ) . B u t t h e s u r f a c e o f t h e e a r t h ( n o d e p t h ) i s t w o - d i m e n s i o n a l , s i n c e a n y p o i n t o n t h e s u r f a c e o f t h e e a r t h c a n b e s p e c i f i e d b y t w o n u m b e r s : i t s l o n g i t u d e a n d i t s l a t i t u d e .
T w o - d i m e n s i o n a l s u r f a c e s i n t h r e e - d i m e n s i o n a l s p a c e c a n b e c l a s s i f i e d a c c o r d i n g t o t h e i r g e n u s . T h e g e n u s i s
t h e n u m b e r o f h o l e s i n t h e s u r f a c e . T h e g e n u s o f a s p h e r e i s z e r o s i n c e t h e r e a r e n o h o l e s i n i t . A d o u g h n u t h a s o n e h o l e i n i t . T h e r e f o r e t h e g e n u s o f a d o u g h n u t ( m a t h e m a t i c a l l y c a l l e d a t o r u s ) i s o n e . A h o l e m e a n s a h o l e t h a t r u n s c o m p l e t e l y t h r o u g h t h e s u r f a c e . A c u p w i t h t w o h a n d l e s h a s t w o h o l e s t h r o u g h i t . T h e r e f o r e i t i s a s u r f a c e o f g e n u s t w o .
A s u r f a c e o f o n e g e n u s c a n b e t r a n s f o r m e d b y a c o n t i n u o u s f u n c t i o n i n t o a n o t h e r s u r f a c e o f t h e s a m e g e n u s . T h e o n l y w a y t o t r a n s f o r m a s u r f a c e o f o n e g e n u s t o o n e o f a d i f f e r e n t g e n u s
is by closing or opening some holes. This cannot be
done by a continuous function, since it will require
some ripping or fusing together, each of which is a
mathematical discontinuity.
Mordell discovered a strange and totally unexpected
connection between the number of holes in the
surface (the genus) of the space of solutions of an
equation and whether or not the
e q u a t i o n h a d a f i n i t e n u m b e r o r a n i n f i n i t e n u m b e r o f s o l u t i o n s . I f t h e s u r f a c e o f s o l u t i o n s , u s i n g c o m p l e x n u m b e r s f o r g r e a t e s t g e n e r a l i t y , h a d t w o o r m o r e h o l e s ( t h a t i s , h a d g e n u s t w o o r h i g h e r ) , t h e n t h e e q u a t i o n h a d o n l y f i n i t e l y m a n y w h o l e - n u m b e r s o l u t i o n s . M o r d e l l w a s u n a b l e t o p r o v e h i s d i s -c o v e r y , a n d i t b e c a m e k n o w n a s M o r d e l l ' s c o n j e c t u r e .
Faltings' Proof
I n 1983, a t w e n t y - s e v e n y e a r o l d G e r m a n m a t h e m a t i c i a n , G e r d F a l t i n g s , a t t h a t t i m e a t t h e U n i v e r s i t y o f W u p p e r t a l , w a s a b l e t o p r o v e t h e M o r d e l l c o n j e c t u r e . F a l t i n g s w a s n o t i n t e r e s t e d i n F e r m a t ' s L a s t T h e o r e m , c o n s i d e r i n g i t a n i s o l a t e d p r o b l e m i n n u m b e r t h e o r y . Y e t h i s p r o o f , w h i c h u s e d g r e a t i n g e n u i t y a l o n g w i t h t h e p o w e r f u l m a c h i n e r y o f a l g e b r a i c g e o m e t r y d e v e l o p e d i n t h i s c e n t u r y , h a d p r o f o u n d i m p l i c a t i o n s f o r t h e s t a t u s o f F e r m a t ' s L a s t T h e o r e m . B e c a u s e t h e g e n u s o f t h e F e r m a t e q u a t i o n f o r n g r e a t e r t h a n 3
w a s 2 o r m o r e , i t b e c a m e c l e a r t h a t i f i n t e g e r s o l u t i o n s t o t h e F e r m a t e q u a t i o n e x i s t e d a t a l l , t h e n t h e s e w e r e f i n i t e ( w h i c h w a s c o m f o r t i n g , s i n c e t h e i r n u m b e r w a s n o w l i m i t e d ) . S o o n a f t e r w a r d s , t w o m a t h e m a t i c i a n s , G r a n v i l l e a n d H e a t h - B r o w n , u s e d F a l t -
i n g s ' r e s u l t t o s h o w t h a t t h e n u m b e r o f s o l u t i o n s o f F e r m a t ' s e q u a t i o n , i f t h e y e x i s t e d , d e c r e a s e d a s t h e e x p o n e n t n i n c r e a s e d . I t w a s s h o w n t h a t t h e p r o p o r t i o n o f e x p o n e n t s f o r w h i c h F e r m a t ' s L a s t T h e o r e m w a s t r u e a p p r o a c h e d o n e h u n d r e d p e r c e n t a s n i n c r e a s e d .
I n o t h e r w o r d s , F e r m a t ' s L a s t T h e o r e m w a s " a l m o s t a l w a y s " t r u e . I f s o l u t i o n s t o F e r m a t ' s e q u a t i o n e x i s t e d ( i n w h i c h c a s e t h e T h e o r e m w a s n o t t r u e ) , t h e n s u c h s o l u t i o n s w e r e f e w a n d
v e r y f a r b e t w e e n . S o t h e s t a t u s o f F e r m a t ' s L a s t T h e o r e m i n 1983 w a s t h e f o l l o w i n g . T h e t h e o r e m w a s p r o v e n f o r n u p t o a m i l l i o n ( a n d i n 1992 t h e l i m i t w a s r a i s e d t o 4 m i l l i o n ) . I n a d d i -t i o n , f o r l a r g e r n, i f s o l u t i o n s e x i s t e d a t a l l t h e n t h e y w e r e v e r y f e w a n d d e c r e a s i n g w i t h n.
The Mysterious Greek General with the Funny Name T h e r e a r e d o z e n s o f e x c e l l e n t b o o k s o n m a t h e m a t i c s p u b -l i s h e d i n F r a n c e , w r i t t e n i n F r e n c h , w i t h t h e a u t h o r l i s t e d a s N i c o l a s B o u r b a k i . T h e r e w a s a G r e e k g e n e r a l n a m e d B o u r b a k i (1816-1897). I n 1862
B o u r b a k i w a s o f f e r e d t h e t h r o n e o f G r e e c e , w h i c h h e d e c l i n e d . T h e g e n e r a l h a d a n i m p o r t a n t r o l e i n t h e F r a n c o - P r u s s i a n w a r , a n d t h e r e i s a s t a t u e o f h i m i n t h e F r e n c h c i t y o f N a n c y . B u t G e n e r a l B o u r b a k i k n e w n o t h i n g a b o u t m a t h e m a t i c s . A n d h e n e v e r w r o t e a b o o k , a b o u t m a t h e m a t i c s o r a n y t h i n g e l s e . W h o w r o t e t h e m a n y v o l u m e s o f m a t h e m a t i c s b e a r i n g h i s n a m e ?
T h e a n s w e r l i e s i n t h e
h a p p y d a y s i n P a r i s o f b e t w e e n t h e t w o W o r l d W a r s . H e m i n g w a y , P i c a s s o a n d M a t i s s e w e r e n o t t h e o n l y p e o p l e w h o l i k e d t o s i t i n c a f e s a n d m e e t t h e i r f r i e n d s a n d s e e p e o p l e a n d b e s e e n . A t t h a t t i m e , a r o u n d t h e s a m e c a f e s o n t h e L e f t B a n k b y t h e U n i v e r s i t y o f P a r i s , t h e r e f l o u r i s h e d a v i b r a n t m a t h e m a t i c a l c o m m u n i t y . P r o f e s s o r s o f m a t h e m a t i c s f r o m t h e u n i v e r s i t y a l s o l i k e d t o m e e t t h e i r f r i e n d s , d r i n k a c a f e a u l a i t o r a p a s t i s i n a g o o d b r a s s e r i e o n t h e B o u l e v a r d S t . M i c h e l b y t h e b e a u t i f u l L u x e m b o u r g G a r d e n s , a n d d i s c u s s . . . m a t h e m a t i c s . S p r i n g t i m e i n P a r i s i n s p i r e d w r i t e r s , a r t i s t s , a n d m a t h e m a t i c i a n s . O n e i m a g i n e s t h a t o n a s u n n y d a y a t a p l e a s a n t c a f e a r o w d y g r o u p o f m a t h e m a t i c i a n s c o n g r e -
ss
g a t e d . F e e l i n g s o f f r a t e r n i t y o v e r c a m e t h e m a s t h e y a r g u e d a n i m a t e d l y o v e r s o m e f i n e p o i n t s o f a t h e o r y . T h e i r r e v e l r y p r o b a b l y d i s t u r b e d H e m i n g w a y w h o w r i t e s h e a l w a y s l i k e d t o w o r k b y h i m s e l f a t a c a f e , a n d h e p r o b a b l y l e f t t o g o t o o n e o f h i s a l t e r n a t e , l e s s f a v o r i t e h a u n t s . B u t t h e m a t h e m a t i c i a n s d i d n ' t c a r e . T h e y v a l u e d e a c h o t h e r ' s c o m p a n y , a n d a c a f e f u l l o f m a t h e m a t i c i a n s — a l l s p e a k i n g t h e s a m e l a n g u a g e o f n u m b e r s a n d s y m b o l s a n d s p a c e s a n d f u n c t i o n s — w a s i n t o x i c a t i n g . " T h i s i s w h a t t h e P y t h a g o r e a n s m u s t h a v e f e l t w h e n t h e y t a l k e d m a t h e m a t i c s , " p e r h a p s o n e o f t h e s e n i o r p e o p l e i n t h e g r o u p s a i d , a s h e l i f t e d h i s g l a s s i n a t o a s t . " Y e s , b u t t h e y d i d n ' t d r i n k P e r n o d , " s a i d a n o t h e r a n d e v e r y o n e l a u g h e d . " B u t w e c o u l d b e l i k e t h e m , " a n s w e r e d t h e f i r s t . " W h y d o n ' t w e f o r m o u r own s o c i e t y ? A s e c r e t o n e , n a t u r a l l y . " T h e r e w e r e v o i c e s o f a p p r o v a l a l l a r o u n d . S o m e o n e s u g g e s t e d t h e y b o r r o w t h e n a m e o f o l d G e n e r a l B o u r b a k i . T h e r e w a s a r e a s o n f o r t h i s s u g g e s t i o n . I n t h o s e d a y s , t h e m a t h e m a t i c s d e p a r t m e n t a t t h e U n i v e r s i t y d f P a r i s h a d a n a n n u a l t r a d i t i o n o f i n v i t i n g a p r o f e s s i o n a l a c t o r w h o p r e s e n t e d h i m s e l f t o t h e a s s e m b l e d f a c u l t y a n d g r a d u a t e s t u d e n t s a s N i c o l a s
B o u r b a k i . H e w o u l d t h e n d o a o n e - m a n s h o w c o n s i s t i n g o f a l o n g m o n o l o g u e o f m a t h e m a t i c a l d o u b l e - t a l k . S u c h p r e s e n t a t i o n s w e r e v e r y e n t e r t a i n i n g s i n c e t h e r i c h n e s s o f m o d e r n m a t h e m a t i c a l t h e o r y m a k e s f o r a v a s t v o c a b u l a r y t h a t i s b o t h d e s c r i p t i v e o f m a t h e m a t i c s a n d h a s d i f f e r e n t m e a n i n g s i n e v e r y d a y l i f e . O n e s u c h w o r d i s " d e n s e . " T h e r a t i o n a l n u m b e r s a r e s a i d t o b e d e n s e w i t h i n t h e r e a l n u m b e r s . T h i s m e a n s t h a t w i t h i n a n y n e i g h b o r h o o d o f r a t i o n a l n u m b e r s a r e i r r a t i o n a l n u m b e r s . B u t " d e n s e " m e a n s m a n y o t h e r t h i n g s i n e v e r y d a y l i f e .
Graduate students today like to play the same
games of double meaning, and they like to tell the
story of beautiful Poly Nomial who meets the smooth
operator Curly Pi (polynomial, smooth operator, and
curly pi are all mathematical terms).
And so the books these mathematicians wrote
together bore the name Nicolas Bourbaki. A Bourbaki
Seminar was initiated, where mathematical ideas and
theories were discussed not infrequently.
Membership in the society was supposed to be
anonymous, and mathematical results were to be
credited to the society in the name of Bourbaki,
rather than to individual members.
But the members of Bourbaki were not the
Pythagoreans. While the author of the textbooks was
Nicolas Bourbaki, research results such as theorems
and their proofs—which are far more prestigious than
books—were credited to the individual
mathematicians who achieved the results. One of the
first members of Bourbaki was Andre Weil (1906- ),
who later moved to the United States and the
Institute of Advanced Study at Princeton. His name
would never be too far away from the important
conjecture leading to the solution of the Fermat
problem.
Another of the founders of Bourbaki was the
French mathematician Jean Dieudonne, who like most
of the other "French only" members of the society
moved on to greener pastures at the universities in
the United States. Dieudonne, who was the principal
author of many of the books bearing the collective
name of Nicolas Bourbaki, epitomizes the clash
between the Bourbakites' quest for individual
anonymity and their individual egos. Dieudonne once
published a paper bearing the name
o f B o u r b a k i . T h e p a p e r w a s f o u n d t o c o n t a i n a n e r r o r , a n d s o D i e u d o n n e p u b l i s h e d a n o t e t i t l e d : " O n a n E r r o r o f N . B o u r b a k i , " a n d s i g n e d i t J . D i e u d o n n e .
T h e s c h i z o p h r e n i c n a t u r e o f t h e s o c i e t y — i t s m e m b e r s w e r e a l l F r e n c h , b u t m o s t o f t h e m l i v e d i n t h e U n i t e d S t a t e s — m a n i f e s t s i t s e l f i n M r . B o u r b a k i ' s a f f i l i a t i o n . U s u a l l y , p u b l i c a t i o n s o f N i c o l a s B o u r b a k i l i s t h i s a f f i l i a t i o n a s " t h e U n i v e r s i t y o f N a n c a g o , " a f u s i o n o f t h e n a m e o f t h e F r e n c h c i t y o f N a n c y w i t h t h a t o f C h i c a g o . B u t B o u r b a k i p u b l i s h e s o n l y i n F r e n c h , a n d w h e n i t s m e m b e r s m e e t , u s u a l l y a t a F r e n c h r e s o r t c i t y , t h e c o n v e r s a t i o n s a r e n o t o n l y i n F r e n c h , t h e y a r e i n t h e d i a l e c t o f t h e P a r i s i a n s t u d e n t s . T h e c h a u v i n i s m c a r r i e s i n t o t h e s e p a r a t e l i v e s o f t h e s e F r e n c h m a t h e -m a t i c i a n s l i v i n g i n A m e r i c a . A n d r e W e i l , a f o u n d i n g B o u r -b a k i t e , p u b l i s h e d m a n y i m p o r t a n t p a p e r s i n E n g l i s h . B u t h i s Collected Works, w h i c h h a d s o m e b e a r i n g o n t h e c o n j e c t u r e r e l a t e d t o t h e F e r m a t p r o b l e m , w a s p u b l i s h e d i n F r e n c h a n d t i t l e d Oeuvres'. W e i l ' s u n u s u a l a c t i o n s w o u l d h u r t o n e o f t h e p r i n c i p a l p l a y e r s i n o u r d r a m a , a n d W e i l w o u l d n o t r e c o v e r f r o m t h i s d e b a c l e .
T h e m e m b e r s o f B o u r b a k i m u s t b e g i v e n c r e d i t f o r t h e i r c o l l e c t i v e s e n s e o f h u m o r . S o m e f o r t y y e a r s a g o , t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y
r e c e i v e d a n a p p l i c a t i o n f o r i n d i v i d u a l m e m b e r s h i p f r o m M r . N i c o l a s B o u r b a k i . T h e s e c r e t a r y o f t h e S o c i e t y w a s i m p e r t u r b a b l e . H e r e p l i e d t h a t i f M r . B o u r b a k i w a n t e d t o j o i n t h e S o c i e t y h e w o u l d h a v e t o a p p l y a s a n i n s t i -t u t i o n a l m e m b e r ( w h i c h w a s m u c h m o r e e x p e n s i v e ) . B o u r b a k i d i d n o t w r i t e b a c k .
F E R M A T ' S L A S T T H E O R E M Elliptic
Curves
D i o p h a n t i n e p r o b l e m s — t h a t i s , p r o b l e m s r a i s e d b y e q u a t i o n s o f t h e f o r m g i v e n b y D i o p h a n t u s i n t h e t h i r d c e n t u r y — b e g a n t o b e s t u d i e d m o r e a n d m o r e i n t h e t w e n t i e t h c e n t u r y u s i n g m a t h e m a t i c a l e n t i t i e s c a l l e d e l l i p t i c c u r v e s . B u t e l l i p t i c c u r v e s h a v e n o t h i n g t o d o w i t h e l l i p s e s . T h e y w e r e o r i g i n a l l y u s e d c e n t u r i e s e a r l i e r i n c o n n e c t i o n w i t h e l l i p t i c f u n c t i o n s , w h i c h i n t u r n w e r e d e v i s e d t o h e l p c a l c u l a t e t h e p e r i m e t e r o f a n e l l i p s e . A s w i t h m a n y i n n o v a t i v e i d e a s i n m a t h e m a t i c s , t h e p i o n e e r i n t h i s f i e l d w a s n o n e o t h e r t h a n G a u s s .
O d d l y , e l l i p t i c c u r v e s a r e n e i t h e r e l l i p s e s n o r e l l i p t i c f u n c t i o n s — t h e y a r e c u b i c p o l y n o m i a l s i n t w o v a r i a b l e s . T h e y l o o k l i k e : y 2
= a x 3 + b x 2 + c x w h e r e a, b, a n d c a r e i n t e g e r s o r r a t i o n a l n u m b e r s ( w e a r e c o n c e r n e d w i t h e l l i p t i c c u r v e s o v e r t h e r a t i o n a l n u m b e r s ) . E x a m p l e s o f s u c h e l l i p t i c c u r v e s a r e s h o w n b e l o w . 1 3
W h e n o n e l o o k s a t t h e r a t i o n a l p o i n t s o n t h e e l l i p t i c c u r v e s — t h a t i s , o n e l o o k s o n l y a t p o i n t s o n t h e c u r v e t h a t a r e r a t i o s o f t w o i n t e g e r s ( n o i r r a t i o n a l n u m b e r s s u c h a s p i o r t h e s q u a r e r o o t o f t w o , e t c . ) , t h e s e n u m b e r s f o r m a g r o u p . T h a t m e a n s t h a t t h e y h a v e n i c e p r o p e r t i e s . T a k e a n y t w o s o l u t i o n s , - t h e y c a n b e " a d d e d " i n a s e n s e t o p r o d u c e a third s o l u t i o n o n t h e c u r v e . N u m b e r t h e o r i s t s h a v e b e c o m e f a s c i n a t e d w i t h t h e e l l i p t i c c u r v e s s i n c e
t h e y c a n a n s w e r m a n y q u e s t i o n s a b o u t e q u a t i o n s a n d t h e i r s o l u t i o n s . E l l i p t i c c u r v e s t h u s b e c a m e o n e o f t h e f o r e m o s t r e s e a r c h t o o l s i n n u m b e r t h e o r y . 1 4
A Strange Conjecture Is about to be Made
I t w a s k n o w n f o r s o m e t i m e b y n u m b e r t h e o r y e x p e r t s t h a t some o f t h e e l l i p t i c c u r v e s t h e y w e r e s t u d y i n g w e r e m o d u l a r .
T h a t i s , t h e s e f e w e l l i p t i c c u r v e s c o u l d b e v i e w e d a s c o n n e c t e d w i t h m o d u l a r f o r m s . S o m e e l l i p t i c c u r v e s c o u l d b e s o m e h o w c o n n e c t e d w i t h t h e c o m p l e x p l a n e a n d t h e s e f u n c t i o n s i n h y p e r b o l i c s p a c e w i t h t h e i r m a n y s y m m e t r i e s . I t w a s n o t c l e a r w h y a n d h o w t h i s w a s h a p p e n i n g . T h e m a t h e m a t i c s w a s e x t r e m e l y c o m p l i c a t e d , e v e n f o r e x p e r t s , a n d t h e i n t e r n a l s t r u c t u r e — t h e b e a u t i f u l h a r m o n i e s t h a t e x i s t e d — w a s l i t t l e u n d e r s t o o d . T h e e l l i p t i c f u n c t i o n s t h a t w e r e i n d e e d m o d u l a r h a d i n t e r e s t i n g p r o p e r t i e s . S o o n s o m e o n e w o u l d m a k e t h e b o l d c o n j e c t u r e t h a t all e l l i p t i c f u n c t i o n s w e r e m o d u l a r .
T o u n d e r s t a n d t h e i d e a o f m o d u l a r i t y , w h i c h e x i s t s
w i t h i n
t h e n o n - E u c l i d e a n s p a c e o f t h e u p p e r c o m p l e x h a l f -p l a n e , w h e r e s y m m e t r i e s a r e s o c o m p l i c a t e d t h a t t h e y c a n h a r d l y b e v i s u a l i z e d , i t i s u s e f u l t o l o o k a t a v e r y s i m p l e e x a m p l e . T h i s i s a n e x a m p l e w h e r e t h e c u r v e o f i n t e r e s t i s n o t a n e l l i p t i c c u r v e , - i n s t e a d o f a c u b i c e q u a t i o n i n t w o v a r i a b l e s , i t i s o n l y a s q u a r e d e q u a t i o n i n t w o v a r i a b l e s : t h e c u r v e i s a s i m p l e c i r c l e . T h e e q u a t i o n o f a c i r c l e w i t h r a d i u s a2
w h o s e c e n t e r i s z e r o i s g i v e n b y : x 2 + y2 =a2. N o w l o o k a t t h e s i m p l e p e r i o d i c f u n c t i o n s : x =a c o s ( , a n d y = a s i n t . T h e s e t w o f u n c t i o n s c a n s t a n d f o r x a n d y i n t h e e q u a t i o n o f t h e c i r c l e . T h e e q u a t i o n o f t h e c i r c l e i s m o d u l a r i n t h i s s e n s e . T h e r e a s o n i s t h e t r i g o n o m e t r i c i d e n t i t y t h a t s a y s t h a t s i n 2 t + c o s 2 t = 1, a n d s u b s t i t u t i n g t h i s f o r -m u l a i n t o t h e e q u a t i o n o f t h e c i r c l e ( e a c h t e r m m u l t i p l i e d b y a) g i v e s a n i d e n t i t y .
A m o d u l a r e l l i p t i c c u r v e i s j u s t a n e x t e n s i o n o f t h i s i d e a t o t h e m o r e c o m p l i c a t e d c o m p l e x p l a n e , w i t h a s p e c i a l n o n - E u c l i d e a n g e o m e t r y . H e r e t h e p e r i o d i c f u n c t i o n s a r e s y m m e t r i e s n o t o n l y w i t h r e s p e c t t o o n e v a r i a b l e , t , a s w i t h t h e s i n e s a n d c o s i n e s o n t h e l i n e— t h e y a r e t h e a u t o m o r p h i c , o r t h e m o d u l a r f o r m s o n t h e c o m p l e x p l a n e , w h i c h h a v e s y m m e t r i e s w i t h r e s p e c t t o m o r e c o m p l i c a t e d
t r a n s f o r m a t i o n s ( f ( z ) —>j(az+b/cz+d)).
Tokyo, Japan, Early 1950s
I n t h e e a r l y 1950s, J a p a n w a s a n a t i o n e m e r g i n g f r o m t h e d e v a s t a t i o n o f w a r . P e o p l e w e r e n o l o n g e r h u n g r y , b u t t h e y w e r e s t i l l p o o r a n d e v e r y d a y s u r v i v a l w a s a s t r u g g l e f o r t h e a v e r a g e J a p a n e s e . Y e t f a c t o r i e s w e r e b e i n g r e b u i l t f r o m t h e r u b b l e , b u s i n e s s e s r e e s t a b l i s h e d , a n d t h e g e n e r a l m o o d w a s h o p e f u l . U n i v e r s i t y l i f e i n J a p a n a t t h a t t i m e w a s a l s o d i f f i c u l t . C o m
p e t i t i o n a m o n g s t u d e n t s w a s f i e r c e : g o o d g r a d e s m e a n t g o o d j o b s a f t e r g r a d u a t i o n . T h i s w a s e s p e c i a l l y t r u e f o r d o c t o r a l s t u d e n t s i n p u r e m a t h e m a t i c s , s i n c e p o s i t i o n s a t u n i v e r s i t i e s w e r e s c a r c e e v e n t h o u g h t h e p a y w a s l o w . Y u t a k a T a n i y a m a w a s o n e s u c h d o c t o r a l s t u d e n t i n m a t h e m a t i c s . H e w a s b o r n o n N o v e m b e r 12, 1927, t h e y o u n g e s t o f e i g h t c h i l d r e n i n t h e f a m i l y o f a c o u n t r y d o c t o r i n t h e t o w n o f K i s a i , a b o u t 3 0 m i l e s n o r t h o f T o k y o . A t a y o u n g a g e , T a n i y a m a b e g a n t o s t u d y t h e a r e a o f m a t h e m a t i c s i n v o l v i n g c o m p l e x m u l t i p l i c a t i o n o f a b e l i a n v a r i e t i e s . N o t m u c h w a s k n o w n a b o u t t h i s f i e l d a n d T a n i y a m a h a d a v e r y d i f f i c u l t t i m e . T o m a k e t h i n g s w o r s e , h e f o u n d t h e a d v i c e o f o l d e r p r o f e s s o r s a t t h e U n i v e r s i t y o f T o k y o v i r t u a l l y u s e l e s s . H e h a d t o d e r i v e e v e r y d e t a i l o n h i s o w n a n d h e u s e d t o d e s c r i b e e v e r y t a s k i n h i s m a t h e m a t i c a l r e s e a r c h u s i n g f o u r C h i n e s e c h a r a c t e r s t h a t m e a n t " h a r d f i g h t i n g " a n d " b i t t e r s t r u g g l e . " N o t h i n g w a s e a s y i n y o u n g Y u t a k a T a n i y a m a ' s l i f e .
T a n i y a m a l i v e d i n a o n e -r o o m a p a r t m e n t o f 81 s q u a r e f e e t . T h e r e w a s o n l y o n e t o i l e t o n e v e r y f l o o r o f t h e b u i l d i n g , s h a r e d b y a l l t h e r e s i d e n t s o f t h e f l o o r . T o t a k e a b a t h , T a n i y a m a h a d t o g o t o a p u b l i c b a t h h o u s e
s o m e d i s t a n c e a w a y f r o m h i s b u i l d i n g . T h e s h a b b y a p a r t m e n t b u i l d i n g w a s n a m e d " V i l l a T r a n q u i l M o u n t a i n s , " i r o n i c a l l y s o s i n c e i t s t o o d o n a b u s y s t r e e t a n d b y a r a i l r o a d t r a c k o n w h i c h t r a i n s t h u n d e r e d b y e v e r y f e w m i n u t e s . P o s s i b l y s o h e c o u l d c o n c e n t r a t e b e t t e r o n h i s r e s e a r c h , y o u n g Y u t a k a w o r k e d m o s t l y a t n i g h t , o f t e n g o i n g t o b e d a t 6 AM
w h e n t h e n o i s y d a y b e g a n . E x c e p t i n t h e h e a t o f s u m m e r , a l m o s t e v e r y d a y T a n i y a m a w o r e t h e s a m e b l u e - g r e e n s u i t w i t h a m e t a l l i c s h e e n . A s h e e x p l a i n e d t o h i s g o o d f r i e n d G o r o S h i m u r a , h i s
f a t h e r b o u g h t t h e m a t e r i a l v e r y c h e a p l y f r o m a p e d d l e r . B u t b e c a u s e o f t h e l u s t e r , n o o n e i n t h e f a m i l y d a r e d w e a r i t . Y u t a k a , w h o d i d n ' t c a r e h o w h e l o o k e d , f i n a l l y v o l u n t e e r e d a n d h a d t h e m a t e r i a l t a i l o r e d f o r a s u i t w h i c h b e c a m e h i s d a i l y o u t f i t .
T a n i y a m a g r a d u a t e d f r o m t h e U n i v e r s i t y o f T o k y o i n 1953 a n d t o o k o n t h e p o s i t i o n o f " s p e c i a l r e s e a r c h s t u d e n t " a t t h e d e p a r t m e n t o f m a t h e m a t i c s t h e r e . H i s f r i e n d S h i m u r a g r a d u a t e d a y e a r e a r l i e r a n d h a d a s i m i l a r p o s i t i o n i n m a t h e m a t i c s a t t h e C o l l e g e o f G e n e r a l E d u c a t i o n a c r o s s c a m p u s . T h e i r f r i e n d s h i p b e g a n a f t e r o n e o f t h e m w r o t e t h e o t h e r a l e t t e r a s k i n g h i m t o r e t u r n t o t h e l i b r a r y a n i s s u e o f a m a t h e m a t i c a l j o u r n a l w h i c h i n t e r e s t e d t h e m b o t h . T h e y w o u l d o f t e n e a t t o g e t h e r a t i n e x p e n s i v e r e s t a u r a n t s s u p p o s e d l y s e r v i n g W e s t e r n -s t y l e f o o d , s u c h a s t o n g u e s t e w , w h i c h w a s b e c o m i n g p o p u l a r i n J a p a n . 1 5
F e w g o o d m a t h e m a t i c i a n s s t a y e d i n J a p a n i n t h o s e d a y s . O n c e a m a t h e m a t i c i a n a c h i e v e d s o m e r e n o w n , h e o r s h e w o u l d t r y t o m o v e t o a u n i v e r s i t y i n A m e r i c a o r E u r o p e , w h e r e t h e m a t h e m a t i c a l c o m m u n i t y w a s m o r e e s t a b l i s h e d a n d w h e r e c o n n e c t i o n s w i t h p e o p l e d o i n g r e s e a r c h i n t h e s a m e f i e l d s w e r e p o s s i b l e . S u c h l i n k s w e r e i m p o r t a n t f o r
c o n d u c t i n g r e s e a r c h i n e s o t e r i c a r e a s a b o u t w h i c h n o t m u c h w a s k n o w n . T o t r y t o f o s t e r r e s e a r c h t i e s w i t h p e o p l e w i t h k n o w l e d g e i n t h e i r f i e l d o f i n t e r e s t , t h e t w o f r i e n d s h e l p e d o r g a n i z e t h e T o k y o - N i k k o S y m p o s i u m o n A l g e b r a i c N u m b e r T h e o r y i n S e p t e m b e r , 1955. S o m e s t a t e m e n t s m a d e a t t h i s s m a l l c o n f e r e n c e , w h i l e d e s t i n e d t o r e m a i n o b s c u r e f o r a l o n g t i m e , w o u l d e v e n t u a l l y l e a d t o m o m e n t o u s r e s u l t s — a n d a s a v a g e c o n t r o v e r s y — a l m o s t f o r t y y e a r s l a t e r .
Goro Shimura, circa 1965, when he first
developed his conjecture
A Hopeful Beginning
T h e t w o f r i e n d s h e l p e d f i l e t h e n e c e s s a r y f o r m s w i t h t h e a d m i n i s t r a t i o n , a r r a n g e d f o r t h e c o n f e r e n c e f a c i l i t i e s , a n d h e l p e d s e n d o u t i n v i t a t i o n s t o l o c a l a n d f o r e i g n m a t h e m a t i c i a n s t h e y h o p e d w o u l d a t t e n d t h e c o n f e r e n c e . A n d r e W e i l , w h o h a d l e f t F r a n c e i n t h e m e a n t i m e a n d w a s a p r o f e s s o r a t t h e U n i v e r s i t y o f C h i c a g o , w a s o n e o f t h o s e i n v i t e d t o a t t e n d . A t t h e I n t e r n a -t i o n a l C o n g r e s s o f M a t h e m a t i c i a n s f i v e y e a r s e a r l i e r , W e i l h a d b r o u g h t t o
t h e a t t e n t i o n o f t h e m a t h e m a t i c a l c o m m u n i t y a n
u n k n o w n c o n j e c t u r e b y a m a t h e m a t i c i a n b y t h e n a m e o f H a s s e a b o u t t h e " z e t a f u n c t i o n o f a v a r i e t y o v e r a n u m b e r f i e l d . " T h e o b s c u r e s t a t e m e n t h e l d s o m e i n t e r e s t t o r e s e a r c h e r s i n n u m b e r t h e o r y . A p p a r e n t l y , W e i l w a s c o l l e c t i n g t h e s e c o n j e c t u r a l i d e a s i n t h e t h e o r y o f n u m b e r s a n d i n c l u d e d t h i s o n e i n h i s Collected Papers, g i v i n g c r e d i t t o H a s s e .
H i s i n t e r e s t i n r e s u l t s i n t h i s a r e a m a d e h i m a t t r a c t i v e t o T a n i y a m a a n d S h i m u r a , a n d t h e y w e r e p l e a s e d w h e n h e a c c e p t e d t h e i n v i t a t i o n t o a t t e n d t h e i r c o n f e r e n c e . A n o t h e r f o r e i g n m a t h e m a t i c i a n t o c o m e t o T o k y o - N i k k o w a s a y o u n g e r F r e n c h m a t h e m a t i c i a n , J e a n - P i e r r e S e r r e . W h i l e h e m a y n o t h a v e b e e n a m e m b e r o f B o u r b a k i a t t h a t t i m e , s i n c e t h e s o c i e t y i n c l u d e d o n l y w e l l -k n o w n m a t h e m a t i c i a n s , h e w o u l d b e c o m e o n e w i t h i n t h e f o l l o w i n g d e c a d e s . S e r r e h a s b e e n d e s c r i b e d b y s o m e m a t h e m a t i c i a n s a s a m b i t i o u s a n d f i e r c e l y c o m p e t i t i v e . H e c a m e t o T o k y o - N i k k o t o l e a r n a s m u c h a s h e c o u l d . T h e J a p a n e s e k n e w s o m e t h i n g s a b o u t n u m b e r t h e o r y , a n d t h e y h a d m a n y r e s u l t s p u b l i s h e d o n l y i n J a p a n e s e a n d t h u s h i d d e n f r o m t h e r e s t o f t h e w o r l d . T h i s w a s a g r e a t o p p o r t u n i t y t o l e a r n o f t h e s e r e s u l t s , s i n c e t h e c o n f e r e n c e w a s t o b e
c o n d u c t e d i n E n g l i s h . H e w o u l d b e o n e o f f e w p e o p l e o u t s i d e J a p a n w i t h k n o w l -e d g e o f t h e m a t h e m a t i c s p r e s e n t e d t h e r e . W h e n t h e p r o c e e d i n g s o f t h e s y m p o s i u m w o u l d b e p u b l i s h e d , i t w o u l d b e i n J a p a n e s e o n l y . T w e n t y y e a r s l a t e r , S e r r e w o u l d d r a w a t t e n t i o n t o s o m e e v e n t s a t T o k y o - N i k k o , a n d t h e w o r l d w o u l d h e a r h i s v e r s i o n — n o t t h e o n e r e c o r d e d i n t h e J a p a n e s e p r o c e e d i n g s .
T h e p r o c e e d i n g s i n c l u d e d t h i r t y - s i x p r o b l e m s . P r o b l e m s 10, 11, 12, a n d 13 w e r e w r i t t e n b y Y u t a k a T a n i y a m a . S i m i l a r t o t h e i d e a s o f H a s s e , T a n i y a m a ' s p r o b l e m s c o n s t i t u t e d a c o n j e c t u r e
a b o u t z e t a f u n c t i o n s . H e s e e m e d t o c o n n e c t P o i n c a r e ' s a u t o m o r p h i c f u n c t i o n s o f t h e c o m p l e x p l a n e w i t h t h e z e t a f u n c t i o n o f a n e l l i p t i c c u r v e . I t w a s m y s t e r i o u s t h a t a n e l l i p t i c c u r v e s h o u l d s o m e h o w b e c o n n e c t e d w i t h s o m e t h i n g i n t h e c o m p l e x p l a n e .
"You Are Saying What...?"
T h e c o n j e c t u r e e m b o d i e d i n t h e f o u r p r o b l e m s w a s n e b u l o u s . T a n i y a m a d i d n o t f o r m u l a t e t h e p r o b l e m s i n a v e r y m e a n i n g f u l w a y , p o s s i b l y b e c a u s e h e w a s n ' t q u i t e s u r e w h a t t h e c o n n e c -t i o n w a s . B u t t h e b a s i c i d e a w a s t h e r e . I t w a s a n i n t u i t i o n , a g u t f e e l i n g t h a t t h e a u t o m o r p h i c f u n c t i o n s w i t h t h e i r m a n y s y m m e t r i e s o n t h e c o m p l e x p l a n e w e r e s o m e h o w c o n n e c t e d w i t h t h e e q u a t i o n s o f D i o p h a n t u s . I t c e r t a i n l y w a s n ' t o b v i o u s . H e w a s p o s i t i n g a h i d d e n c o n n e c t i o n b e t w e e n t w o v e r y d i f f e r e n t b r a n c h e s o f m a t h e m a t i c s .
A n d r e W e i l w a n t e d t o k n o w e x a c t l y w h a t T a n i y a m a h a d i n m i n d . A c c o r d i n g t o t h e w r i t t e n r e c o r d o f t h e c o n f e r e n c e , t h e p r o c e e d i n g s p u b l i s h e d i n J a p a n e s e , t h e f o l l o w i n g e x c h a n g e t o o k p l a c e b e t w e e n W e i l a n d T a n i y a m a : 1 6
Weil: Do you think all elliptic functions are
uniformized by modular functions?
Taniyama: Modular functions alone will not be
enough. I think other special types of automorphic
functions are necessary. Weil: Of course some of
them can probably be handled that way. But in the
general case, they look completely different and
mysterious...
T w o t h i n g s a r e e v i d e n t f r o m t h e c o n v e r s a t i o n . F i r s t , T a n i y a m a
w a s r e f e r r i n g t o " a u t o m o r p h i c f u n c t i o n s " r a t h e r t h a n " m o d u l a r f u n c t i o n s a l o n e " a s b e i n g a s s o c i a t e d w i t h t h e e l l i p t i c c u r v e s . A n d s e c o n d , W e i l d i d n o t b e l i e v e t h a t i n g e n e r a l t h e r e w a s s u c h a c o n n e c t i o n . L a t e r h e w o u l d b e m o r e s p e c i f i c a b o u t t h i s d i s b e l i e f , a l l o f w h i c h w o u l d m a k e i t a s t o u n d i n g t h a t his n a m e , o f a l l p e o p l e ' s , s h o u l d e n d u p b e i n g a s s o c i a t e d w i t h a c o n j e c t u r e h e n e i t h e r f o r m u l a t e d n o r e v e n b e l i e v e d w a s t r u e . B u t f a t e s o m e t i m e s t a k e s s t r a n g e , i m p l a u s i b l e t u r n s , a n d e v e n m o r e b i z a r r e o c c u r r e n c e s w e r e g o i n g t o t r a n s p i r e .
A l l t h i s w o u l d m a t t e r d e c a d e s l a t e r . T o k n o w e x a c t l y w h a t Y u t a k a T a n i y a m a m e a n t , t h o u g h t , a n d s a i d w o u l d b e a b o o n t o m o d e r n h i s t o r i a n s . B u t , u n f o r t u n a t e l y , t r a g e d y s t a l k e d T a n i y a m a , a s i t d i d s o m a n y o t h e r y o u n g m a t h e m a t i c a l g e n i u s e s .
W i t h i n a c o u p l e o f y e a r s , G o r o S h i m u r a l e f t T o k y o , f i r s t f o r P a r i s , t h e n f o r t h e I n s t i t u t e f o r A d v a n c e d S t u d y a n d P r i n c e t o n U n i v e r s i t y . T h e t w o f r i e n d s c o n t i n u e d t o c o m m u n i c a t e b y m a i l . I n S e p t e m b e r , 1958, G o r o S h i m u r a r e c e i v e d t h e l a s t l e t t e r w r i t t e n b y Y u t a k a T a n i y a m a . I n t h e m o r n i n g o f N o v e m b e r 17, 1958, f i v e d a y s a f t e r h i s t h i r t y - f i r s t b i r t h d a y , Y u t a k a T a n i y a m a w a s f o u n d d e a d i n
h i s a p a r t m e n t , a s u i c i d e n o t e o n h i s d e s k .
Shimura's Conjecture
A d e c a d e p a s s e d s i n c e t h e T o k y o - N i k k o c o n f e r e n c e a n d G o r o S h i m u r a , n o w a t P r i n c e t o n , c o n t i n u e d h i s r e s e a r c h o n n u m b e r t h e o r y , z e t a f u n c t i o n s , a n d e l l i p t i c c u r v e s . H e u n d e r s t o o d w h e r e h i s l a t e f r i e n d h a d b e e n w r o n g , a n d h i s o w n r e s e a r c h a n d q u e s t f o r h i d d e n h a r m o n i e s i n t h e r e a l m s o f m a t h e m a t i c s l e d h i m t o f o r m u l a t e a d i f f e r e n t , b o l d e r a n d m o r e p r e c i s e c o n j e c t u r e . H i s c o n j e c t u r e w a s t h a t e v e r y e l l i p t i c c u r v e o v e r t h e
r a t i o n a l n u m b e r s i s u n i f o r m i z e d b y a m o d u l a r f o r m . M o d u l a r f o r m s a r e m o r e s p e c i f i c e l e m e n t s o v e r t h e c o m p l e x p l a n e t h a n a r e t h e a u t o m o r p h i c f u n c t i o n s o f T a n i y a m a . A n d s p e c i f y i n g t h e d o m a i n a s t h e r a t i o n a l n u m b e r s , a n d o t h e r m o d i f i c a t i o n s , w e r e i m p o r t a n t c o r r e c t i o n s a s w e l l .S h i m u r a ' s c o n j e c t u r e c a n b e
e x p l a i n e d u s i n g a p i c t u r e :
I f w e " f o l d " t h e c o m p l e x p l a n e a s a t o r u s ( t h e d o u g h n u t i n t h e p i c t u r e ^ , t h e n t h i s s u r f a c e w i l l h o l d a l l s o l u t i o n s t o e l l i p t i c e q u a t i o n s o v e r t h e r a t i o n a l n u m b e r s , t h e s e i n t u r n a r i s i n g f r o m t h e e q u a t i o n s o f D i o p h a n t u s . W h a t w o u l d l a t e r b e i m p o r t a n t t o t h e s o l u t i o n o f F e r m a t ' s L a s t T h e o r e m i s t h a t i f a s o l u t i o n t o F e r m a t ' s e q u a t i o n x n + y n
= z n e x i s t e d , t h i s s o l u t i o n w o u l d a l s o h a v e t o l i e o n t h a t t o r u s . N o w , S h i m u r a c o n j e c t u r e d t h a t e v e r y e l l i p t i c c u r v e w i t h r a t i o n a l c o e f f i c i e n t s ( t h a t i s , a n e q u a t i o n w i t h c o e f f i c i e n t s o f
t h e f o r m alb w h e r e b o t h a a n d b
a r e i n t e g e r s ) h a s a " m a t e " o n t h e c o m p l e x h a l f - p l a n e o f P o i n c a r e , w i t h i t s n o n -E u c l i d e a n , h y p e r b o l i c g e o m e t r y . T h e p a r t i c u l a r m a t e o f e a c h r a t i o n a l e l l i p t i c c u r v e w a s a v e r y s p e c i f i c f u n c t i o n o n t h e
c o m p l e x h a l f - p l a n e , w h i c h w a s i n v a r i a n t u n d e r c o m p l i c a t e d t r a n s f o r m a t i o n s o f t h e p l a n e — t h e o n e s m e n t i o n e d e a r l i e r :
/ ( z ) >f{az+b/cz+d), t h e c o e f f i c i e n t s f o r m i n g a g r o u p
w i t h m a n y u n e x p e c t e d s y m m e t r i e s . A l l o f t h i s w a s v e r y c o m p l e x , v e r y t e c h n i c a l , a n d — a s m o s t m a t h e m a t i c i a n s w o u l d b e l i e v e f o r s e v e r a l d e c a d e s —i m p o s s i b l e t o p r o v e i n t h e f o r e s e e a b l e f u t u r e .W h a t S h i m u r a ' s c o n j e c t u r e
w a s s a y i n g w a s t h a t e v e r y e l l i p t i c c u r v e w a s t h e p a r t o f a n i c e b e r g l y i n g a b o v e t h e w a t e r l i n e . B e l o w t h e s u r f a c e l a y a w h o l e i n t r i c a t e s t r u c t u r e . T o p r o v e t h e c o n j e c t u r e , o n e w o u l d h a v e t o s h o w t h a t every i c e b e r g h a d a n u n d e r w a t e r p a r t . S o m e s p e c i a l g r o u p s o f i c e b e r g s w e r e k n o w n t o h a v e t h e u n d e r w a t e r p a r t , b u t s i n c e t h e r e w e r e i n f i n i t e l y m a n y i c e b e r g s , o n e c o u l d n ' t j u s t g o l o o k u n d e r e a c h o n e o f t h e m . A g e n e r a l p r o o f w a s n e c e s s a r y t o s h o w t h a t a n i c e b e r g c o u l d n ' t e x i s t w i t h o u t p a r t o f i t b e i n g u n d e r w a t e r . T h e f o r m u l a t i o n o f s u c h a p r o o f w a s c o n s i d e r e d e x c e e d i n g l y d i f f i c u l t .
Intrigue and a Betrayal
A t a p a r t y a t t h e I n s t i t u t e f o r A d v a n c e d S t u d y a t P r i n c e t o n i n t h e e a r l y 1960s,
S h i m u r a a g a i n m e t J e a n -P i e r r e S e r r e . A c c o r d i n g t o
S h i m u r a , S e r r e a p p r o a c h e d h i m r a t h e r a r r o g a n t l y . " I d o n ' t t h i n k t h a t y o u r r e s u l t s o n m o d u l a r c u r v e s a r e a n y g o o d , " h e s a i d . " W h y , t h e y d o n ' t e v e n a p p l y t o a n a r b i t r a r y e l l i p t i c c u r v e . " I n r e s p o n s e , S h i m u r a s t a t e d h i s c o n j e c t u r e e x a c t l y : " S u c h a c u r v e s h o u l d always b e u n i f o r m i z e d b y a m o d u l a r c u r v e . " 1 7 S e r r e w e n t t o W e i l , w h o w a s n o t a t t h e p a r t y b u t w a s a m e m b e r o f t h e I n s t i -t u t e a n d t h e r e f o r e i n t h e i m m e d i a t e a r e a , a n d t o l d h i m o f h i s c o n v e r s a t i o n w i t h S h i m u r a . I n r e s p o n s e , A n d r e W e i l c a m e t o
Shimura. "Did you really say that?" he asked him, puzzled. "Yes,"
answered Shimura, "don't you think it plausible?" Ten years after
Taniyama's related conjecture, Andre Weil still did not believe
either conjecture. He answered: "I don't see any reason against it,
since one and the other of these sets are denumerable, but I don't
see any reason either for this hypothesis." What Weil said on this
occasion would later be described as "stupid," and "inane" by Serge
Lang of Yale University, who would circulate these comments
together with copies of two dozen letters he named collectively
"The Taniyama-Shimura File," among about fifty mathematicians
worldwide. What Weil meant in his response to Shimura was
tantamount to the following: If in a room you have seven men and
seven women and you conjecture that these are seven married
couples, then 1 see no reason against it, since the number of men
is the same as the number of women. But I don't see any reason for
your hypothesis, either. It could be that they are all single. What
made Lang describe the statement as stupid was that the counting
argument didn't really apply in any simple way here, because
"denumerable" means infinite and countable (such as the number
of all the positive integers: 1, 2, 3, 4,...) and matching such
infinite collections is no simple task. At any rate, it is clear that
Andre Weil did not believe Shimura's theory was necessarily true.
He would later admit that the conversation took place and, stupid,
inane, or otherwise, would quote it. But this would happen only in
1979, when he would write:18
Quelques annees plus tard, a Princeton, Shimura me demanda si
je trouvais plausible que toute courbe elliptique sur Q hit
contenue dans le jacobienne d'une courbe definie par une sous-
groupe de
congruence du groupe modulaire, je lui repondis, il
me semble, que je n'y voyais pas d'empechement,
puisque l'un et l'autre ensemble est denombrable,
mais je ne voyais rien non plus qui parlat en faveur
de cette hypothese.
[ " S o m e y e a r s l a t e r , a t P r i n c e t o n , S h i m u r a a s k e d m e i f I f o u n d i t p l a u s i b l e t h a t e v e r y e l l i p t i c c u r v e o v e r Q w a s c o n t a i n e d i n t h e j a c o b i a n o f a c u r v e d e f i n e d b y a c o n g r u e n c e s u b g r o u p o f a m o d u l a r g r o u p , - 1 r e s p o n d e d t o h i m , i t s e e m s t o m e , t h a t I d o n ' t s e e a n y t h i n g a g a i n s t i t , s i n c e o n e s e t a n d t h e o t h e r a r e d e n u m e r a b l e , b u t n e i t h e r d o 1 s e e a n y t h i n g t h a t s p e a k s i n f a v o r o f t h i s h y p o t h e s i s . " ]
B u t e v e n t h e n , W e i l w o u l d w r i t e " S h i m u r a a s k e d m e " ( m e demanda), r a t h e r t h a n " S h i m u r a t o l d m e , " w h e n r e f e r r i n g t o t h e s t a t e m e n t t h a t i s S h i m u r a ' s c o n j e c t u r e . W e i l p u b l i s h e d s o m e r e l a t e d p a p e r s , a n d w h i l e h e d i d n o t b e l i e v e S h i m u r a ' s t h e o r y , h i s o w n n a m e b e c a m e a s s o c i a t e d w i t h i t . T h e e r r o r w a s p e r p e t u a t e d w h e n m a t h e m a t i c i a n s m a d e r e f e r e n c e s i n t h e i r p a p e r s t o t h e w o r k s o f o t h e r s , a n d t h e m i s q u o t a t i o n i s p r e s e n t t o t h i s d a y w h e n w r i t e r s i g n o r a n t o f t h e h i s t o r y r e f e r t o t h e W e i l - T a n i y a m a c o n j e c t u r e i n s t e a d o f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e . W e i l s e e m e d t o e n j o y h i s a s s o c i a t i o n w i t h a n
i m p o r t a n t t h e o r y w h i c h —w h i l e h e h i m s e l f d i d n o t b e l i e v e i n i t — m o s t m a t h e m a t i c i a n s t h o u g h t w o u l d b e p r o v e d s o m e d a y i n t h e d i s t a n t f u t u r e .
W i t h t h e p a s s i n g d e c a d e s , t h e r e w a s m o r e a n d m o r e r e a s o n f o r t h e c o n n e c t i o n t o e x i s t . I f a n d w h e n t h e c o n j e c t u r e w a s p r o v e d , i t w o u l d b e a s u b s t a n t i a l m a t h e m a t i c a l t h e o r y . W e i l w o r k e d a r o u n d t h e c o n j e c t u r e , n e v e r l e a v i n g m a t h e m a t i c a l
results he obtained too far away from the possible
connection between modular forms of the complex
plane and the elliptic curves of Diophantine
equations. And while he certainly knew better, he
held back references to Shimura and his crucial role
until almost two decades had passed. Then he gave
offhand praise to Shimura in casual conversation and
mentioned him— almost in passing—in a published
paper. Meanwhile, in France, Serre was actively
contributing to the false attribution. He made every
effort to associate the name of Andre Weil with the
conjecture, instead of that of Goro Shimura.
"An Exercise for the Interested Reader"
In 1967, Andre Weil wrote a paper in German, in
which he said:19
Ob sich diese Dinge immer, d.h. fur jede iiber Q
definierte Kurve C, so verhalten, scheint im
Moment noch problematisch zu sein und mag dem
interessierten Leser als Ubungsaufgabe empfohlen
werden. x
["Whether these things, that is for every curve C
defined over Q, so behave, at this moment is still
seen as problematic and will be recommended as an
exercise for the interested reader."] This paragraph
refers to elliptic curves over the rational numbers
(which mathematicians denote by Q), and "sich so
verhalten" here refers to being modular, that is, it
states Shimura's conjecture. But, here again, Weil
did not attribute the theory to its originator. He did
so only 12 years later, and even then as "Shimura
asked me..." as we have just seen. In this
p a p e r i n G e r m a n , a b o v e , W e i l c a l l s t h e c o n j e c t u r e " p r o b l e m a t i c . " A n d t h e n h e d o e s s o m e t h i n g s t r a n g e . H e s i m p l y a s s i g n s t h e c o n j e c t u r e a s a n exercise for the interested
reader ( " u n d m a g d e m i n t e r e s s i e r t e n L e s e r a l s U b u n g s a u f g a b e e m p f o h l e n w e r d e n " ) . T h i s e x e r c i s e f o r t h e " i n t e r e s t e d r e a d e r " w o u l d t a k e o n e o f t h e w o r l d ' s f i n e s t m a t h e m a t i c i a n s s e v e n y e a r s o f w o r k i n s o l i t u d e t o a t t e m p t t o p r o v e . W h e n a m a t h e m a t i c i a n a s s i g n s a h o m e w o r k p r o b l e m (Ubungsaufgabe),
u s u a l l y h e o r s h e k n o w s t h e p r o o f t h r o u g h a n d t h r o u g h , a n d b e l i e v e s — k n o w s f o r c e r -t a i n — t h a t t h e t h e o r e m i s t r u e , n o t " p r o b l e m a t i c " a s W e i l d e s c r i b e s i t .
T h e r e i s a n o l d s t o r y a b o u t a m a t h p r o f e s s o r w h o t e l l s h i s c l a s s " t h i s i s o b v i o u s , " w h e n r e f e r r i n g t o s o m e c o n c e p t . T h e c l a s s l o o k s c o n f u s e d s i n c e i t i s n o t a t a l l o b v i o u s , a n d f i n a l l y a b o l d s t u d e n t r a i s e s a h a n d a n d a s k s , " W h y ? " T h e p r o f e s s o r t h e n s t a r t s d r a w i n g a n d w r i t i n g o n t h e e d g e o f t h e b o a r d w i t h o n e h a n d , c o v e r i n g t h e w r i t i n g w i t h h i s o t h e r h a n d , a n d e r a s i n g e v e r y t h i n g a s h e i s d o n e . A f t e r a b o u t t e n m i n u t e s o f t h i s f u r t i v e s c r i b b l i n g , t h e p r o f e s s o r e r a s e s t h e b o a r d c o m p l e t e l y a n d a n n o u n c e s t o t h e b e f u d d l e d c l a s s : " Y e s , i t ' s
o b v i o u s . "
Tbe Lie
I n t h e 1970s, T a n i y a m a ' s p r o b l e m s f r o m t h e T o k y o -N i k k o m e e t i n g r e c e i v e d w i d e r d i s t r i b u t i o n . I n t h e m e a n t i m e , s i n c e W e i l h a d w r i t t e n a b o u t t h e c o n j e c t u r e h e d o u b t e d , m o d u l a r e l l i p t i c c u r v e s b e c a m e k n o w n a s " W e i l C u r v e s . " W h e n T a n i y a m a ' s p r o b l e m s b e c a m e b e t t e r k n o w n i n t h e W e s t , t h e c o n j e c t u r e a b o u t s u c h c u r v e s c a m e t o b e c a l l e d t h e " T a n i y a m a - W e i l c o n j e c t u r e . " S h i m u r a ' s n a m e w a s l e f t o u t c o m p l e t e l y . B u t
s i n c e T a n i y a m a ' s n a m e c a m e i n , W e i l s t a r t e d t o i n v e i g h a g a i n s t c o n j e c t u r e s a l t o g e t h e r . I n 1979, a m e r e f i v e y e a r s b e f o r e i t w a s p r o v e d b y G e r d F a l t i n g s , W e i l e v e n s p o k e a g a i n s t " t h e s o - c a l l e d ' M o r d e l l c o n j e c t u r e ' o n D i o p h a n t i n e e q u a t i o n s . " H e c o n t i n u e d , " I t w o u l d b e n i c e i f t h i s w e r e s o , a n d I w o u l d r a t h e r b e t f o r i t t h e n a g a i n s t i t . B u t i t i s n o m o r e t h a n w i s h f u l t h i n k i n g b e c a u s e t h e r e i s n o t a s h r e d o f e v i d e n c e f o r i t , a n d a l s o n o n e a g a i n s t i t . " B u t W e i l w a s w r o n g e v e n t h e n . A n u m b e r o f R u s s i a n m a t h e m a t i c i a n s , a m o n g t h e m S h a f a r e v i c h a n d P a r s h i n , w e r e a l r e a d y o b t a i n i n g r e s u l t s t h a t w o u l d s u p p l y e v i d e n c e f o r t h e M o r d e l l c o n j e c t u r e a s e a r l y a s t h e e a r l y 1970s. I n 1984, o f c o u r s e , G e r d F a l t i n g s w o u l d p r o v e t h e c o n j e c t u r e o u t r i g h t , m a k i n g F e r m a t ' s L a s t T h e o r e m " a l m o s t a l w a y s t r u e . "
W h i l e A n d r e W e i l w a s t u r n i n g a g a i n s t a l l c o n j e c t u r e s a t a t i m e w h e n h i s n a m e w a s n o l o n g e r b e i n g e x c l u s i v e l y a s s o c i a t e d w i t h t h e c o n j e c t u r e n o w c a l l e d T a n i y a m a - W e i l b y m a n y m a t h e m a t i c i a n s , S e r r e i n P a r i s w a s w o r k i n g t o k e e p S h i m u r a ' s n a m e d i s s o c i a t e d f r o m t h e c o n j e c t u r e . I n 1986,
a t a p a r t y a t t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y , a n d w i t h i n e a r s h o t o f a
n u m b e r o f p e o p l e , Jean- P i e r r e S e r r e t o l d S e r g e L a n g t h a t A n d r e W e i l h a d t o l d h i m o f a c o n v e r s a t i o n h e h a d h a d with
G o r o S h i m u r a . A c c o r d i n g t o S e r r e , this i s what W e i l t o l d h i m t o o k p l a c e :
Weil: Why did Taniyama think that all elliptic
curves are modular?
Shimura: You told him so yourself, and you have
forgotten.
A t this m o m e n t , L a n g , w h o u n k n o w i n g l y h a d b e e n u s i n g t h e t e r m s "Weil c u r v e " a n d " T a n i y a m a -Weil c o n j e c t u r e , " b e c a m e
s u s p i c i o u s . H e t o o k i t u p o n h i m s e l f t o f i n d t h e t r u t h . L a n g i m m e d i a t e l y w r o t e t o b o t h W e i l a n d S h i m u r a , t h e n t o S e r r e . S h i m u r a c a t e g o r i c a l l y d e n i e d t h a t s u c h a c o n v e r s a t i o n e v e r t o o k p l a c e , a n d g a v e a m p l e e v i d e n c e f o r t h i s c l a i m . W e i l d i d n o t r e p l y r i g h t a w a y . A n d S e r r e , i n h i s r e s p o n s e , c r i t i c i z e d L a n g ' s a t t e m p t t o f i n d t h e t r u t h . I n h i s B o u r b a k i S e m i n a r i n J u n e 1995,
S e r r e s t i l l r e f e r r e d t o t h e c o n j e c t u r e a s t h a t o f " T a n i y a m a - W e i l , " l e a v i n g o u t t h e n a m e o f i t s o r i g i n a t o r , w h o t r u s t e d h i m w i t h h i s c o n j e c t u r e 3 0 y e a r s e a r l i e r . W e i l r e s p o n d e d a f t e r a s e c -o n d a t t e m p t t o c o n t a c t h i m b y L a n g . H i s l e t t e r f o l l o w s . 2 0
3 December 1986.
Dear Lang,
I do not recall when and where your letter of
August 9 first reached me. When it did, I had (and
still have) far more serious matters to think about.
I cannot but resent strongly any suggestion that
I ever sought to diminish the credit due to
Taniyama and to Shimura. I am glad to see that
you admire them. So do 1.
Reports of conversations held long ago are open
to misunderstandings. You choose to regard them
as "history",- they are not. At best they are
anecdotes. Concerning the controversy which you
have found fit to raise, Shimura's letters seem to
me to put an end to it, once and for all.
As to attaching names to concepts, theorems, or
(?) conjectures, I have often said: (a) that, when a
proper name gets attached to (say) a concept, this
should never be taken as a sign that the author in
question had anything to do with the concept;
more often than not, the opposite is true.
Pythagoras had
nothing to do with "his" theorem, nor Fuchs with the Fonctions
fuchsiennes, any more than Auguste Comte with rue Auguste-
Comte,- (b) proper names tend, quite properly, to get replaced
by more appropriate ones,- the Leray-Koszul sequence is now a
spectral sequence (and as Siegel once told Erdos, abelian is now
written with a small a).
Why shouldn't I have made "stupid" remarks sometimes, as
you are pleased to say? But indeed, 1 was "out of it" in 1979
when expressing some skepticism about Mordell's conjecture,
since at that time 1 was totally ignorant of the work of the
Russians (Parshin, etc.) in that direction. My excuse, if it is one,
is that I had had long conversations with Shafarevich in 1972,
and he never mentioned any of that work.
Sincerely,
A. Weil
AW:ig
RS. Should you wish to run this letter through your Xerox
machine, do feel free to do so. I wonder what the Xerox Co.
would do without you and the like of you.
Deep in the Black Forest, Fall i 984
W h i l e t h e c o n t r o v e r s y a b o u t w h o o r i g i n a t e d t h e S h i m u r a -T a n i y a m a c o n j e c t u r e w a s r a g i n g i n B e r k e l e y , N e w H a v e n , P r i n c e t o n , a n d a c r o s s t h e A t l a n t i c i n P a r i s , s o m e t h i n g t o t a l l y u n e x p e c t e d w a s h a p p e n i n g d e e p i n t h e B l a c k F o r e s t o f s o u t h w e s t G e r m a n y .
G e r h a r d F r e y r e c e i v e d h i s D i p l o m a f r o m t h e U n i v e r s i t y o f T u b i n g e n , a n d h i s P h . D . f r o m t h e U n i v e r s i t y o f H e i d e l b e r g ,
w h e r e h e s t u d i e d n u m b e r t h e o r y a n d w a s i n f l u e n c e d b y t h e w o r k s o f H a s s e a n d W e i l . F r e y w a s f a s c i n a t e d b y t h e i n t e r p l a y b e t w e e n t h e t h e o r y o f n u m b e r s a n d a l g e b r a i c g e o m e t r y , a n a r e a o f m a t h e m a t i c s w h i c h w a s d e v e l o p e d i n t h e l a s t f i f t y y e a r s . H e w a s a l s o i n t e r e s t e d i n a r i t h m e t i c g e o m e t r y . I t w a s t h e c o n n e c t i o n s b e t w e e n n u m b e r t h e o r y a n d t h e n e w e r f i e l d s o f a l g e b r a i c a n d a r i t h m e t i c g e o m e t r y t h a t w o u l d l e a d h i m t o m a k e a n u n e x p e c t e d m a t h e m a t i c a l s t a t e m e n t . I n t h e 1970s, F r e y d i d a l o t o f w o r k o n e l l i p t i c c u r v e s a n d D i o p h a n t i n e e q u a t i o n s , a n d i n 1978 h e r e a d t h e p a p e r " M o d u l a r c u r v e s a n d t h e E i s e n s t e i n i d e a l , " b y B a r r y M a z u r o f H a r v a r d U n i v e r s i t y . F r e y w a s s t r o n g l y i n f l u e n c e d b y t h e p a p e r , a s w e r e m a n y n u m b e r t h e o r i s t s , a m o n g t h e m B e r k e l e y ' s K e n n e t h R i b e t a n d P r i n c e t o n ' s A n d r e w W i l e s . F r e y b e c a m e c o n v i n c e d b y M a z u r ' s p a p e r t h a t h e s h o u l d t h i n k v e r y s e r i o u s l y a b o u t a p p l i c a t i o n s o f m o d u l a r c u r v e s a n d G a l o i s r e p r e s e n t a t i o n s t o t h e t h e o r y o f e l l i p t i c c u r v e s . H e f o u n d t h a t t h i s l e d h i m
a l m o s t u n a v o i d a b l y t o D i o p h a n t i n e q u e s t i o n s c l o s e l y r e l a t e d t o e q u a t i o n s o f F e r m a t ' s t y p e . T h i s w a s a p o w e r f u l i n s i g h t , w h i c h F r e y t r i e d t o m a k e m o r e p r e c i s e .
I n 1981, G e r h a r d F r e y s p e n t a f e w w e e k s a t H a r v a r d U n i -v e r s i t y a n d h a d a n u m b e r o f d i s c u s s i o n s w i t h B a r r y M a z u r . T h e s e d i s c u s s i o n s w e r e c l e a r i n g t h i n g s i n h i s m i n d . T h e h e a v y f o g s u r r o u n d i n g t h e d i f f i c u l t c o n n e c t i o n s h e e n v i s i o n e d b e t w e e n F e r m a t - l i k e e q u a t i o n s a n d t h e r e l a t i o n b e t w e e n m o d u l a r f o r m s a n d e l l i p t i c c u r v e s w a s s l o w l y l i f t i n g . F r e y w e n t o n t o B e r k e l e y , w h e r e h e s p o k e w i t h K e n R i b e t , a b r i g h t n u m b e r t h e o r i s t w h o w a s a g r a d u a t e o f H a r v a r d a n d h a d w o r k e d w i t h M a z u r o n r e l a t e d i s s u e s . F r e y r e t u r n e d t o h i s n a t i v e G e r m a n y .
no
T h r e e y e a r s l a t e r , h e w a s i n v i t e d t o g i v e a l e c t u r e a t t h e O b e r - w o l f a c h c e n t e r d e e p i n t h e B l a c k F o r e s t .
O b e r w o l f a c h w a s d e s i g n e d a s a c o n f e r e n c e a n d w o r k s h o p c e n t e r i n m a t h e m a t i c s , s e t i n b e a u t i f u l a n d p e a c e f u l s u r r o u n d i n g s f a r f r o m c i t i e s a n d c r o w d s . E v e r y y e a r , t h e c e n t e r s p o n s o r s a b o u t f i f t y i n t e r n a t i o n a l m e e t i n g s o n d i f f e r e n t t o p i c s o f m a t h e -m a t i c s . L e c t u r e s , a n d e v e n j u s t a t t e n d a n c e a t t h e m e e t i n g s , a r e e x c l u s i v e l y b y i n v i t a t i o n . E v e r y e f f o r t i s m a d e t o a l l o w f o r t h e e a s y e x c h a n g e o f i d e a s a m o n g e x p e r t s f r o m d i f f e r e n t c o u n -t r i e s . I n 1984, G e r h a r d F r e y g a v e a t a l k a t a n u m b e r t h e o r y c o n f e r e n c e t h e r e . H e m a d e w h a t l o o k e d l i k e a c r a z y a s s e r t i o n . T h e m i m e o g r a p h e d s h e e t s f i l l e d w i t h m a t h e m a t i c a l f o r m u l a s h e p a s s e d a r o u n d a t t h e c o n f e r e n c e s e e m e d t o i m p l y t h a t i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w e r e i n d e e d t r u e , F e r m a t ' s L a s t T h e o r e m w o u l d b e p r o v e d . T h i s m a d e n o s e n s e a t a l l . W h e n K e n R i b e t f i r s t h e a r d o f F r e y ' s s t a t e m e n t , h e t h o u g h t i t w a s a j o k e . W h a t c o u l d m o d u l a r i t y o f e l l i p t i c c u r v e s p o s s i b l y h a v e t o d o w i t h F e r m a t ' s L a s t T h e o r e m ? h e a s k e d h i m s e l f . H e g a v e t h i s s t r a n g e a s s e r t i o n n o f u r t h e r t h o u g h t a n d w e n t a b o u t h i s u s u a l w o r k . B u t a n u m b e r o f p e o p l e i n P a r i s a n d
e l s e w h e r e w e r e i n t e r e s t e d i n F r e y ' s u n p r o v e n , a n d s o m e w h a t i n c o m p l e t e s t a t e m e n t . J e a n - P i e r r e S e r r e w r o t e a p r i v a t e l e t t e r t o a m a t h e m a t i c i a n b y t h e n a m e o f J . - F . M e s t r e . T h i s l e t t e r b e c a m e p u b l i c , a n d S e r r e s u b s e q u e n t l y p u b l i s h e d a p a p e r r e p e a t i n g h i s o w n c o n j e c t u r e s f r o m t h e l e t t e r t o M e s t r e . 2 1
Ribet's Theorem
K e n R i b e t , w h o f i r s t t h o u g h t t h i s s t a t e m e n t w a s a j o k e , s t a r t e d t h i n k i n g a b o u t S e r r e ' s c o n j e c t u r e s , a n d i n f a c t r e c o g n i z e d i n
t h e m s o m e t h i n g h e h a d a l r e a d y f o r m u l a t e d f o r h i m s e l f w h e n h e f o u n d t i m e t o t h i n k a b o u t F r e y ' s " j o k e . " T h e s e w e r e c e r t a i n c l a r i f i c a t i o n s o f G e r h a r d F r e y ' s s t a t e m e n t s , w h i c h , i f p r o v e n , w o u l d e s t a b l i s h t h e f o l l o w i n g i m p l i c a t i o n :
Shimura-Taniyama conjecture > Fermat's Last Theorem
T h e w a y t h e F r e y i d e a w o r k e d w a s i n g e n i o u s . F r e y r e a s o n e d a s f o l l o w s : S u p p o s e t h a t F e r m a t ' s L a s t T h e o r e m i s not t r u e . T h e n , f o r s o m e p o w e r n t h a t i s g r e a t e r t h a n 1 t h e r e is a s o l u t i o n t o F e r m a t ' s e q u a t i o n : x n + yn = z",
w h e r e x , y, a n d z a r e i n t e g e r s . T h i s p a r t i c u l a r s o l u t i o n , a , b , a n d c, w o u l d t h e n g i v e r i s e t o a s p e c i f i c e l l i p t i c c u r v e . N o w F r e y w r o t e ' d o w n t h e g e n e r a l e q u a t i o n o f t h i s c u r v e t h a t w o u l d r e s u l t f r o m t h e s o l u t i o n o f F e r m a t ' s e q u a t i o n . H i s c o n j e c t u r e p r e s e n t e d a t O b e r w o l f a c h s t a t e d t h a t t h i s v e r y c u r v e , n o w c a l l e d t h e F r e y c u r v e , w a s a v e r y s t r a n g e a n i m a l . I t w a s s o s t r a n g e , i n f a c t , t h a t i t c o u l d n ' t p o s s i b l y e x i s t . A n d , m o s t i m p o r t a n t , t h e e l l i p t i c c u r v e t h a t w o u l d a r i s e i f F e r m a t ' s L a s t T h e o r e m w e r e f a l s e w a s d e f i n i t e l y not m o d u l a r . S o , i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s i n d e e d t r u e ,
t h e n a l l e l l i p t i c c u r v e s m u s t b e m o d u l a r . T h e r e f o r e , a n e l l i p t i c c u r v e t h a t w a s n o t m o d u l a r c o u l d n ' t p o s s i b l y e x i s t . A n d i t w o u l d f o l l o w t h a t F r e y ' s c u r v e , a n e l l i p t i c c u r v e t h a t w a s n o t m o d u l a r ( i n a d d i t i o n t o a l l i t s o t h e r s t r a n g e c h a r a c t e r i s t i c s ) c o u l d n o t e x i s t . T h e r e f o r e , t h e s o l u t i o n s t o F e r m a t ' s e q u a t i o n c o u l d n o t e x i s t e i t h e r . W i t h o u t t h e e x i s t e n c e o f s o l u t i o n s t o t h e F e r m a t e q u a t i o n , F e r m a t ' s L a s t T h e o r e m ( w h i c h s t a t e s t h a t t h e r e a r e no s o l u t i o n s t o t h e e q u a t i o n f o r a n y n>2),
w o u l d b e p r o v e d . T h i s w a s a c o m p l i c a t e d s e q u e n c e o f i m p l i c a t i o n s , b u t i t f o l l o w e d b e a u t i f u l l y t h e l o g i c o f m a t h e m a t i c a l
Ken Ribet presenting his important
theorems.
Andrew Wiles being
interviewed.
Barry Mazur of Harvard
University— the "granddaddy" of
them all, a mathematician whose
geometrical insights inspired
everyone who contributed to the
final proof of Fermat's Last
Theorem.
Gerhard Frey, who had the
"crazy idea" that an elliptic
curve resulting from a
solution of Fermat's
equation simply could not
exist.
p r o o f . T h e l o g i c w a s : A i m p l i e s B ; t h e r e f o r e , i f B i s not t r u e , t h e n A c a n n o t b e t r u e e i t h e r . H o w e v e r , t h e F r e y s t a t e m e n t w a s , i t s e l f , a conjecture. I t w a s a c o n j e c t u r e w h i c h s a i d t h a t i f a n o t h e r c o n j e c t u r e ( S h i m u r a -T a n i y a m a ) w a s t r u e , t h e n F e r m a t ' s L a s t T h e o r e m w o u l d b e e s t a b l i s h e d . T h e p a i r o f s u b s e q u e n t c o n j e c t u r e s i n S e r r e ' s l e t t e r t o M e s t r e f u r t h e r a l l o w e d K e n R i b e t t o t h i n k a b o u t t h e F r e y c o n j e c t u r e i n c l e a r t e r m s .
K e n R i b e t h a d n e v e r b e f o r e b e e n i n t e r e s t e d i n F e r m a t ' s L a s t T h e o r e m . H e h a d s t a r t e d o u t a s a c h e m i s t r y m a j o r a t B r o w n U n i v e r s i t y . U n d e r t h e i n f l u e n c e a n d t u t e l a g e o f K e n n e t h F . I r e -l a n d , R i b e t w a s s t e e r e d t o m a t h e m a t i c s a n d g o t i n t e r e s t e d i n z e t a f u n c t i o n s , e x p o n e n t i a l s u m s , a n d n u m b e r t h e o r y . H e h a d d i s m i s s e d F e r m a t ' s L a s t T h e o r e m a s " o n e o f t h o s e p r o b l e m s a b o u t w h i c h n o t h i n g f u r t h e r o f r e a l i m p o r t a n c e c o u l d b e s a i d . " T h i s w a s a v i e w h e l d b y m a n y m a t h e m a t i c i a n s , b e c a u s e p r o b l e m s i n n u m b e r t h e o r y t e n d t o b e i s o l a t e d ,
w i t h n o u n i f y i n g s c h e m e o r u n d e r l y i n g g e n e r a l p r i n c i p l e b e h i n d t h e m . W h a t i s i n t e r e s t i n g a b o u t F e r m a t ' s L a s t T h e o r e m , h o w e v e r , i s t h a t i t s p a n s m a t h e m a t i c a l h i s t o r y f r o m t h e d a w n o f c i v i l i z a t i o n t o o u r o w n t i m e . A n d t h e t h e o r e m ' s u l t i m a t e s o l u t i o n a l s o s p a n s t h e b r e a d t h o f m a t h e m a t i c s , i n v o l v i n g f i e l d s o t h e r t h a n n u m b e r t h e o r y : a l g e b r a , a n a l y s i s , g e o m e t r y , a n d t o p o l o g y — v i r t u a l l y a l l o f m a t h e m a t i c s .
R i b e t w e n t o n t o p u r s u e a P h . D . i n m a t h e m a t i c s a t H a r v a r d U n i v e r s i t y . T h e r e , f i r s t i n d i r e c t l y a n d t o w a r d h i s g r a d u a t i o n m o r e d i r e c t l y , h e f e l l u n d e r t h e i n f l u e n c e o f t h e g r e a t n u m b e r t h e o r i s t a n d g e o m e t e r B a r r y M a z u r , w h o s e v i s i o n i n s p i r e d e v e r y m a t h e m a t i c i a n i n v o l v e d e v e n i n t h e s m a l l e s t w a y i n e f f o r t s t o p r o v e F e r m a t ' s L a s t T h e o r e m . M a z u r ' s p a p e r o n t h e
I 14
Eisenstein ideal acted as an abstraction of the theory
of ideals developed in the last century by Ernst
Kummer into the modern fields of mathematics,
algebraic geometry and new approaches to the
theory of numbers through geometry.22
Ken Ribet eventually became a professor of
mathematics at the University of California at
Berkeley and did research in number theory. In 1985,
he heard about Frey's "crazy" notion that if a solution
of Fermat's equation existed, that is, if Fermat's Last
Theorem were false, then this solution would give
rise to a very weird curve. This Frey Curve would be
associated with an elliptic curve that could not be
modular. The pair of associated conjectures in
Serre's letter to Mestre made him interested in trying
to prove Frey's conjecture. While he wasn't really
interested in Fermat's Last Theorem, Ribet
recognized that this had become a hot problem, and
it happened to be in an area he knew well. During the
week of August 18-24, 1985, Ribet was at a meeting
on arithmetic algebraic geometry in Areata,
California. He began thinking about Frey's statement,
and the problem remained on his mind for the next
year. When he was freed of his teaching obligations
at Berkeley early in the summer of 1986, Ribet flew
to Germany where he was to do research at the
world-famous center for mathematics, the Max
Planck Institute. Just as he arrived at the Institute,
Ribet made his great breakthrough. He was now
almost able to prove Frey's conjecture.
But he was not quite there. When he returned to
Berkeley, Ribet ran into Barry Mazur, who was
visiting from Harvard. "Barry, let's go for a cup of
coffee," Ribet suggested. The two retreated to a
popular cafe by the University of California campus.
While sipping a cappuccino, Ribet confided to
M a z u r : " I ' m t r y i n g t o g e n e r a l i z e w h a t I ' v e d o n e , s o t h a t I ' l l b e a b l e t o p r o v e t h e F r e y c o n j e c t u r e . I j u s t d o n ' t s e e m t o g e t t h i s o n e t h i n g t o g e n e r a l i z e i t . . . " M a z u r l o o k e d a t w h a t h e w a s s h o w i n g h i m . " B u t y o u ' v e d o n e i t a l r e a d y K e n , " h e s a i d , " a l l y o u n e e d t o d o i s a d d o n s o m e e x t r a g a m m a z e r o o f N s t r u c t u r e , a n d r u n t h r o u g h y o u r a r g u m e n t , a n d y o u ' r e t h e r e ! " R i b e t l o o k e d a t M a z u r , h e l o o k e d b a c k a t h i s c a p p u c c i n o , t h e n b a c k a t M a z u r w i t h d i s b e l i e f . " M y G o d , y o u ' r e a b s o l u t e l y r i g h t ! " h e s a i d . L a t e r h e w e n t b a c k t o h i s o f f i c e t o f i n i s h o f f t h e p r o o f . " K e n ' s i d e a w a s b r i l l i a n t , " M a z u r e x c l a i m e d w h e n d e s c r i b i n g K e n R i b e t ' s i n g e n i o u s p r o o f a f t e r i t w a s p u b l i s h e d a n d b e c a m e k n o w n i n t h e w o r l d o f m a t h e m a t i c i a n s .
R i b e t f o r m u l a t e d a n d p r o v e d a t h e o r e m w h i c h e s t a b l i s h e d a s f a c t t h a t i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s t r u e , t h e n F e r m a t ' s L a s t T h e o r e m w o u l d f a l l o u t o f i t a s a d i r e c t c o n s e q u e n c e . T h e m a n w h o o n l y a y e a r e a r l i e r t h o u g h t t h a t F r e y ' s s u g g e s t i o n w a s a j o k e n o w p r o v e d t h a t t h e " j o k e " w a s a c t u a l l y a m a t h e m a t i c a l r e a l i t y . T h e d o o r t o t h e a t t a c k o n F e r m a t ' s p r o b l e m u s i n g t h e m o d e r n m e t h o d s o f a r i t h m e t i c a l g e b r a i c g e o m e t r y w a s n o w w i d e o p e n . A l l t h e w o r l d n e e d e d
n o w w a s s o m e o n e w h o w o u l d p r o v e t h e s e e m i n g l y -i m p o s s i b l e S h i m u r a - T a n i y a m a c o n j e c t u r e . T h e n F e r m a t ' s L a s t T h e o r e m w o u l d a u t o m a t i c a l l y b e t r u e .
A Child's Dream
T h e p e r s o n w h o w a n t e d d o j u s t t h a t w a s A n d r e w W i l e s . W h e n h e w a s t e n y e a r s o l d , A n d r e w W i l e s w e n t t o t h e p u b l i c l i b r a r y i n h i s t o w n i n E n g l a n d a n d l o o k e d a t a b o o k o n m a t h e m a t i c s . I n t h a t b o o k h e r e a d a b o u t F e r m a t ' s L a s t T h e o r e m . T h e t h e o r e m , a s d e s c r i b e d i n t h e b o o k , s e e m e d s o s i m p l e , t h a t a n y
c h i l d c o u l d u n d e r s t a n d i t . I n W i l e s ' o w n w o r d s : " I t s a i d t h a t y o u w i l l n e v e r f i n d n u m b e r s , x, y , a n d z , s o t h a t x3 + y 3 = z 3 . N o m a t t e r h o w h a r d y o u t r i e d , y o u w i l l n e v e r , e v e r f i n d s u c h n u m b e r s . A n d i t s a i d t h a t t h e s a m e w a s t r u e f o r x4 + y 4 = z 4 , a n d f o r x5
+ y5 = z 5 , a n d s o o n . . . I t s e e m e d s o s i m p l e . A n d i t s a i d t h a t n o b o d y h a s e v e r f o u n d a p r o o f o f t h i s f o r o v e r t h r e e h u n d r e d y e a r s . I w a n t e d t o p r o v e i t . . . "
I n t h e 1970s, A n d r e w W i l e s w e n t t o t h e u n i v e r s i t y . W h e n h e f i n i s h e d h i s d e g r e e h e w a s a d m i t t e d a s a r e s e a r c h s t u d e n t i n m a t h e m a t i c s t o C a m b r i d g e . H i s a d v i s e r w a s P r o f e s s o r J o h n C o a t e s . W i l e s h a d t o d r o p h i s c h i l d h o o d d r e a m o f p r o v i n g F e r m a t ' s L a s t T h e o r e m . R e s e a r c h o n t h e p r o b l e m w o u l d h a v e t u r n e d i n t o s u c h a w a s t e o f t i m e t h a t n o g r a d u a t e s t u d e n t c o u l d a f f o r d i t . B e s i d e s , w h a t d o c t o r a l a d v i s e r w o u l d h a v e a c c e p t e d a s t u d e n t w o r k i n g o n s u c h a n a n c i e n t p u z z l e , o n e t h a t h a d k e p t t h e w o r l d ' s b r i g h t e s t m i n d s f r o m a s o l u t i o n f o r t h r e e c e n t u r i e s ? I n t h e 1970s,
F e r m a t w a s n o t i n f a s h i o n . W h a t w a s " i n " a t t h e t i m e , t h e r e a l h o t t o p i c f o r r e s e a r c h i n n u m b e r t h e o r y , w a s e l l i p t i c c u r v e s . S o A n d r e w W i l e s s p e n t h i s t i m e d o i n g r e s e a r c h o n e l l i p t i c c u r v e s a n d i n a n a r e a c a l l e d I w a s a w a t h e o r y . H e c o m p l e t e d h i s d o c t o r a l d i s s e r t a t i o n , a n d w h e n h e w a s a w a r d e d h i s P h . D . , h e
g o t a p o s i t i o n i n m a t h e m a t i c s a t P r i n c e t o n U n i v e r s i t y a n d m o v e d t o t h e U n i t e d S t a t e s . T h e r e , h e c o n t i n u e d d o i n g r e s e a r c h o n e l l i p t i c c u r v e s a n d I w a s a w a t h e o r y .
An Old Flame Rekindled
I t w a s a w a r m s u m m e r e v e n i n g , a n d A n d r e w W i l e s w a s s i p p i n g i c e d t e a a t a f r i e n d ' s h o u s e . S u d d e n l y , i n t h e m i d d l e o f t h e c o n -v e r s a t i o n , h i s f r i e n d s a i d : " B y t h e w a y , d i d y o u h e a r t h a t K e n
R i b e t j u s t p r o v e d t h e E p s i l o n C o n j e c t u r e ? " T h e E p s i l o n C o n j e c t u r e i s w h a t n u m b e r t h e o r i s t s w e r e i n f o r m a l l y c a l l i n g F r e y ' s c o n j e c t u r e , a s m o d i f i e d b y S e r r e , a b o u t t h e c o n n e c t i o n b e t w e e n F e r m a t ' s L a s t T h e o r e m a n d t h e S h i m u r a - T a n i y a m a C o n j e c t u r e . W i l e s w a s e l e c t r i f i e d . A t t h a t m o m e n t , h e k n e w t h a t h i s l i f e w a s c h a n g i n g . T h e c h i l d h o o d d r e a m h e h a d o f p r o v i n g F e r m a t ' s L a s t T h e o r e m — a d r e a m h e h a d h a d t o g i v e u p t o u n d e r t a k e m o r e f e a s i b l e r e s e a r c h — c a m e a l i v e a g a i n w i t h i n c r e d i b l e f o r c e . H e w e n t h o m e a n d s t a r t e d t h i n k i n g a b o u t h o w h e w o u l d p r o v e t h e S h i m u r a - T a n i y a m a c o n j e c t u r e .
" F o r t h e f i r s t f e w y e a r s , " h e l a t e r c o n f i d e d , "I k n e w I
h a d n o c o m p e t i t i o n , s i n c e I
k n e w t h a t n o b o d y — m e i n c l u d e d — h a d a n y i d e a w h e r e t o s t a r t . " H e d e c i d e d t o w o r k i n c o m p l e t e s e c r e c y , a n d i n i s o l a t i o n . " T o o m a n y s p e c t a t o r s w o u l d s p o i l t h e c o n c e n t r a t i o n , a n d I
d i s c o v e r e d e a r l y o n t h a t j u s t a m e n t i o n o f F e r m a t i m m e d i a t e l y g e n e r a t e s t o o m u c h i n t e r e s t . " O f c o u r s e , g i f t e d , a b l e m a t h e m a t i c i a n s a b o u n d , e s p e c i a l l y a t a p l a c e l i k e P r i n c e t o n , a n d t h e d a n g e r o f s o m e o n e c o m p l e t i n g y o u r w o r k f o r y o u— a n d e v e n d o i n g i t b e t t e r — i s v e r y r e a l .
W h a t e v e r t h e r e a s o n , W i l e s l o c k e d h i m s e l f u p i n h i s a t t i c o f f i c e a n d w e n t t o w o r k . H e a b a n d o n e d a l l o t h e r
r e s e a r c h p r o j e c t s t o d e v o t e h i s t i m e c o m p l e t e l y t o F e r m a t . W i l e s w o u l d u s e a l l t h e p o w e r o f t h e m o d e r n m a c h i n e r y o f a l g e b r a , g e o m e t r y , a n a l y s i s , a n d t h e o t h e r a r e a s o f m a t h e m a t i c s . H e w o u l d a l s o m a k e u s e o f t h e i m p o r t a n t m a t h e m a t i c a l r e s u l t s o f h i s c o n t e m p o -r a r i e s , a n d o f h i s h i s t o r i c a l p r e d e c e s s o r s . H e w o u l d m a k e u s e o f R i b e t ' s c l e v e r m e t h o d s o f p r o o f , a n d h i s r e s u l t s , - h e w o u l d u s e t h e t h e o r i e s o f B a r r y M a z u r , a n d t h e i d e a s o f S h i m u r a , F r e y , S e r r e , A n d r e W e i l , a n d t h o s e o f m a n y , m a n y o t h e r m a t h e m a t i c i a n s .
W i l e s ' g r e a t n e s s , G e r h a r d F r e y w o u l d l a t e r s a y , w a s t h a t h e b e l i e v e d i n w h a t h e w a s d o i n g a t a t i m e w h e n v i r t u a l l y e v e r y m a t h e m a t i c i a n i n t h e w o r l d b e l i e v e d t h a t t h e S h i m u r a -T a n i y a m a c o n j e c t u r e c o u l d n o t b e p r o v e n i n t h e t w e n t i e t h c e n t u r y .
To p r o v e t h e S h i m u r a -T a n i y a m a C o n j e c t u r e , A n d r e w W i l e s k n e w h e h a d t o p r o v e t h a t e v e r y e l l i p t i c c u r v e i s m o d u l a r . H e h a d t o s h o w t h a t e v e r y e l l i p t i c c u r v e , w h o s e s o l u t i o n s l i e o n a d o u g h n u t , w a s r e a l l y a m o d u l a r f o r m i n d i s g u i s e . T h e d o u g h n u t w a s , i n a s e n s e , a l s o t h i s s p a c e o f i n t r i c a t e l y s y m m e t r i c f u n c t i o n s o n t h e c o m p l e x p l a n e c a l l e d m o d u l a r f o r m s . N o b o d y h a d a n y i d e a h o w t o s h o w s u c h a w e i r d c o n n e c t i o n b e t w e e n t h e s e t w o s e e m i n g l y v e r y d i f f e r e n t e n t i t i e s .
W i l e s r e a l i z e d t h a t t h e b e s t i d e a w a s t o t r y t o count
t h e n u m b e r o f e l l i p t i c c u r v e s , a n d t o c o u n t t h e n u m b e r o f modular e l l i p t i c c u r v e s , a n d t h e n t o s h o w t h a t t h e i r " n u m b e r " w a s t h e s a m e . T h i s c o n s t r u c t i o n w o u l d p r o v e t h a t t h e e l l i p t i c c u r v e s a n d t h e m o d u l a r e l l i p t i c c u r v e s w e r e o n e a n d t h e s a m e , a n d h e n c e e v e r y e l l i p t i c c u r v e i s i n d e e d m o d u l a r , a s t h e S h i m u r a -T a n i y a m a c o n j e c t u r e c l a i m s .
W i l e s r e a l i z e d t w o t h i n g s . O n e w a s t h a t h e d i d n ' t h a v e t o p r o v e t h e e n t i r e S h i m u r a -T a n i y a m a c o n j e c t u r e , b u t
o n l y a s p e c i a l c a s e : semistable
e l l i p t i c c u r v e s w i t h r a t i o n a l n u m b e r s a s t h e c o e f f i c i e n t s . P r o v i n g t h a t t h e c o n j e c t u r e w a s t r u e f o r t h i s s m a l l e r c l a s s o f e l l i p t i c c u r v e s w o u l d b e e n o u g h t o e s t a b l i s h F e r m a t ' s L a s t T h e o r e m . T h e o t h e r t h i n g W i l e s k n e w w a s t h a t " c o u n t i n g " w o u l d n o t w o r k h e r e b e c a u s e h e w a s d e a l i n g w i t h infinite s e t s . T h e s e t o f s e m i s t a b l e e l l i p t i c c u r v e s w a s i n f i n i t e . A n y r a t i o n a l n u m b e r alb, w h e r e a a n d b a r e i n t e g e r s , w o u l d
g i v e y o u a n o t h e r e l l i p t i c c u r v e ( w e s a y a n e l l i p t i c c u r v e over t h e r a t i o n a l s ) . S i n c e t h e r e a r e i n f i n i t e l y m a n y s u c h n u m b e r s — a a n d b c a n b e a n y o f t h e i n f i n i t e l y m a n y n u m b e r s 1,2,3,4..., t o i n f i n i t y , t h e r e a r e i n f i n i t e l y m a n y e l l i p t i c c u r v e s . S o c o u n t i n g a s w e k n o w i t w o u l d n ' t w o r k .
Breaking Down a Big Problem into Smaller Ones W i l e s t h o u g h t h e m i g h t t r y t o w o r k o n s m a l l e r p r o b l e m s , o n e a t a t i m e . M a y b e h e c o u l d l o o k a t s e t s o f e l l i p t i c f u n c t i o n s a n d s e e w h a t h e c o u l d d o a b o u t t h e m . T h i s w a s a g o o d a p p r o a c h s i n c e i t b r o k e d o w n t h e t a s k s o t h a t , s t e p b y s t e p , h e c o u l d u n d e r s t a n d e a c h s e t . F i r s t o f a l l , s o m e e l l i p t i c c u r v e s w e r e a l r e a d y k n o w n t o b e m o d u l a r . T h e s e w e r e v e r y i m p o r t a n t r e s u l t s , d e v e l o p e d b y m a n y o t h e r n u m b e r t h e o r i s t s . B u t s o o n A n d r e w W i l e s r e a l i z e d t h a t l o o k i n g o n l y a t e l l i p t i c c u r v e s a n d t r y i n g t o c o u n t t h e m o f f a g a i n s t m o d u l a r f o r m s m i g h t n o t b e a g o o d a p p r o a c h — h e w a s d e a l i n g w i t h t w o i n f i n i t e s e t s . I n f a c t , h e w a s n o c l o s e r t o a s o l u t i o n t h a n w a s t h e s k e p t i c a l A n d r e W e i l w h e n h e s a i d : " I s e e n o r e a s o n a g a i n s t t h e c o n j e c t u r e s i n c e o n e a n d t h e o t h e r o f t h e t w o s e t s a r e d e n u m e r a b l e [ i n f i -n i t e b u t o f t h e o r d e r o f i n f i n i t y o f t h e i n t e g e r s a n d r a t i o n a l n u m b e r s , n o t t h e h i g h e r o r d e r o f i n f i n i t y o f t h e i r r a t i o n a l n u m b e r s a n d t h e c o n t i n u u m ] , b u t I d o n ' t s e e a n y r e a s o n f o r i t ,
e i t h e r . . . " A f t e r t w o y e a r s o f g e t t i n g n o w h e r e , W i l e s t r i e d a n e w a p p r o a c h . H e t h o u g h t h e m i g h t transform t h e e l l i p t i c c u r v e s i n t o G a l o i s r e p r e s e n t a t i o n s , a n d t h e n c o u n t t h e s e G a l o i s r e p -r e s e n t a t i o n s a g a i n s t t h e m o d u l a r f o r m s .
T h e i d e a w a s a n e x c e l l e n t o n e , a l t h o u g h i t w a s n o t o r i g i n a l . T h e p r i n c i p l e b e h i n d t h i s m o v e i s i n t e r e s t i n g . N u m b e r t h e o r i s t s
a r e c o n c e r n e d w i t h f i n d i n g s o l u t i o n s o f e q u a t i o n s , s u c h a s t h e F e r m a t e q u a t i o n . T h e m a t h e m a t i c a l t h e o r y o f f i e l d s o f n u m b e r s s e t s t h i s p r o b l e m i n t h e c o n t e x t o f f i e l d e x t e n s i o n s . F i e l d s a r e l a r g e , i n f i n i t e c o l l e c t i o n s w h i c h a r e d i f f i c u l t t o a n a l y z e . T h e r e f o r e , w h a t n u m b e r t h e o r i s t s h a v e o f t e n d o n e i s t o u s e t h e t h e o r i e s o f E v a r i s t e G a l o i s , c a l l e d G a l o i s t h e o r y , i n o r d e r t o t r a n s l a t e t h e s e p r o b l e m s f r o m t h e c o m p l i c a t e d f i e l d s t o w h a t a r e k n o w n a s g r o u p s . O f t e n a g r o u p i s g e n e r a t e d b y a f i n i t e ( r a t h e r t h a n a n i n f i n i t e ) s e t o f e l e m e n t s . U s i n g G a l o i s t h e o r y t h u s a l l o w s t h e n u m b e r t h e o r i s t t o m o v e f r o m a n i n f i n i t e c o l l e c t i o n t o o n e t h a t i s r e p r e s e n t e d b y a f i n i t e s e t . T h i s t r a n s l a t i o n o f a p r o b l e m c o n s t i t u t e s a n i m m e n s e s t e p f o r w a r d , s i n c e a f i n i t e s e t o f e l e m e n t s i s s o m u c h e a s i e r t o h a n d l e t h a n a n i n f i n i t e s e t . C o u n t i n g m a k e s s e n s e o n l y f o r a f i n i t e n u m b e r o f e l e m e n t s . T h e a p p r o a c h s e e m e d t o w o r k f o r s o m e s e t s o f e l l i p t i c c u r v e s . T h i s w a s a g o o d b r e a k t h r o u g h . B u t a f t e r a n o t h e r y e a r , W i l e s w a s s t u c k a g a i n .
The Flack Paper
W h a t A n d r e w W i l e s w a s t r y i n g t o d o n o w w a s t o c o u n t s e t s o f G a l o i s r e p r e s e n t a t i o n s , c o r r e s p o n d i n g t o t h e ( s e m i -s t a b l e ) e l l i p t i c c u r v e s a g a i n s t t h e m o d u l a r f o r m s ,
a n d t o s h o w t h a t t h e y a r e t h e s a m e . I n d o i n g s o , h e w a s u s i n g h i s a r e a o f e x p e r -t i s e , i n w h i c h h e h a d d o n e h i s d i s s e r t a t i o n , c a l l e d H o r i z o n t a l I w a s a w a T h e o r y . W i l e s w a s t r y i n g t o u s e t h i s t h e o r y t o a t t a i n t h e C l a s s N u m b e r F o r m u l a , a r e s u l t w h i c h h e n e e d e d f o r t h e " c o u n t i n g . " B u t h e r e h e w a s u p a g a i n s t a b r i c k w a l l . N o t h i n g h e c o u l d d o b r o u g h t h i m n e a r e r t o t h e a n s w e r .
I n t h e s u m m e r o f 1991, W i l e s w a s a t a c o n f e r e n c e i n B o s t o n , w h e r e h e m e t h i s f o r m e r d o c t o r a l a d v i s e r a t C a m b r i d g e , J o h n
C o a t e s . P r o f e s s o r C o a t e s t o l d W i l e s t h a t o n e o f C o a t e s ' s t u d e n t s , M a t t h i a s F l a c h , u s i n g e a r l i e r w o r k b y a R u s s i a n m a t h e m a t i c i a n n a m e d K o l y v a g i n , h a d d e v i s e d a n E u l e r S y s t e m ( n a m e d a f t e r L e o n h a r d E u l e r ) i n a n a t t e m p t t o p r o v e t h e C l a s s N u m b e r F o r -m u l a . T h i s w a s e x a c t l y w h a t W i l e s n e e d e d f o r h i s p r o o f o f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e — i f , i n d e e d , h e c o u l d e x t e n d F l a c h ' s p a r t i a l r e s u l t s t o t h e f u l l C l a s s N u m b e r F o r m u l a . W i l e s w a s e l a t e d o n h e a r i n g f r o m C o a t e s a b o u t t h i s w o r k o f F l a c h . " T h i s w a s t a i l o r -m a d e " f o r h i s o w n p r o b l e m , W i l e s s a i d , a s i f M a t t h i a s F l a c h h a d d o n e a l l t h i s w o r k j u s t f o r h i m . A n d W i l e s i m m e d i a t e l y a b a n d o n e d a l l h i s H o r i z o n t a l I w a s a w a T h e o r y w o r k a n d i m m e r s e d h i m s e l f d a y a n d n i g h t i n t h e w o r k o f K o l y v a g i n a n d F l a c h . I f t h e i r " E u l e r S y s t e m " r e a l l y w o r k e d , W i l e s w o u l d h o p e f u l l y a c h i e v e t h e C l a s s N u m b e r r e s u l t a n d t h e S h i m u r a - T a n i y a m a C o n j e c t u r e w o u l d b e p r o v e d f o r s e m i s t a b l e e l l i p t i c c u r v e s —e n o u g h t o p r o v e F e r m a t ' s L a s t T h e o r e m .
T h i s w a s h a r d w o r k , h o w e v e r , a n d i t w a s o u t s i d e t h e I w a s a w a r e a l m w h i c h W i l e s k n e w s o w e l l . I n c r e a s i n g l y , W i l e s s t a r t e d t o f e e l t h e n e e d t o f i n d s o m e b o d y t o t a l k t o . H e w a n t e d s o m e o n e w h o c o u l d c h e c k h i s p r o g r e s s i n t h e s e u n c h a r t e d w a t e r s , b u t
s o m e o n e w h o w o u l d n o t r e v e a l a t h i n g t o a n y o n e e l s e .
A Good Friend
W i l e s f i n a l l y h a d t o m a k e a d e c i s i o n : s h o u l d h e c o n t i n u e t o k e e p e v e r y t h i n g s e c r e t a s h e h a d d o n e f o r s o l o n g , o r s h o u l d h e b r e a k d o w n a n d t a l k t o s o m e o n e w i t h g o o d k n o w l e d g e o f n u m b e r t h e o r y ? H e f i n a l l y d e c i d e d h e c o u l d p r o b a b l y n o t d o v e r y w e l l b y k e e p i n g s e c r e c y f o r e v e r . A s h e h i m s e l f s a i d , o n e
c o u l d w o r k o n a p r o b l e m f o r a n e n t i r e l i f e t i m e a n d n o t s e e a n y r e s u l t s . T h e n e e d t o c o m p a r e n o t e s w i t h a n o t h e r p e r s o n f i n a l l y o u t w e i g h e d t h a t i n t e n s e n e e d t o k e e p i t a l l t o h i m s e l f . B u t n o w t h e q u e s t i o n w a s : w h o ? W h o m c o u l d h e t r u s t t o k e e p h i s s e c r e t ?
I n J a n u a r y o f 1993, a f t e r s i x y e a r s o f w o r k i n g a l o n e , W i l e s m a d e h i s c o n t a c t . H e c a l l e d i n P r o f e s s o r N i c k K a t z , o n e o f h i s c o l l e a g u e s a t t h e P r i n c e t o n m a t h e m a t i c s d e p a r t m e n t . K a t z w a s a n e x p e r t o n m a n y o f t h e t h e o r i e s t h a t w e n t i n t o a t t e m p t s t o p r o v e t h e C l a s s N u m b e r F o r m u l a . B u t m o r e i m p o r t a n t l y , K a t z w a s c o m p l e t e l y t r u s t w o r t h y . H e w o u l d n e v e r r e v e a l w h a t A n d r e w W i l e s w a s u p t o . T h i s a s s e s s m e n t o n W i l e s ' p a r t t u r n e d o u t t o b e c o r r e c t . N i c k K a t z k e p t h i s m o u t h s h u t t h r o u g h o u t h i s w o r k w i t h W i l e s o v e r t h e m a n y r e m a i n i n g m o n t h s o f t h e p r o j e c t . T h e i r m u t u a l c o l l e a g u e s a t t h e t i g h t l y -k n i t m a t h e m a t i c a l c o m m u n i t y a t P r i n c e t o n n e v e r s u s p e c t e d a t h i n g , e v e n a f t e r w e e k s o f s e e i n g t h e t w o s p e n d h o u r s h u d d l e d t o g e t h e r o v e r c o f f e e a t t h e f a r e n d o f t h e C o m m o n s R o o m .
B u t A n d r e w W i l e s s t i l l w o r r i e d t h a t s o m e o n e m i g h t s u s p e c t w h a t h e w a s w o r k i n g o n . H e c o u l d n ' t t a k e a c h a n c e . S o h e c a m e u p w i t h a s c h e m e t o h i d e t h e f a c t t h a t h e w a s w o r k i n g v e r y i n t e n s e l y o n " s o m e t h i n g " w i t h N i c k K a t z . W i l e s w o u l d
o f f e r a n e w g r a d u a t e c o u r s e i n m a t h e m a t i c s f o r t h e s p r i n g o f 1993, a c o u r s e w h i c h N i c k K a t z w o u l d a t t e n d a s o n e o f t h e s t u d e n t s , a n d t h i s w o u l d a l l o w t h e t w o o f t h e m t o w o r k t o g e t h e r w i t h o u t o t h e r s s u s p e c t i n g w h a t t h e y w e r e d o i n g . O r a t l e a s t t h i s i s w h a t W i l e s h a s s a i d . T h e g r a d u a t e s t u d e n t s c o u l d n o t s u s p e c t t h a t b e h i n d t h e s e l e c t u r e s w a s a r o a d t o F e r m a t ' s L a s t T h e o r e m , a n d W i l e s w o u l d b e a b l e t o p i c k t h e i r b r a i n s f o r a n y p o s s i b l e h o l e s i n h i s t h e o r y , w i t h t h e h e l p o f h i s g o o d f r i e n d K a t z .
T h e c o u r s e w a s a n n o u n c e d . I t w a s c a l l e d " C a l c u l a t i o n s w i t h E l l i p t i c C u r v e s , " w h i c h w a s i n n o c e n t e n o u g h s o n o o n e c o u l d s u s p e c t a n y t h i n g . A n d a t t h e s t a r t o f t h e c o u r s e , P r o f e s s o r W i l e s s a i d t h a t t h e p u r p o s e o f t h e l e c t u r e s w a s t o s t u d y s o m e r e c e n t w o r k o f M a t t h i a s F l a c h o n t h e C l a s s N u m b e r F o r m u l a . T h e r e w a s n o m e n t i o n o f F e r m a t , n o m e n t i o n o f S h i m u r a o r T a n i y a m a , a n d n o o n e c o u l d s u s p e c t t h a t t h e C l a s s N u m b e r F o r m u l a t h e y w e r e g o i n g t o s t u d y w o u l d b e t h e k e y s t o n e t o p r o v i n g F e r m a t ' s L a s t T h e o r e m . A n d n o o n e h a d a n y i d e a t h a t t h e t r u e p u r p o s e o f t h e l e c t u r e s w a s n o t t o t e a c h g r a d u a t e s t u -d e n t s m a t h e m a t i c s b u t t o a l l o w W i l e s a n d K a t z t o w o r k t o g e t h e r o n t h i s p r o b l e m w i t h o u t s u s p i c i o n b y a n y o f t h e i r c o l l e a g u e s , w h i l e a t t h e s a m e t i m e g e t t i n g u n s u s p e c t i n g g r a d u a t e s t u d e n t s t o c h e c k t h e w o r k f o r t h e m .
B u t w i t h i n a f e w w e e k s a l l t h e g r a d u a t e s t u d e n t s d r i f t e d a w a y . T h e y c o u l d n ' t k e e p u p w i t h a c o u r s e t h a t w a s n ' t r e a l l y g o i n g a n y w h e r e . T h e o n l y " s t u d e n t " w h o s e e m e d t o k n o w a n y t h i n g a n d t o p a r t i c i p a t e i n c l a s s w a s t h e o t h e r m a t h p r o f e s s o r w h o w a s s i t t i n g t h e r e w i t h t h e m . S o a f t e r a w h i l e , N i c k K a t z w a s t h e o n l y o n e i n t h e a u d i e n c e . B u t W i l e s j u s t w e n t o n u s i n g t h e " c l a s s " t o w r i t e h i s l o n g p r o o f o f t h e
C l a s s N u m b e r T h e o r e m o n t h e b o a r d , c o n t i n u i n g t o t h e n e x t s t e p e v e r y c l a s s m e e t i n g , w i t h N i c k K a t z v e r i f y i n g e a c h s t e p .
T h e l e c t u r e s r e v e a l e d n o e r r o r s . I t s e e m e d t h a t t h e C l a s s N u m b e r F o r m u l a w a s w o r k i n g , a n d W i l e s w a s o n h i s w a y t o t h e s o l u t i o n o f t h e F e r m a t p r o b l e m . A n d s o i n t h e l a t e s p r i n g o f 1993, a s t h e c o u r s e w a s c o m i n g t o a n e n d , A n d r e w W i l e s w a s a l m o s t f i n i s h e d . S t i l l , h e w a s w r e s t l i n g w i t h j u s t o n e f i n a l o b s t a c l e . H e w a s a b l e t o p r o v e t h a t m o s t o f t h e e l l i p t i c c u r v e s
w e r e m o d u l a r , b u t a f e w o f t h e m r e m a i n e d u n p r o v a b l e . H e t h o u g h t h e c o u l d o v e r c o m e t h e s e d i f f i c u l t i e s , a n d h e w a s g e n e r a l l y o p t i m i s t i c . W i l e s f e l t i t w a s t i m e t o t a l k t o o n e m o r e p e r s o n , t o t r y t o g a i n a l i t t l e m o r e i n s i g h t i n t o t h i s l a s t d i f f i c u l t y f a c i n g h i m . S o h e c a l l e d i n a n o t h e r o n e o f h i s c o l l e a g u e s a t t h e P r i n c e t o n m a t h e m a t i c s d e p a r t m e n t , P r o f e s s o r P e t e r S a r n a k , a n d s w o r e h i m t o s e c r e c y a s w e l l . " 1 t h i n k I a m a b o u t t o p r o v e F e r m a t ' s L a s t T h e o r e m , " h e t o l d t h e s t u n n e d S a r n a k .
" T h i s w a s i n c r e d i b l e , " S a r n a k l a t e r r e c a l l e d . " 1 w a s f l a b b e r g a s t e d , e l a t e d , d i s t u r b e d — I m e a n . . . I r e m e m b e r f i n d i n g i t d i f f i c u l t t o s l e e p t h a t n i g h t . " S o n o w t h e r e w e r e t w o c o l l e a g u e s t r y i n g t o h e l p W i l e s f i n i s h h i s p r o o f . W h i l e n o b o d y s u s p e c t e d w h a t i t w a s t h a t t h e y w e r e d o i n g , p e o p l e w e r e n o t i c i n g s o m e t h i n g . A n d w h i l e h e m a i n t a i n e d t h a t n o o n e e v e r f o u n d a n y -t h i n g t h r o u g h h i m , S a r n a k l a t e r a d m i t t e d t h a t h e d r o p p e d " m a y b e a f e w h i n t s . . . "
The Last Piece oj the Puzzle
I n M a y , 1993, A n d r e w W i l e s w a s s i t t i n g a l o n e a t h i s d e s k . H e w a s g e t t i n g s o m e w h a t f r u s t r a t e d . I t s e e m e d t h a t t h o s e f e w e l l i p t i c c u r v e s t h a t g o t a w a y f r o m h i m w e r e n o t c o m i n g a n y n e a r e r . H e
s i m p l y c o u l d n ' t p r o v e t h a t t h e y w e r e m o d u l a r . A n d h e n e e d e d t h e m , t o o , i f h e w e r e t o p r o v e t h a t all ( s e m i s t a b l e ) e l l i p t i c c u r v e s w e r e m o d u l a r s o t h a t F e r m a t ' s L a s t T h e o r e m w o u l d f o l l o w . D o i n g i t f o r m o s t o f t h e s e m i s t a b l e e l l i p t i c c u r v e s w a s a g r e a t m a t h e m a t i c a l r e s u l t o n i t s o w n r i g h t , b u t n o t e n o u g h t o r e a c h h i s g o a l . To r e s t a l i t t l e h o r n h i s i n t e n s e s e a r c h l e a d i n g n o w h e r e , W i l e s p i c k e d u p a n o l d p a p e r o f t h e g r e a t m a s t e r , B a r r y M a z u r o f H a r v a r d U n i v e r s i t y . M a z u r h a d m a d e s o m e
g r o u n d b r e a k i n g d i s c o v e r i e s i n n u m b e r t h e o r y — r e s u l t s t h a t h a d i n s p i r e d m a n y o f t h e e x p e r t s i n t h e f i e l d , i n c l u d i n g R i b e t a n d F r e y , w h o s e w o r k p a v e d t h e w a y f o r W i l e s ' e f f o r t . M a z u r ' s p a p e r t h a t W i l e s w a s n o w r e r e a d i n g w a s a n e x t e n s i o n o f t h e t h e o r y o f i d e a l s , s t a r t i n g w i t h K u m m e r a n d D e d e k i n d , a n d c o n t i n u i n g w i t h y e t a t h i r d n i n e t e e n t h c e n t u r y m a t h e m a t i c i a n —F e r d i n a n d G o t t h o l d E i s e n s t e i n (1823-1852). A l t h o u g h h e d i e d y o u n g , E i s e n s t e i n m a d e g r e a t s t r i d e s i n n u m b e r t h e o r y . G a u s s , i n f a c t , i s q u o t e d a s h a v i n g s a i d : ' T h e r e h a v e b e e n o n l y t h r e e e p o c h - m a k i n g m a t h e m a t i c i a n s , A r c h i m e d e s , N e w t o n , a n d E i s e n s t e i n . "
M a z u r ' s p a p e r o n t h e E i s e n s t e i n I d e a l h a d o n e l i n e i n i t t h a t n o w c a u g h t W i l e s ' a t t e n t i o n . M a z u r w a s s a y i n g t h a t i t w a s p o s s i b l e t o switch
f r o m o n e s e t o f e l l i p t i c c u r v e s t o a n o t h e r . T h e s w i t c h h a d t o d o w i t h p r i m e n u m b e r s . W h a t M a z u r w a s s a y i n g w a s t h a t i f o n e w a s d e a l i n g w i t h e l l i p t i c c u r v e s t h a t w e r e b a s e d o n t h e p r i m e n u m b e r t h r e e , i t w a s p o s s i b l e t o t r a n s f o r m t h e m s o t h a t o n e c o u l d s t u d y t h e m u s i n g t h e p r i m e n u m b e r f i v e i n s t e a d . T h i s 3- t o -5 s w i t c h w a s e x a c t l y w h a t W i l e s w o u l d n e e d . H e w a s s t u c k w i t h n o t b e i n g a b l e t o p r o v e t h a t c e r t a i n c l a s s e s o f e l l i p t i c c u r v e s b a s e d o n t h e
p r i m e n u m b e r t h r e e w e r e m o d u l a r . A n d h e r e M a z u r w a s s a y i n g t h a t h e c o u l d s w i t c h t h e m t o c u r v e s b a s e d o n t h e p r i m e n u m b e r f i v e . B u t t h e s e c u r v e s b a s e d o n f i v e W i l e s h a d a l r e a d y p r o v e d w e r e m o d u l a r . S o t h e 3- t o -5 s w i t c h w a s t h e f i n a l t r i c k . I t t o o k t h e d i f f i c u l t e l l i p t i c c u r v e s b a s e d o n t h r e e , t r a n s f o r m e d t h e m i n t o b a s e f i v e a n d t h e s e w e r e k n o w n t o b e m o d u l a r . O n c e a g a i n , s o m e o t h e r m a t h e m a t i c i a n ' s b r i l l i a n t i d e a h e l p e d W i l e s o v e r c o m e a s e e m i n g l y i n s u r m o u n t a b l e h u r d l e . A n d r e w W i l e s w a s f i n a l l y d o n e .
H i s t i m i n g w a s p e r f e c t , t o o . I n t h e n e x t m o n t h , J u n e , h i s
f o r m e r a d v i s e r J o h n C o a t e s w o u l d b e h o s t i n g a n u m b e r t h e o r y c o n f e r e n c e i n C a m b r i d g e . A l l t h e b i g n a m e s i n n u m b e r t h e o r y w o u l d b e t h e r e . A n d C a m b r i d g e w a s W i l e s ' o l d h o m e t o w n a n d w h e r e h e w e n t t o g r a d u a t e s c h o o l . W o u l d n ' t i t b e p e r f e c t t o p r e s e n t h i s p r o o f o f F e r m a t ' s L a s t T h e o r e m t h e r e ? W i l e s w a s n o w r a c i n g a g a i n s t t h e c l o c k . H e h a d t o p u t t o g e t h e r h i s e n t i r e p r o o f t h a t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s t r u e f o r s e m i s t a b l e e l l i p t i c c u r v e s . T h i s m e a n t t h a t t h e F r e y c u r v e c o u l d n o t e x i s t . A n d i f t h e F r e y c u r v e c o u l d n o t " e x i s t , t h a t m e a n t t h a t s o l u t i o n s t o F e r m a t ' s e q u a t i o n c o u l d n o t e x i s t f o r n>2 a n d t h e r e f o r e F e r m a t ' s L a s t T h e o r e m w a s p r o v e n . T h e w r i t e - u p t o o k A n d r e w W i l e s 200 p a g e s . H e w a s d o n e j u s t i n t i m e t o c a t c h h i s p l a n e f o r E n g l a n d . A n d a t t h e e n d o f h i s l a s t l e c t u r e t h e r e , h e e m e r g e d v i c t o r i o u s t o t h e s w e e p i n g a p p l a u s e , t h e f l a s h i n g c a m e r a s , a n d t h e r e p o r t e r s .
The Aftermath
N o w i t w a s t i m e f o r t h e p e e r - r e v i e w . U s u a l l y , a m a t h e m a t i c a l r e s u l t — o r a n y a c a d e m i c f i n d i n g , f o r t h a t m a t t e r — i s s u b m i t t e d t o a " r e f e r e e d j o u r n a l . " S u c h r e f e r e e d j o u r n a l s a r e t h e s t a n d a r d v e h i c l e f o r s c h o l a r s t o s u b m i t t h e i r w o r k f o r p o s s i b l e p u b l i c a t i o n . T h e
j o u r n a l ' s c h a r g e i s t h e n t o s e n d t h e p r o p o s e d p a p e r t o o t h e r e x p e r t s i n t h e a p p r o p r i a t e a r e a o f s t u d y w h o r e v i e w t h e p a p e r ' s c o n t e n t a n d d e t e r m i n e w h e t h e r i t i s c o r r e c t , a n d w h e t h e r t h e p a p e r m a k e s a c o n t r i b u t i o n w o r t h y o f p u b l i c a t i o n . R e f e r e e d j o u r n a l p u b l i c a t i o n s a r e t h e b r e a d a n d b u t t e r o f a c a d e m i a . T e n u r e a n d p r o m o t i o n , a n d o f t e n s a l a r y l e v e l s a n d r a i s e s , a r e a l l d e p e n d e n t o n a r e s e a r c h e r ' s o u t p u t o f r e f e r e e d j o u r n a l a r t i c l e s .
B u t A n d r e w W i l e s c h o s e a d i f f e r e n t a p p r o a c h . I n s t e a d o f s u b m i t t i n g h i s p r o o f t o a p r o f e s s i o n a l m a t h e m a t i c s j o u r n a l — a s a l m o s t a n y o n e e l s e w o u l d h a v e d o n e — h e p r e s e n t e d i t a t a c o n f e r e n c e . T h e r e a s o n w a s p r o b a b l y t w o f o l d . T h r o u g h o u t t h e y e a r s o f w o r k o n t h e p r o o f , W i l e s w a s o b s e s s e d w i t h s e c r e c y . I f h e s u b m i t t e d t h e p r o o f t o a j o u r n a l , t h e p r o o f w o u l d h a v e b e e n s e n t t o a n u m b e r o f r e f e r e e s c h o s e n b y t h e j o u r n a l a n d o n e o f t h e m , o r t h e e d i t o r s , m i g h t h a v e s a i d s o m e t h i n g t o t h e w o r l d a t l a r g e . W i l e s p r o b a b l y w a s a l s o w o r r i e d t h a t s o m e o n e w h o r e a d t h e p r o p o s e d p r o o f m i g h t s o m e h o w s t e a l i t a n d s e n d i t o u t u n d e r h i s o r h e r o w n n a m e . T h i s , u n f o r t u n a t e l y , d o e s h a p p e n i n a c a d e m i a . T h e o t h e r r e a s o n , l i n k e d w i t h t h e f i r s t , w a s t h a t W i l e s w a n t e d t o m a i n t a i n t h e b u i l d u p o f s u s p e n s e a s h e p r e s e n t e d h i s p r o o f a t C a m b r i d g e .
B u t e v e n s o , h a v i n g p r e s e n t e d t h e r e s u l t s a t a c o n f e r e n c e , t h e w o r k w o u l d s t i l l h a v e t o b e r e f e r e e d . T h e s t e p s w o u l d s t i l l h a v e t o b e p e e r - r e v i e w e d , t h a t i s , o t h e r e x p e r t s i n n u m b e r t h e o r y w o u l d h a v e t o g o t h r o u g h W i l e s ' p r o o f , l i n e b y l i n e , t o a s c e r t a i n t h a t h e h a d i n d e e d e s t a b l i s h e d w h a t h e s e t o u t t o p r o v e .
The Deep Gulf Materializes
W i l e s ' 200- p a g e p a p e r w a s
s e n t t o a n u m b e r o f l e a d i n g e x p e r t s i n n u m b e r t h e o r y . S o m e o f t h e m q u i c k l y e x p r e s s e d c o n c e r n s , b u t g e n e r a l l y m a t h e m a t i c i a n s t h o u g h t t h e p r o o f w a s p r o b a b l y c o r r e c t . T h e y w o u l d w a i t t o h e a r t h e e x p e r t s ' v e r d i c t , h o w e v e r . " O h y e s ! " s a i d K e n R i b e t w h e n I a s k e d h i m i f h e b e l i e v e d W i l e s ' p r o o f . " I w a s u n a b l e t o s e e w h a t s o m e p e o p l e w e r e s a y -i n g s o o n a f t e r t h e y r e a d t h e p r o o f — n a m e l y t h a t t h e r e w a s n o E u l e r S y s t e m h e r e . "
O n e o f t h e e x p e r t s c h o s e n t o g o o v e r W i l e s ' p r o o f w a s h i s P r i n c e t o n f r i e n d , N i c k K a t z . P r o f e s s o r K a t z s p e n t t w o w h o l e m o n t h s , J u l y a n d A u g u s t o f 1993, d o i n g n o t h i n g b u t s t u d y i n g t h e e n t i r e p r o o f . E v e r y d a y , h e w o u l d s i t a t h i s d e s k a n d s l o w l y r e a d e v e r y l i n e , e v e r y m a t h e m a t i c a l s y m b o l , e v e r y l o g i c a l i m p l i c a t i o n , t o m a k e s u r e t h a t i t m a d e p e r f e c t s e n s e a n d t h a t i t w o u l d i n d e e d b e a c c e p t a b l e t o a n y m a t h e m a t i c i a n w h o m i g h t r e a d t h e p r o o f . O n c e o r t w i c e a d a y , K a t z w o u l d s e n d a n e - m a i l m e s s a g e t o A n d r e w W i l e s , w h o s t a y e d a w a y f r o m P r i n c e t o n t h a t s u m m e r , a s k i n g h i m : " W h a t d o y o u m e a n i n t h i s l i n e o n t h i s p a g e ? " o r " I d o n ' t s e e h o w t h i s i m p l i c a t i o n f o l l o w s f r o m t h e o n e a b o v e , " e t c . I n r e s p o n s e , W i l e s w o u l d e - m a i l b a c k , a n d i f t h e p r o b l e m r e q u i r e d m o r e d e t a i l s , h e w o u l d f a x t h e a n s w e r t o K a t z .
O n e d a y , w h e n K a t z w a s a b o u t t w o t h i r d s o f t h e w a y t h r o u g h W i l e s ' l o n g m a n u s c r i p t , h e c a m e a c r o s s a p r o b l e m . I t s e e m e d i n n o c e n t e n o u g h a t f i r s t , o n e o f m a n y W i l e s h a d a n s w e r e d e a r l i e r t o K a t z ' s c o m p l e t e s a t i s f a c t i o n . B u t n o t t h i s t i m e . I n r e s p o n s e t o K a t z ' s q u e s t i o n s , W i l e s e -m a i l e d b a c k a n a n s w e r . B u t K a t z h a d t o e - m a i l b a c k : " I s t i l l d o n ' t u n d e r s t a n d i t , A n d r e w . " S o t h i s t i m e W i l e s s e n t a f a x t r y i n g t o m a k e t h e l o g i c a l c o n n e c t i o n .
A g a i n K a t z w a s n o t s a t i s f i e d . S o m e t h i n g w a s s i m p l y n o t r i g h t . T h i s w a s s u p p o s e d t o b e e x a c t l y o n e o f t h e a r g u m e n t s t h a t W i l e s a n d K a t z w e n t o v e r c a r e f u l l y i n t h e s p r i n g w h e n W i l e s w a s t e a c h i n g h i s " c o u r s e . " A n y d i f f i c u l t i e s s h o u l d h a v e a l r e a d y b e e n i r o n e d o u t . B u t a p p a r e n t l y t h e h o l e i n W i l e s ' l o g i c e l u d e d t h e m b o t h . P o s s i b l y i f t h e g r a d u a t e s t u -d e n t s h a d s t a y e d o n , o n e o f t h e m m i g h t h a v e a l e r t e d t h e t w o t o t h e p r o b l e m .
B y t h e t i m e K a t z f o u n d t h e e r r o r , o t h e r m a t h e m a t i c i a n s t h r o u g h o u t t h e w o r l d w e r e a w a r e o f t h e e x a c t s a m e p r o b l e m w i t h W i l e s ' p r o o f . T h e r e s i m p l y w a s n o E u l e r S y s t e m h e r e , a n d t h e r e w a s n o t h i n g d o i n g . A n d w i t h o u t t h e E u l e r S y s t e m — s u p -p o s e d l y a g e n e r a l i z a t i o n o f t h e e a r l i e r w o r k o f F l a c h a n d K o l y - v a g i n — t h e r e w a s n o C l a s s N u m b e r F o r m u l a . W i t h o u t t h e C l a s s N u m b e r F o r m u l a i t w a s i m p o s s i b l e t o " c o u n t " t h e G a l o i s r e p r e s e n t a t i o n s o f t h e e l l i p t i c c u r v e s a g a i n s t t h e m o d u l a r f o r m s , a n d S h i m u r a -T a n i y a m a w a s n o t e s t a b l i s h e d . A n d w i t h o u t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e p r o v e d a s c o r r e c t , t h e r e w a s n o p r o o f o f F e r m a t ' s L a s t T h e o r e m . I n s h o r t , t h e h o l e i n t h e E u l e r S y s t e m m a d e e v e r y t h i n g c o l l a p s e l i k e a h o u s e o f c a r d s .
The Agony
A n d r e w W i l e s r e t u r n e d t o P r i n c e t o n i n t h e f a l l o f 1993.
H e w a s e m b a r r a s s e d , h e w a s u p s e t , h e w a s a n g r y , f r u s t r a t e d , h u m i l i a t e d . H e h a d p r o m i s e d t h e w o r l d a p r o o f o f F e r m a t ' s L a s t T h e o r e m — b u t h e c o u l d n ' t d e l i v e r . I n m a t h e m a t i c s , a s i n a l m o s t a n y o t h e r f i e l d , t h e r e a r e n o r e a l " s e c o n d p r i z e s " o r " a l s o r a n " a w a r d s . T h e c r e s t f a l l e n W i l e s w a s b a c k i n h i s a t t i c t r y i n g t o f i x t h e p r o o f . " A t t h i s p o i n t , h e w a s h i d i n g a s e c r e t f r o m
t h e w o r l d , " r e c a l l e d N i c k K a t z , " a n d I t h i n k h e m u s t h a v e f e l t p r e t t y u n c o m f o r t a b l e a b o u t i t . " O t h e r c o l l e a g u e s h a d t r i e d t o h e l p W i l e s , i n c l u d i n g h i s f o r m e r s t u d e n t R i c h a r d T a y l o r w h o w a s t e a c h i n g a t C a m b r i d g e b u t j o i n e d W i l e s a t P r i n c e t o n t o h e l p h i m t r y t o f i x t h e p r o o f .
" T h e f i r s t s e v e n y e a r s , w o r k i n g a l l a l o n e , I e n j o y e d e v e r y m i n u t e o f i t , " W i l e s r e c a l l e d , " n o m a t t e r h o w h a r d o r s e e m i n g l y i m p o s s i b l e a h u r d l e I f a c e d . B u t n o w , d o i n g m a t h e m a t i c s
i n t h i s o v e r - e x p o s e d w a y w a s c e r t a i n l y n o t m y s t y l e . I h a v e n o w i s h t o e v e r r e p e a t t h i s e x p e r i e n c e . " A n d t h e b a d e x p e r i e n c e l a s t e d a n d l a s t e d . R i c h a r d T a y l o r , h i s s a b b a t i c a l l e a v e o v e r , r e t u r n e d t o C a m b r i d g e a n d s t i l l W i l e s s a w n o e n d i n s i g h t . H i s c o l l e a g u e s l o o k e d a t h i m w i t h a m i x t u r e o f a n t i c i p a t i o n , h o p e , a n d p i t y , a n d h i s s u f f e r i n g w a s c l e a r t o e v e r y o n e a r o u n d h i m . P e o p l e w a n t e d t o k n o w . T h e y w a n t e d t o h e a r g o o d n e w s , b u t n o n e o f h i s c o l l e a g u e s d a r e d a s k h i m h o w h e w a s d o i n g w i t h t h e p r o o f . O u t s i d e h i s d e p a r t m e n t , t h e r e s t o f t h e w o r l d w a s c u r i o u s , t o o . S o m e t i m e i n t h e n i g h t o f D e c e m b e r 4, 1993,
A n d r e w W i l e s p o s t e d a n e -m a i l m e s s a g e t o t h e c o m p u t e r n e w s g r o u p S c i . m a t h , t o w h i c h s e v e r a l n u m b e r t h e o r i s t s a n d o t h e r m a t h e m a t i c i a n s b e l o n g e d :
In view of the speculation on the status of my
work on the Taniyama-Shimura conjecture and
Fermat's Last Theorem 1 will give a brief account
of the situation. During the review process a
number of problems emerged, most of which have
been resolved, but one in particular 1 have not yet
settled... I believe that 1 will be able to finish this
in the near future using the ideas explained in my
Cambridge lectures. The fact that a lot of work
remains to be done on the manuscript makes it
unsuitable for release as a preprint. In my course
in Princeton beginning in February I will give a full
account of this work.
Andrew Wiles
The Post-Mortem
B u t A n d r e w W i l e s w a s
p r e m a t u r e l y o p t i m i s t i c . A n d w h a t e v e r c o u r s e h e m a y h a v e p l a n n e d t o o f f e r a t P r i n c e t o n w o u l d n o t
y i e l d a n y s o l u t i o n . W h e n m o r e t h a n a y e a r p a s s e d s i n c e h i s s h o r t - l i v e d t r i u m p h i n C a m b r i d g e , A n d r e w W i l e s w a s a b o u t t o g i v e u p a l l h o p e a n d t o f o r g e t h i s c r i p p l e d p r o o f .
O n M o n d a y m o r n i n g , S e p t e m b e r 19, 1994, W i l e s w a s s i t t i n g a t h i s d e s k a t P r i n c e t o n U n i v e r s i t y , p i l e s o f p a p e r s t r e w n a l l a r o u n d h i m . H e d e c i d e d h e w o u l d t a k e o n e l a s t l o o k a t h i s p r o o f b e f o r e c h u c k i n g i t a l l a n d a b a n d o n i n g a l l h o p e t o p r o v e F e r m a t ' s L a s t T h e o r e m . H e w a n t e d t o s e e e x a c t l y w h a t i t w a s t h a t w a s p r e v e n t i n g h i m f r o m c o n s t r u c t i n g t h e E u l e r S y s t e m . H e w a n t e d t o k n o w —j u s t f o r h i s o w n s a t i s f a c t i o n— w h y h e h a d f a i l e d . W h y w a s t h e r e n o E u l e r S y s t e m ? — h e w a n t e d t o b e a b l e t o p i n p o i n t p r e c i s e l y w h i c h t e c h n i c a l f a c t w a s m a k i n g t h e w h o l e t h i n g f a i l . I f h e w a s g o i n g t o g i v e u p , h e f e l t , t h e n a t l e a s t h e w a s o w e d a n a n s w e r t o w h y h e h a d b e e n w r o n g .
W i l e s s t u d i e d t h e p a p e r s i n f r o n t o f h i m , c o n c e n t r a t i n g v e r y h a r d f o r a b o u t t w e n t y m i n u t e s . A n d t h e n h e s a w e x a c t l y w h y h e w a s u n a b l e t o m a k e t h e s y s t e m w o r k . F i n a l l y , h e u n d e r s t o o d w h a t w a s w r o n g . " I t w a s t h e m o s t i m p o r t a n t m o m e n t i n m y e n t i r e w o r k i n g l i f e , " h e l a t e r d e s c r i b e d t h e f e e l i n g . " S u d d e n l y , t o t a l l y u n e x p e c t e d l y , I h a d t h i s i n c r e d i b l e r e v e l a t i o n .
N o t h i n g I ' l l e v e r d o a g a i n w i l l . . . " a t t h a t m o m e n t t e a r s w e l l e d u p a n d W i l e s w a s c h o k i n g w i t h e m o t i o n . W h a t W i l e s r e a l i z e d a t t h a t f a t e f u l m o m e n t w a s " s o i n d e s c r i b a b l y b e a u t i f u l , i t w a s s o s i m p l e a n d s o e l e g a n t . . . a n d I j u s t s t a r e d i n d i s b e l i e f . " W i l e s r e a l i z e d t h a t e x a c t l y w h a t w a s m a k i n g t h e E u l e r S y s t e m f a i l i s w h a t w o u l d m a k e t h e H o r i z o n t a l I w a s a w a T h e o r y a p p r o a c h h e h a d a b a n d o n e d t h r e e y e a r s e a r l i e r work. W i l e s s t a r e d a t h i s p a p e r f o r a l o n g t i m e . H e m u s t b e d r e a m i n g , h e t h o u g h t , t h i s w a s j u s t t o o g o o d t o b e t r u e . B u t l a t e r h e s a i d i t w a s s i m p l y t o o g o o d t o b e
false. T h e d i s c o v e r y w a s s o p o w e r f u l , s o b e a u t i f u l , t h a t i t bad t o b e t r u e .
W i l e s w a l k e d a r o u n d t h e d e p a r t m e n t f o r s e v e r a l h o u r s . H e d i d n ' t k n o w w h e t h e r h e w a s a w a k e o r d r e a m i n g . E v e r y o n c e i n a w h i l e , h e w o u l d r e t u r n t o h i s d e s k t o s e e i f h i s f a n t a s t i c f i n d i n g w a s s t i l l t h e r e — a n d i t w a s . H e w e n t h o m e . H e h a d t o s l e e p o n i t — m a y b e i n t h e m o r n i n g h e w o u l d f i n d s o m e f l a w i n t h i s n e w a r g u m e n t . A y e a r o f p r e s s u r e f r o m t h e e n t i r e w o r l d , a y e a r o f o n e f r u s t r a t e d a t t e m p t a f t e r a n o t h e r h a d s h a k e n W i l e s ' c o n f i d e n c e . H e c a m e b a c k t o h i s d e s k i n t h e m o r n i n g , a n d t h e i n c r e d i b l e g e m h e h a d f o u n d t h e d a y b e f o r e w a s s t i l l t h e r e , w a i t i n g f o r h i m .
W i l e s w r o t e u p h i s p r o o f u s i n g t h e c o r r e c t e d H o r i z o n t a l I w a s a w a T h e o r y a p p r o a c h . F i n a l l y , e v e r y t h i n g f e l l p e r f e c t l y i n t o p l a c e . T h e a p p r o a c h h e h a d u s e d t h r e e y e a r s e a r l i e r w a s t h e c o r r e c t o n e . A n d t h a t k n o w l e d g e c a m e t o h i m f r o m t h e f a i l i n g o f t h e F l a c h a n d K o l y v a g i n r o u t e h e h a d c h o s e n i n m i d s t r e a m . T h e m a n u s c r i p t w a s r e a d y t o b e s h i p p e d o u t . E l a t e d , A n d r e w
W i l e s l o g g e d i n t o h i s c o m p u t e r a c c o u n t . H e s e n t e -m a i l m e s s a g e s a c r o s s t h e I n t e r n e t t o a s c o r e o f m a t h e m a t i c i a n s a r o u n d t h e w o r l d : " E x p e c t a F e d e r a l E x p r e s s p a c k a g e i n t h e n e x t f e w d a y s , " t h e m e s s a g e s r e a d .
A s h e h a d p r o m i s e d h i s f r i e n d R i c h a r d T a y l o r , w h o h a d c o m e f r o m E n g l a n d e s p e c i a l l y t o h e l p h i m c o r r e c t h i s p r o o f , t h e n e w p a p e r c o r r e c t i n g t h e I w a s a w a t h e o r y b o r e b o t h o f t h e i r n a m e s e v e n t h o u g h W i l e s o b t a i n e d t h e a c t u a l r e s u l t a f t e r T a y l o r ' s d e p a r t u r e . W i t h i n t h e n e x t f e w w e e k s , t h e m a t h e m a t i c i a n s w h o r e c e i v e d W i l e s ' c o r r e c t i o n t o h i s C a m b r i d g e p a p e r s w e n t o v e r a l l t h e d e t a i l s . T h e y c o u l d f i n d n o t h i n g w r o n g .
I 33
W i l e s n o w u s e d t h e c o n v e n t i o n a l a p p r o a c h t o t h e p r e s e n t a t i o n o f m a t h e m a t i c a l r e s u l t s . I n s t e a d o f d o i n g w h a t h e h a d d o n e i n C a m b r i d g e a y e a r a n d a h a l f e a r l i e r , h e s e n t t h e p a p e r s t o a p r o f e s s i o n a l j o u r n a l , t h e Annals oj Mathematics, w h e r e t h e y c o u l d b e p e e r - r e v i e w e d b y o t h e r m a t h e m a t i c i a n s . T h e r e v i e w p r o c e s s t o o k a f e w m o n t h s , b u t n o f l a w s w e r e f o u n d t h i s t i m e . T h e M a y , 1995, i s s u e o f t h e j o u r n a l c o n t a i n e d W i l e s ' o r i g i n a l C a m b r i d g e p a p e r a n d t h e c o r r e c t i o n b y T a y l o r a n d W i l e s . F e r m a t ' s L a s t T h e o r e m w a s f i n a l l y l a i d t o r e s t .
Did Fermat Possess the Prooj?
A n d r e w W i l e s d e s c r i b e d h i s p r o o f a s " a t w e n t i e t h c e n t u r y p r o o f . " I n d e e d , W i l e s u s e d t h e w o r k o f m a n y t w e n t i e t h - c e n t u r y m a t h e m a t i c i a n s . H e a l s o u s e d t h e w o r k o f e a r l i e r m a t h e m a t i c i a n s . A l l t h e m y r i a d e l e m e n t s o f W i l e s ' c o n s t r u c t i o n s c a m e f r o m t h e w o r k o f o t h e r s , m a n y o t h e r s . S o t h e p r o o f o f F e r m a t ' s L a s t T h e o r e m w a s r e a l l y t h e a c h i e v e m e n t o f a l a r g e n u m b e r o f m a t h e m a t i c i a n s l i v i n g i n t h e t w e n t i e t h c e n t u r y — a n d a l l t h e p r e c e d i n g o n e s t o t h e t i m e o f F e r m a t h i m s e l f . A c c o r d i n g t o W i l e s , F e r m a t c o u l d n o t p o s s i b l y h a v e h a d t h i s p r o o f i n m i n d w h e n h i s w r o t e h i s f a m o u s n o t e i n t h e m a r g i n . T h i s m u c h i s t r u e , o f c o u r s e , b e c a u s e t h e S h i m u r a -
T a n i y a m a c o n j e c t u r e d i d n o t e x i s t u n t i l t h e t w e n t i e t h c e n t u r y . B u t c o u l d F e r m a t h a v e h a d a n o t h e r p r o o f i n m i n d ?
T h e a n s w e r i s p r o b a b l y n o t . B u t t h i s i s n o t a c e r t a i n t y . We w i l l n e v e r k n o w . O n t h e o t h e r h a n d , F e r m a t l i v e d a n o t h e r 28 y e a r s a f t e r h e w r o t e h i s t h e o r e m i n t h e m a r g i n . A n d h e n e v e r s a i d a n y t h i n g m o r e a b o u t i t . P o s s i b l y h e k n e w h e c o u l d n ' t p r o v e t h e t h e o r e m . O r h e m a y h a v e e r r o n e o u s l y t h o u g h t t h a t
h i s m e t h o d o f i n f i n i t e d e s c e n t u s e d i n p r o v i n g t h e s i m p l e c a s e « = 3 c o u l d a p p l y t o a g e n e r a l s o l u t i o n . O r m a y b e h e s i m p l y f o r g o t a b o u t t h e t h e o r e m a n d w e n t o n t o d o o t h e r t h i n g s .
P r o v i n g t h e t h e o r e m t h e w a y i t w a s f i n a l l y d o n e i n t h e 1990s r e q u i r e d a l o t m o r e m a t h e m a t i c s t h a n F e r m a t h i m s e l f c o u l d h a v e k n o w n . T h e p r o f o u n d n a t u r e o f t h e t h e o r e m i s t h a t n o t o n l y d o e s i t s h i s t o r y s p a n t h e l e n g t h o f h u m a n c i v i l i z a t i o n , b u t t h e f i n a l s o l u t i o n o f t h e p r o b l e m c a m e a b o u t b y h a r n e s s i n g — a n d i n a s e n s e u n i f y i n g — t h e e n t i r e b r e a d t h o f m a t h e m a t i c s . I t w a s t h i s u n i f i c a t i o n o f s e e m i n g l y d i s p a r a t e a r e a s o f m a t h e m a t i c s t h a t f i n a l l y n a i l e d t h e t h e o r e m . A n d d e s p i t e t h e f a c t t h a t A n d r e w W i l e s w a s t h e p e r s o n w h o d i d t h e i m p o r t a n t f i n a l w o r k o n t h e t h e o r e m b y p r o v i n g a f o r m o f t h e S h i m u r a -T a n i y a m a c o n j e c t u r e n e e d e d t o p r o v e F e r m a t ' s t h e o r e m , t h e e n t i r e e n t e r p r i s e w a s t h e w o r k o f m a n y p e o p l e . A n d i t i s a l l t h e i r c o n t r i b u -t i o n s , t a k e n t o g e t h e r , w h i c h b r o u g h t a b o u t t h e f i n a l s o l u t i o n . W i t h o u t t h e w o r k o f E r n s t K u m m e r t h e r e w o u l d h a v e b e e n n o t h e o r y o f i d e a l s , a n d w i t h o u t i d e a l s t h e w o r k o f B a r r y M a z u r w o u l d n o t h a v e e x i s t e d . W i t h o u t M a z u r t h e r e w o u l d h a v e b e e n n o c o n j e c t u r e b y F r e y , a n d w i t h o u t t h e c r u c i a l c o n j e c t u r e a n d i t s s y n t h e s i s
b y S e r r e t h e r e w o u l d n o t h a v e b e e n t h e p r o o f b y R i b e t t h a t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w o u l d e s t a b l i s h F e r m a t ' s L a s t T h e o r e m . A n d i t s e e m s t h a t n o p r o o f o f F e r m a t ' s L a s t T h e o r e m w o u l d b e p o s s i b l e w i t h o u t t h a t c o n j e c t u r e p u t f o r w a r d b y Y u t a k a T a n i y a m a a t T o k y o -N i k k o i n 1955 a n d t h e n r e f i n e d a n d m a d e s p e c i f i c b y G o r o S h i m u r a . O r w o u l d i t ?
F e r m a t , o f c o u r s e , c o u l d n o t h a v e f o r m u l a t e d s u c h a n o v e r a r c h i n g c o n j e c t u r e t h a t w o u l d u n i f y t w o v e r y d i f f e r e n t b r a n c h e s o f m a t h e m a t i c s . O r c o u l d h e h a v e d o n e s o ? N o t h i n g
i s c e r t a i n . We o n l y k n o w t h a t t h e t h e o r e m h a s f i n a l l y b e e n e s t a b l i s h e d a n d t h a t t h e p r o o f h a s b e e n c h e c k e d a n d v e r i f i e d t o i t s f i n e s t d e t a i l s b y s c o r e s o f m a t h e m a t i c i a n s t h r o u g h o u t t h e w o r l d . B u t j u s t b e c a u s e a p r o o f e x i s t s a n d i t i s a v e r y c o m p l i c a t e d , a d v a n c e d o n e d o e s n o t m e a n t h a t a s i m p l e r p r o o f i s n o t p o s s i b l e . R i b e t , i n f a c t , p o i n t s o u t i n o n e o f h i s p a p e r s a d i r e c t i o n w h e r e a p r o o f o f F e r m a t ' s t h e o r e m m i g h t b e p o s s i b l e w i t h o u t a p r o o f o f t h e S h i m u r a -T a n i y a m a c o n j e c t u r e . A n d p e r h a p s F e r m a t d i d k n o w a l o t o f p o w e r f u l " m o d e r n " m a t h e m a t i c s , n o w l o s t ( a c t u a l l y , t h e c o p y o f B a c h e t ' s D i o p h a n t u s i n w h i c h h e s u p p o s e d l y w r o t e h i s m a r g i n - s t a t e m e n t h a s n e v e r b e e n f o u n d ) . S o w h e t h e r o r n o t F e r m a t d i d p o s s e s s a " t r u l y m a r v e l o u s p r o o f " o f h i s t h e o r e m , o n e t h a t c o u l d n o t f i t i n t h e m a r g i n o f h i s b o o k , w i l l f o r e v e r r e m a i n h i s s e c r e t .
Gerd Faltings. He had a
totally different approach
to Fermat's Last Theorem.
When Wiles failed in his
first attempt in 1993,
many feared that Faltings
would now beat him to the
true proof.
Andrew Wiles at the crucial
moment of his third lecture
at Cambridge, June 1993,
when it was clear to
everyone Fermat was just
around the corner.
Ken Ribet at the famous
cafe where he finished the
proof that Shimura-
Taniyama would imply that
Fermat's Last Theorem had
to be true.
Endnotes
1. E.T. Bell, Men oj Mathematics, New York: Simon and Schuster, 1937, p. 56 .
I . Barry Mazur, "Number Theory as Gadfly," American Mathematical
Monthly, Vol. 98 , 1991, p. 593.
3 . Plimpton 322 and its implications about the advanced level of
Babylonian mathematics were brought to the attention of the scientific com-
munity by Otto Neugebauer in 1934. An account in English can be found in
his book The Exact Sciences in Antiquity (Princeton University Press, 1957 ) .
23.Actually, Cantor went much farther. He hypothesized.that the order of
infinity of the irrational numbers immediately follows that of the rationals.
That is, he believed that there is no order of infinity that is both higher
than that of the rational numbers and lower than that of the irrational
numbers. This statement became known as the Continuum Hypothesis,
and the work of Kurt Godel and Paul Cohen in the twentieth century
established that it is impossible to prove this hypothesis within the rest of
mathematics. The Continuum Hypothesis stands alone (with some
equivalent restatements) opposite the rest of mathematics, their
respective truths independent of each other. This remains one of the most
bizarre truths in the foundations of mathematics.
24.D. Wells, Curious and Interesting Numbers, London: Penguin Books, 1987,
p. 81 .
25.C. Boyer, A History of Mathematics, New York: Wiley, 1968 , p. 9 .
26.Reprinted in B. Mazur, op. cit.
27.Ian Stewart, Nature's Numbers, New York: Basic Books, 1995, p. 140.
9 . Michael Mahoney, The Mathematical Career oj Pierre de Fermat, 2d ed.,
Princeton University Press, 1994, p. 4 .
10 . Harold M. Edwards, Fermat's Last Theorem, New York: Springer-Verlag,
1977, pp. 61-73 .
I I . Much of what is publicly known about the secret society comes from
Paul R. Halmos, "Nicolas Bourbaki," Scientific American, 196, May 1957, pp.
88 -97 .
12 . Andre Weil, Oeuvres, Vols. Mil, Paris: Springer-Verlag, 1979.
13 . Adapted from Kenneth A. Ribet and Brian Hayes, "Fermat's Last
Theorem and Modern Arithmetic," American Scientist, Vol. 82 , March-April
1994, pp. 144-156 .
28.A good introduction to this topic is the book by Joseph H . Silverman and
John Tate, Rational Points on Elliptic Curves, New York: Springer-Verlag,
1992.
29.Most of the information on Yutaka Taniyama's life is from Goro Shimura,
"Yutaka Taniyama and His Time: Very Personal Recollections," Bulletin oj
the London Mathematical Society, Vol. 21 , 1989, pp. 184-196 .
30.Reprinted in the Japanese journal Sugaku, May 1956, pp. 227 -231 .
31.Shimura thus stated to Serre his actual conjecture, sharing it for the
first time, and implicitly trusting that Serre would acknowledge him as its
originator.
32.Andre Weil, Oeuvres, op. cit., Vol. Ill, p. 450.
33.Andre Weil, "Liber die Bestimmung Dirichletscher Reihen durch
Funktionalgleichungen," Math. Annalen, Vol. 168 , 1967, pp. 165-172.
34.Weil's letter to Lang, along with much of the chronology of events
described here, including private conversations and letters, are
reproduced in Serge Lang, "Some History of the Shimura-Taniyama
Conjecture," Notices oj the American Mathematical Society, November 1995,
pp. 1301-1307. It is to Lang's credit that his article and the "Taniyama-
Shimura File" he has been circulating among mathematicians for ten
years now are finally helping bring Goro Shimura the recognition he
rightly deserves.
35.Jean-Pierre Serre, "Lettre a J.-F. Mestre," reprinted in Current Trends in
Arithmetical Algebraic Geometry, Providence: American Mathematical
Society, 1987, pp. 263 -268 .
22 . Barry Mazur, "Modular Curves and the Eisenstein Ideal," Paris,
France: The Mathematical Publications ojI.H.E.S., Vol. 47 , 1977, pp. 33-186.
36.Barry Mazur, op. cit.
37.The first and more important of the two papers, Andrew Wiles, "Modular
Elliptic Curves and Fermat's Last Theorem," Annals of Mathematics, Vol.
142, 1995, pp. 443-551, begins with Fermat's actual margin-
statement of his theorem in Latin: Cubum autem in duos cubos, aut
quadratocjuadratum in duos quadratoquadratos, et generaliter nullam in infinitum
ultra quadratum potestatem in duos ejus-dem nominis fas est dividere: cujus rei
demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.
Pierre de Fermat. The journal sold out even before the publication date, and
for the first time imposed a charge of $14 per individual issue.
Author's Note
I n p r e p a r i n g t h i s b o o k I d r e w m u c h o f t h e h i s t o r i c a l b a c k -g r o u n d f r o m a v a r i e t y o f s o u r c e s . M y f a v o r i t e , a n d t h e m o s t c o m p l e t e a n d o r i g i n a l s o u r c e , i s E . T . B e l l ' s b o o k , Men oj Mathematics
( a l t h o u g h I d i s l i k e t h e s e x i s t t i t l e , w h i c h i s a l s o m i s l e a d i n g s i n c e t w o o f t h e m a t h e m a t i c i a n s a r e w o m e n , - t h e b o o k w a s w r i t t e n i n 1937).
A p p a r e n t l y o t h e r h i s t o r i a n s o f m a t h e m a t i c s h a v e d r a w n t h e i r i n f o r m a t i o n f r o m B e l l , s o I w i l l n o t m e n t i o n t h e m b y n a m e h e r e . A l l m y i m p o r t a n t s o u r c e s a r e r e f e r e n c e d i n t h e e n d n o t e s . A d d i t i o n a l l y , I f o u n d t h e a r t i c l e s o f J a c q u e l y n S a v a n i o f P r i n c e t o n U n i v e r s i t y (Princeton
Weekly Bulletin, S e p t e m b e r 6, 1993)
u s e f u l , a n d I t h a n k h e r f o r s e n d i n g m e a c o p y o f a p r o g r a m a i r e d o n t h e B B C a b o u t F e r m a t ' s L a s t T h e o r e m .
I a m i n d e b t e d t o C . J . M o z z o c h i f o r a n u m b e r o f p h o t o g r a p h s o f m a t h e m a t i c i a n s i n v o l v e d i n t h e p r o o f o f F e r m a t ' s L a s t T h e o r e m . V e r y w a r m t h a n k s t o P r o f e s s o r K e n n e t h A . R i b e t o f t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y f o r i n f o r m a t i v e i n t e r v i e w s a n d m u c h i m p o r t a n t i n f o r m a t i o n a b o u t h i s w o r k l e a d i n g t o t h e p r o o f o f F e r m a t ' s t h e o r e m . M y d e e p g r a t i t u d e t o P r o f e s s o r G o r o S h i m u r a o f P r i n c e t o n U n i v e r s i t y f o r h i s g e n e r o s i t y w i t h h i s t i m e i n g i v i n g m e a c c e s s t o s o m u c h
i m p o r t a n t i n f o r m a t i o n a b o u t h i s w o r k , a n d h i s c o n j e c t u r e w i t h o u t w h i c h t h e r e w o u l d b e n o p r o o f o f F e r m a t ' s t h e o r e m . I a m g r a t e f u l t o P r o f e s s o r G e r d F a l t i n g s o f t h e M a x P l a n c k I n s t i t u t e i n B o n n a n d t o P r o f e s s o r G e r h a r d F r e y o f t h e U n i v e r s i t y o f E s s e n , G e r m a n y , f o r p r o v o c a t i v e i n t e r v i e w s a n d i n s i g h t f u l o p i n i o n s . T h a n k s a l s o t o P r o f e s s o r B a r r y M a z u r o f
H a r v a r d U n i v e r s i t y f o r e x p l a i n i n g t o m e i m p o r t a n t c o n c e p t s i n t h e g e o m e t r y o f n u m b e r t h e o r y . A n y e r r o r s t h a t r e m a i n a r e c e r t a i n l y m i n e .
I t h a n k m y p u b l i s h e r , J o h n O a k e s , f o r h i s e n c o u r a g e m e n t a n d s u p p o r t . T h a n k s a l s o t o J i l l E l l y n R i l e y a n d K a t h r y n B e l d e n o f F o u r W a l l s E i g h t W i n d o w s . A n d f i n a l l y , m y d e e p g r a t i t u d e t o m y w i f e , D e b r a .
Index
Abel, Niels Henrik, 59 , 76 -77 abelian
groups, 77 abelian varieties, 77
abstract algebra, 71 algebra, 34-35,
40-42 abstract, 71
Diophantine equations, 33 , 92 , 110
algebraic geometry, 35, 110 algebraic
groups, 83 algebraic numbers, 78
algorists, 45 algorithms, 34
American Mathematical Society,
91-92
analysis, 60
Analysis Situs (Poincare), 81 analytic
functions, 55-56 modular forms, 56 ,
83 -84 ,
93 -94 , 99 -100 , 101 -6 , 110
Annals of Mathematics, 134 Archimedes,
5 , 29 , 30 -32 archimedes' screw, 31
architecture, Golden Section in,
39-40 Aristotle, 26 , 27
arithmetic geometry, 110 Arithmetica
(Diophantus), 6 , 9 , 33 , 43 Ars Magna
(Cardano), 42 astronomy, 58 -59
automorphic forms, 83 , 99 -100
axioms, 29
Babylonian Era, 12-15 , 18 , 21 Bachet,
Claude, 42 -43 Bell, E, T„ 5
Bernoulli, Daniel, 45, 46 Bernoulli,
Johannes, 45 Bernoulli, Nicolaus, 45,
46 Bessel, Friedrich Wilhelm, 56
Bolyai, Janos, 70 Bourbaki, Nicolas, 88 -
91 , 98 Bridges, of Konigsberg, problem
of, 49-50
calculus, 5 , 30 differential, 30
integral, 29 , 30 Cantor, Georg, 23
Cardano, Geronimo, 41-42 Catherine
the Great, empress of
Russia, 48 Cauchy, Augustin-Louis, 68 ,
72 , 76 , 77
Chevalier, Auguste, 76 Cicero, 31
Class Number Formula, 121 -24 , 130
Coates, John, 1 ,4 , 117 , 122, 127
comets, 58 -59 complex analysis, 49 ,
54-56 complex number field, 54
complex plane, 55
functions on, 55-56
periodic functions on, 65-66 , 82 -
84
computers
attempts to prove Fermat's Last
Theorem, 10
numerical analysis, 65
conjectures, 43 Conway, John, 1
cosine, 64 Cossists, 41 -42 cubic
equations, 41-42
Dedekind, Richard, 77 -79 Diderot,
Denis, 48 Dieudonne, Jean, 90—91
differential calculus, 30 Diophantine
equations, 33 , 92 , 1 10 Diophantus,
6 , 9 , 32 -33 , 43 Dirichlet, Peter
Gustav Lejeune, 44 ,
59-61 , 67 , 78 DisQuisitiones
Aritbmeticae (Gauss), 53 ,
59 -60 , 76
Eisenstein, Ferdinand Gotthold, 126
Eisenstein Ideal, 110, 126 Elements
(Euclid), 27 elliptic curves, 92 -94 ,
117 connection with modular forms,
93 -94 , 99 -100 , 101 -6 ,
110 , 119 -27
rational, 101-2
semistable, 119 -27
Shimura conjecture, 100-106
switching, 126 -27
zeta functions of, 99-100 Epsilon
Conjecture, 118
See also Shimura-Taniyama conjecture
equations
cubic, 41-42
Diophantine, 33 , 92 , 110 Mordell's
conjecture, 8 5-87 solution surfaces,
86-87 solving, 35 , 40-42 Euclid,
27 , 29
Eudoxus of Cnidus, 5, 29 -30 Euler,
Leonhard, 43 , 44 , 45-50 Euler
System, 122, 130, 132-33
factorization, 66 Faltings, Gerd, 87 ,
107 Fermat, Clement Samuel de, 7
Fermat, Dominique, 5 -7 Fermat,
Pierre de, 4 -8 , 9 , 43 , 44 ,
134-36 Fermat's Last
Theorem
attempts to prove, 9 -10 , 43-45,
66 , 68 -69 corollaries, 43
Gauss on, 53-54 implication of
Mordell's
conjecture, 87 marginal note, 6 ,
9 -10 , 33 , 136 partial proofs,
44 -45 , 51 , 59 , 87 -88
prizes offered for proof, 52 , 69 ,
70 related to Diophantine
equations, 1 10 Shimura-
Taniyama conjecture
and, 4 , 101 , 1 1 1 -14 , 115-
16,
118 -27
Sophie Germain's Theorem, 57 Wiles'
proof, 4 , 10-11 , 118 -27, 132-34
Ferro, Scippione del, 42 Fibonacci
(Leonardo of Pisa), 35-36 Fibonacci
numbers, 25 , 36-37 See also Golden
Section
Fibonacci Society, 39 , 40 fields,
121
first law of hydrostatics, 30 -31
Flach, Matthias, 122, 124 Four
Color Map problem, 50 Fourier,
Joseph, 61 -65 Fourier analysis, 65
Fourier Series, 62 , 64 -65 French
Academy of Sciences, 69 Frey,
Gerhard, 109-14, 115, 119 Frey
curve, 112, 115 functions
analytic, 55-56
automorphic forms, 83 , 99-100
on complex plane, 55-56
Fourier Series, 62 , 64 -65
modular forms, 56 , 83-84 ,
93 -94 , 99 -100 , 101 -6 ,
110 periodic, 62-66 , 82 -84
trigonometric, 64 zeta, 98 , 99 -
100
Galois, Evariste, 71 -76 , 121 Galois
Theory, 72 , 74 , 79 , 1 10 , 120 -21
Gauss, Carl Friedrich, 51 -54 ,
55-59, 61 , 77 , 78 , 92 ,
126 genus, 85-87 geometry
algebraic, 35 , 110
arithmetic, 110
Euclidean, 29
method of exhaustion, 29 -30
non-Euclidean, 70 -71 , 83
origins, 27-28
Pythagorean Theorem, 18 -19 ,
21 , 36 theorems, 28 -29
Germain, Sophie, 56 -58 God, proof
of existence, 48 Golden Section, 25-
26 , 36-40 Guthrie, Francis, 50
Heiberg, J. L., 32 Herodotus, 27-28
Hieron II, king of Syracuse, 30 -31
Histories (Herodotus), 27 -28
Horizontal Iwasawa Theory, 121,
132 -33
hydrostatics, first law of, 30-31
ideal numbers, 68 , 78 ideals, 70 ,
78 -79 , 115, 126 imaginary
numbers, 48 , 49 , 54-56 infinite
descent, 44 integral calculus, 29 ,
30 Ireland, Kenneth F, 114
irrational numbers, 21-23 Iwasawa
Theory, 1 ,117 Horizontal, 121,
132 -33
Al Jabr Wa'l Mucfabalab (Al
Khowarizmi), 35
Katz, Nick, 123-24, 129-30 Al
Khowarizmi, Mohammed Ibn
Musa, 34 -35 Konigsberg,
Seven Bridges of,
49 -50 Kronecker, Leopold,
23 Kummer, Ernst Eduard, 66 -70
Lame, Gabriel, 44 , 66 Lang, Serge,
103, 107-9 Lebesgue, Henri, 44
Legendre, Adrien-Marie, 44 , 59
Leibniz, Gottfried Wilhelm von, 5
lemas, 43
Leonardo of Pisa (Fibonacci), 35 -36
Liber Abaci (Fibonacci), 36 Liber
Quadratorum (Fibonacci), 35 Liouville,
Joseph, 66 , 76 Lobachevsky, Nicolas
Ivanovitch, 70
map-coloring problems, 50 Marcellus,
31
Mazur, Barry, 8 , 79 , 110, 113,
114 -16 , 126 Mesopotamia,
12-15 Mestre, J.-F., 111 method of
exhaustion, 29-30 The Method
(Archimedes), 32 modular forms, 56 ,
83 -84 connection to elliptic curves,
93-94 , 99-100, 101-6 ,
110,
119-27
Shimura conjecture, 100-106
modularity, 94 Mohammed, 34
Monge, Gaspard, 61 , 62 Mordell,
Louis J., 84 -87 Mordell's conjecture,
8 5-87 , 107 music, harmony in, 26-
27
Newton, Sir Isaac, 5
number systems
Babylonian, 13 -15 base 10 , 26
number theory, 52 -53 , 58 , 1 10 ,
114 algebraic numbers, 78
Fermat's theorems in, 9 Fourier
series used in, 65 prime numbers,
8 , 44 , 60 , 68 -70
numerical analysis, 65
Olbers, H. W. M., 53-54 Ovid, 27
Pacioli, Luca, 41 Parthenon, 39
pentagons, 24 -25 , 26 perfect
numbers, 17 periodic functions, 62-
65 extended to complex plane, 65-
66 , 82-84 Philolaos of Tarentum, 24
pi, 12 , 21 Plato, 24 , 29
Poincare, Henri, 79 -84 , 101 Poisson,
Simeon-Denis, 74 prime numbers, 8 ,
60
as exponents in Fermat's Theorem,
44
irregulars, 68 -70
proofs
of Fermat's Last Theorem, 4 ,
10 -1 1 , 118 -27 , 132 -34 of
Frey's conjecture, 115-16 of
God's existence, 48 of Shimura-
Taniyama
conjecture, 118 -27 of theorems, 28 -
29 pyramids, 16 , 21 , 40
Pythagorean Pentagon, 24 -25 , 26
Pythagorean Theorem, 18 -19 , 21 ,
36 Pythagorean triples, 13 -16 , 19 ,
35 Pythagoreans, 17 , 18 -22 , 24-
25, 26 -27
Pythagorus, 13 , 15-18 , 22 , 24
rational numbers, 22 -23 Ribet,
Kenneth, 3 -4 , 110 , 111 -12, 113 ,
1 14 -16 , 129, 136
Sarnak, Peter, 10-1 1 , 125
semistable elliptic curves, 119 -27
Serre, Jean-Pierre, 97 , 98 , 102,
105,
107, 108 , 1 1 1 , 1 1 8
Seven Bridges of Konigsberg,
problem of, 49-50 Shimura,
Goro, 65 , 96-97 ,
100-101, 107-8 Shimura-
Taniyama conjecture, 65 , 100-106
proof of, 118 -27 used to prove
Fermat's Last
Theorem, 4 , 101, 1 1 1 -14 , 115-
16, 118 -27 Weil's association with,
102-9 sine, 64 , 82 square roots
computing, 65 of five, 26
of negative one, 54-55
of two, 21-22 squared numbers,
13-16 , 19 , 20 , 35 statistics, 53
Stewart, Ian, 37
Taniyama, Yutaka, 95-100
Taniyama-Shimura conjecture. See
Shimura-Taniyama
conjecture Tartaglia, Niccolo,
41-42 Taylor, Richard, 130-31,
133 -34 tetraktys, 26-27
theorems, 28 -29 Tokyo-Nikko
Symposium on
Algebraic Number Theory, 96-100
topology, 49 , 81-82
map-coloring problems, 50 Mordell's
conjecture, 8 5-87 torus, 85
triangles, Pythagorean Theorem,
18 -19 , 2 1 , 36
trigonometric functions, 64
Weil, Andre, 90 , 91 , 97-98 , 99 -
100 , 102 -9 , 120 Weil Curves, 106
Weil-Taniyama conjecture, 104, 106
See also Shimura-Taniyama
conjecture Wiles, Andrew, 110,
1 1 3 interest in Fermat's Last
Theorem, 116 -18 lectures at
Cambridge
conference, 1 -4 , 127 peer
review of proof, 10 , 127 -30
problem found in proof, 10-1 1 ,
129 -32
proof developed by, 118 -27 , 132 -
34 Wolfskehl Prize, 70
zeta functions, 98 , 99 -100