Man and Mystery Vol 10 - Math Wonders [Rev06]

8/10/2019 Man and Mystery Vol 10 - Math Wonders [Rev06]

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A collection of intriguing topics and fascinating stories

about the rare, the paranormal, and the strange

Math Wonders

Volume 1

Discover whats intriguing about this most hated subject

Pablo C. Agsalud Jr.Revision 6


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Foreword

In the past, things like television, and words andideas like advertising, capitalism, microwave and

cancer all seemed too strange for the ordinaryman.

As man walks towards the future, overloaded withinformation, more mysteries have been solvedthrough the wonders of science. Although somethings remained too odd for science to reproduceor disprove, man had placed them in the grayareas between truth and skepticism and labeledthem with terminologies fit for the modern age.

But the truth is, as long as the strange andunexplainable cases keep piling up, the more likelyit would seem normal or natural. Answers arealways elusive and far too fewer than questions.And yet, behind all the wonderful and frighteningphenomena around us, it is possible that what wecall mysterious today wont be too strangetomorrow.

This book might encourage you to believe or refutewhat lies beyond your own understanding.Nonetheless, I hope it will keep you entertainedand astonished.

The content of this book remains believable for aslong as the sources and/or the references from thespecified sources exist and that the validity of theinformation remains unchallenged.


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Intriguing NumbersWikipedia.org

Explore the world of mathematics and discover the most bafflingnumber theories.


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Fibonacci numberWikipedia.org

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:

0,1,1,2,3,5,8,13,21,34,55,89,144,

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and eachsubsequent number is the sum of the previous two.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrencerelation

with seed values

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics,although the sequence had been described earlier in Indian mathematics. (By modernconvention, the sequence begins with F0 = 0. The Liber Abaci began the sequence with F1 =1, omitting the initial 0, and the sequence is still written this way by some.)

A tiling with squares whose sides are successive Fibonacci numbers in length

A Fibonacci spiral created by drawing circular arcs connecting the opposite corners of squaresin the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pairof Lucas sequences. They are intimately connected with the golden ratio, for example theclosest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications includecomputer algorithms such as the Fibonacci search technique and the Fibonacci heap datastructure, and graphs called Fibonacci cubes used for interconnecting parallel and distributedsystems. They also appear in biological settings, such as branching in trees, arrangement of


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leaves on a stem, the fruit spouts of a pineapple, the flowering of artichoke, an uncurlingfernand the arrangement of a pine cone.

Origins

The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with theshort (S), and counting the different patterns of L and S within a given fixed length results inthe Fibonacci numbers; the number of patterns that are m short syllables long is the Fibonaccinumber Fm + 1.

Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed inpart to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopla (c.1135AD), and Hemachandra (c.1150)". Parmanand Singh cites Pingala's cryptic formula misrau cha("the two are mixed") and cites scholars who interpret it in context as saying that the casesfor m beats (Fm+1) is obtained by adding a [S] to Fmcases and [L] to the Fm1cases. Hedates Pingala before 450 BCE.

However, the clearest exposition of the series arises in the work of Virahanka (c. 700AD),

whose own work is lost, but is available in a quotation by Gopala (c.1135):

Variations of two earlier meters [is the variation]... For example, for [a meter oflength] four, variations of meters of two [and] three being mixed, five happens.[works out examples 8, 13, 21]... In this way, the process should be followed in allmAtrA-vr.ttas (prosodic combinations).

The series is also discussed by Gopala (before 1135AD) and by the Jain scholar Hemachandra(c. 1150AD).

In the West, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardoof Pisa, known as Fibonacci. Fibonacci considers the growth of an idealized (biologicallyunrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, onefemale, are put in a field; rabbits are able to mate at the age of one month so that at the end

of its second month a female can produce another pair of rabbits; rabbits never die and amating pair always produces one new pair (one male, one female) every month from thesecond month on. The puzzle that Fibonacci posed was: how many pairs will there be in oneyear?

At the end of the first month, they mate, but there is still only 1 pair. At the end of the second month the female produces a new pair, so now there are 2

pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3

pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair,

the female born two months ago produces her first pair also, making 5 pairs.

At the end of the nth month, the number of pairs of rabbits is equal to the number of new

pairs (which is the number of pairs in month n 2) plus the number of pairs alive last month(n 1). This is the nth Fibonacci number.

The name "Fibonacci sequence" was first used by the 19th-century number theorist douardLucas.

The first 21 Fibonacci numbers Fnfor n = 0, 1, 2, ..., 20 are:


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The bee ancestry code

Fibonacci numbers also appear in the description of the reproduction of a population ofidealized honeybees, according to the following rules:

If an egg is laid by an unmated female, it hatches a male or drone bee. If, however, an egg was fertilized by a male, it hatches a female.

Thus, a male bee will always have one parent, and a female bee will have two.

If one traces the ancestry of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers ofparents is the Fibonacci sequence. The number of ancestors at each level, Fn, is the number offemale ancestors, which is Fn1, plus the number of male ancestors, which is Fn2. (This isunder the unrealistic assumption that the ancestors at each level are otherwise unrelated.)


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Fermat's Last TheoremWikipedia.org

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and ccan satisfy the equation an+ bn= cnfor any integer value of ngreater than two.

This theorem was first conjectured by Pierre deFermat in 1637, famously in the margin of a copy ofArithmetica where he claimed he had a proof thatwas too large to fit in the margin. No successfulproof was published until 1995 despite the efforts ofcountless mathematicians during the 358intervening years. The unsolved problem stimulatedthe development of algebraic number theory in the19th century and the proof of the modularitytheorem in the 20th. It is among the most famoustheorems in the history of mathematics and prior toits 1995 proof was in the Guinness Book of WorldRecords for "most difficult math problems".

Fermat's conjecture (History)

Fermat left no proof of the conjecture for all n, buthe did prove the special case n = 4. This reducedthe problem to proving the theorem for exponents nthat are prime numbers. Over the next two centuries(16371839), the conjecture was proven for onlythe primes 3, 5, and 7, although Sophie Germainproved a special case for all primes less than 100. Inthe mid-19th century, Ernst Kummer proved thetheorem for regular primes. Building on Kummer'swork and using sophisticated computer studies,

other mathematicians were able to prove the conjecture for all odd primes up to four million.The final proof of the conjecture for all n came in the late 20th century. In 1984, Gerhard Freysuggested the approach of proving the conjecture through a proof of the modularity theoremfor elliptic curves. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enoughof the modularity theorem to prove Fermat's Last Theorem, with the assistance of RichardTaylor. Wiles's achievement was reported widely in the popular press, and has beenpopularized in books and television programs.

Mathematical context

Pythagorean triples

Pythagorean triples are a set of three integers (a, b, c) that satisfy a special case of Fermat'sequation (n = 2)

Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). There are infinitely manysuch triples, and methods for generating such triples have been studied in many cultures,beginning with the Babylonians and later ancient Greek, Chinese and Indian mathematicians.The traditional interest in Pythagorean triples connects with the Pythagorean theorem; in its


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converse form, it states that a triangle with sides of lengths a, b and c has a right anglebetween the a and b legs when the numbers are a Pythagorean triple. Right angles havevarious practical applications, such as surveying, carpentry, masonry and construction.Fermat's Last Theorem is an extension of this problem to higher powers, stating that nosolution exists when the exponent 2 is replaced by any larger integer.

Diophantine equations

Fermat's equation xn + yn = zn is an example of a Diophantine equation. A Diophantineequation is a polynomial equation in which the solutions must be integers. Their name derivesfrom the 3rd-century Alexandrian mathematician, Diophantus, who developed methods fortheir solution. A typical Diophantine problem is to find two integers x and y such that theirsum, and the sum of their squares, equal two given numbers A and B, respectively:

Diophantus's major work is the Arithmetica, of which only a portion has survived. Fermat'sconjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica,which was translated into Latin and published in 1621 by Claude Bachet.

Diophantine equations have been studied for thousands of years. For example, the solutions tothe quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples,originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations,such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).Many Diophantine equations have a form similar to the equation of Fermat's Last Theoremfrom the point of view of algebra, in that they have no cross terms mixing two letters, withoutsharing its particular properties. For example, it is known that there are infinitely manypositive integers x, y, and z such that xn+ yn= zmwhere n and m are relatively prime naturalnumbers.

Fermat's conjecture

Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the famousmargin which was too small to contain Fermat's alleged proof of his "last theorem".

Problem II.8 of the Arithmetica asks how a given square number is split into two othersquares; in other words, for a given rational number k, find rational numbers uand vsuchthat k2= u2+ v2. Diophantus shows how to solve this sum-of-squares problem for k = 4(thesolutions being u = 16/5 and v = 12/5).

Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica nextto Diophantus' sum-of-squares problem:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et

generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominisfas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginisexiguitas non caperet.

it is impossible to separate a cube into two cubes, or a fourth power into two fourthpowers, or in general, any power higher than the second, into two like powers. I havediscovered a truly marvelous proof of this, which this margin is too narrow to contain.

Although Fermat's general proof is unknown, his proof of one case (n = 4) by infinite descenthas survived. Fermat posed the cases of n = 4and of n = 3as challenges to his mathematical


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correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis. However, in the lastthirty years of his life, Fermat never again wrote of his "truly marvellous proof" of the generalcase.

After Fermat's death in 1665, his son Clment-Samuel Fermat produced a new edition of thebook (1670) augmented with his father's comments. The margin note became known asFermat's Last Theorem, as it was the last of Fermat's asserted theorems to remain unproven.

Proofs for specific exponents

Only one mathematical proof by Fermat has survived, in which Fermat uses the technique ofinfinite descent to show that the area of a right triangle with integer sides can never equal thesquare of an integer. His proof is equivalent to demonstrating that the equation

has no primitive solutions in integers (no pairwise coprime solutions). In turn, this provesFermat's Last Theorem for the case n=4, since the equation a4+ b4= c4can be written as c4 b4= (a2)2. For a version of Fermat's proof by infinite descent, see Infinite descent#Non-solvability of r2 + s4 = t4. For various proofs by infinite descent, see Grant and Perella(1999), Barbara (2007), and Dolan (2011).

Alternative proofs of the case n = 4 were developed later by Frnicle de Bessy (1676),Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830),Schopis (1825), Terquem (1846), Joseph Bertrand (1851), Victor Lebesgue (1853, 1859,1862), Theophile Pepin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901),Gambioli (1901), Leopold Kronecker (1901), Bang (1905), Sommer (1907), Bottari (1908),Karel Rychlk (1910), Nutzhorn (1912), Robert Carmichael (1913), Hancock (1931), andVrnceanu (1966).

After Fermat proved the special case n =4, the general proof for all n required only that thetheorem be established for all odd prime exponents. In other words, it was necessary to prove

only that the equation an+ bn= cnhas no integer solutions (a, b, c) when n is an odd primenumber. This follows because a solution (a, b, c) for a given n is equivalent to a solution for allthe factors of n. For illustration, let nbe factored into dand e, n = de. The general equation

implies that (ad, bd, cd) is a solution for the exponent e

Thus, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that ithas no solutions for at least one prime factor of every n. All integers n > 2contain a factor of

4, or an odd prime number, or both. Therefore, Fermat's Last Theorem can be proven for all nif it can be proven for n = 4and for all odd primes (the only even prime number is the number2) p.

In the two centuries following its conjecture (16371839), Fermat's Last Theorem was provenfor three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect. In1770, Leonhard Euler gave a proof of p = 3, but his proof by infinite descent contained amajor gap. However, since Euler himself had proven the lemma necessary to complete theproof in other work, he is generally credited with the first proof. Independent proofs were


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published by Kausler (1802), Legendre (1823, 1830), Calzolari (1855), Gabriel Lam (1865),Peter Guthrie Tait (1872), Gnther (1878), Gambioli (1901), Krey (1909), Rychlk (1910),Stockhaus (1910), Carmichael (1915), Johannes van der Corput (1915), Axel Thue (1917),and Duarte (1944). The case p = 5was proven independently by Legendre and Peter Dirichletaround 1825. Alternative proofs were developed by Carl Friedrich Gauss (1875, posthumous),Lebesgue (1843), Lam (1847), Gambioli (1901), Werebrusow (1905), Rychlk (1910),van derCorput (1915), and Guy Terjanian (1987). The case p = 7was proven by Lam in 1839. His

rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs werepublished by Angelo Genocchi in 1864, 1874 and 1876. Alternative proofs were developed byThophile Ppin (1876) and Edmond Maillet (1897).

Fermat's Last Theorem has also been proven for the exponents n = 6, 10, and 14. Proofs for n= 6have been published by Kausler, Thue, Tafelmacher, Lind, Kapferer, Swift, and Breusch.Similarly, Dirichlet and Terjanian each proved the case n = 14, while Kapferer and Breuscheach proved the case n = 10. Strictly speaking, these proofs are unnecessary, since thesecases follow from the proofs for n = 3, 5, and 7, respectively. Nevertheless, the reasoning ofthese even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof forn = 14was published in 1832, before Lam's 1839 proof for n = 7.

Many proofs for specific exponents use Fermat's technique of infinite descent, which Fermatused to prove the case n = 4, but many do not. However, the details and auxiliary arguments

are often ad hoc and tied to the individual exponent under consideration. Since they becameever more complicated as p increased, it seemed unlikely that the general case of Fermat'sLast Theorem could be proven by building upon the proofs for individual exponents. Althoughsome general results on Fermat's Last Theorem were published in the early 19th century byNiels Henrik Abel and Peter Barlow, the first significant work on the general theorem was doneby Sophie Germain.

Sophie Germain

In the early 19th century, Sophie Germain developed several novel approaches to proveFermat's last theorem for all exponents. First, she defined a set of auxiliary primes constructed from the prime exponent pby the equation = 2hp+1, where h is any integer not

divisible by three. She showed that if no integers raised to the pth power were adjacentmodulo (the non-consecutivity condition), then must divide the product xyz. Her goal wasto use mathematical induction to prove that, for any given p, infinitely many auxiliary primes satisfied the non-consecutivity condition and thus divided xyz; since the product xyzcan haveat most a finite number of prime factors, such a proof would have established Fermat's LastTheorem. Although she developed many techniques for establishing the non-consecutivitycondition, she did not succeed in her strategic goal. She also worked to set lower limits on thesize of solutions to Fermat's equation for a given exponent p, a modified version of which waspublished by Adrien-Marie Legendre. As a byproduct of this latter work, she proved SophieGermain's theorem, which verified the first case of Fermat's Last Theorem for every odd primeexponent less than 100. Germain tried unsuccessfully to prove the first case of Fermat's LastTheorem for all even exponents, specifically for n = 2p, which was proven by Guy Terjanian in1977. In 1985, Leonard Adleman, Roger Heath-Brown and tienne Fouvry proved that the firstcase of Fermat's Last Theorem holds for infinitely many odd primes p.


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Ernst Kummer and the theory of ideals

In 1847, Gabriel Lam outlined a proof of Fermat's Last Theorem based on factoring theequation xp+ yp= zpin complex numbers, specifically the cyclotomic field based on the rootsof the number 1. His proof failed, however, because it assumed incorrectly that such complexnumbers can be factored uniquely into primes, similar to integers. This gap was pointed out

immediately by Joseph Liouville, who later read a paper that demonstrated this failure ofunique factorisation, written by Ernst Kummer.

Kummer set himself the task of determining whether the cyclotomic field could be generalizedto include new prime numbers such that unique factorisation was restored. He succeeded inthat task by developing the ideal numbers. Using the general approach outlined by Lam,Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However,he could not prove the theorem for the exceptional primes (irregular primes) whichconjecturally occur approximately 39% of the time; the only irregular primes below 100 are37, 59 and 67.

Mordell conjecture

In the 1920s, Louis Mordell posed a conjecture that implied that Fermat's equation has atmost a finite number of nontrivial primitive integer solutions if the exponent n is greater thantwo. This conjecture was proven in 1983 by Gerd Faltings, and is now known as Faltings'theorem.

Computational studies

In the latter half of the 20th century, computational methods were used to extend Kummer'sapproach to the irregular primes. In 1954, Harry Vandiver used a SWAC computer to proveFermat's Last Theorem for all primes up to 2521. By 1978, Samuel Wagstaff had extended thisto all primes less than 125,000. By 1993, Fermat's Last Theorem had been proven for allprimes less than four million.

Connection with elliptic curves

The ultimately successful strategy for proving Fermat's Last Theorem was by proving themodularity theorem. The strategy was first described by Gerhard Frey in 1984. Frey noted thatif Fermat's equation had a solution (a, b, c) for exponent p > 2, the corresponding ellipticcurve

would have such unusual properties that the curve would likely violate the modularity

theorem. This theorem, first conjectured in the mid-1950s and gradually refined through the1960s, states that every elliptic curve is modular, meaning that it can be associated with aunique modular form.

Following this strategy, the proof of Fermat's Last Theorem required two steps. First, it wasnecessary to show that Frey's intuition was correct, that the above elliptic curve is always non-modular. Frey did not succeed in proving this rigorously; the missing piece was identified byJean-Pierre Serre. This missing piece, the so-called "epsilon conjecture", was proven by KenRibet in 1986. Second, it was necessary to prove a special case of the modularity theorem.This special case (for semistable elliptic curves) was proven by Andrew Wiles in 1995.


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Thus, the epsilon conjecture showed that any solution to Fermat's equation could be used togenerate a non-modular semistable elliptic curve, whereas Wiles' proof showed that all suchelliptic curves must be modular. This contradiction implies that there can be no solutions toFermat's equation, thus proving Fermat's Last Theorem.

Wiles' general proof

Ribet's proof of the epsilon conjecture in 1986 accomplished the first half of Frey's strategy forproving Fermat's Last Theorem. Upon hearing of Ribet's proof, Andrew Wiles decided tocommit himself to accomplishing the second half: proving a special case of the modularitytheorem (then known as the TaniyamaShimura conjecture) for semistable elliptic curves.Wiles worked on that task for six years in almost complete secrecy. He based his initialapproach on his area of expertise, Horizontal Iwasawa theory, but by the summer of 1991,this approach seemed inadequate to the task. In response, he exploited an Euler systemrecently developed by Victor Kolyvagin and Matthias Flach. Since Wiles was unfamiliar withsuch methods, he asked his Princeton colleague, Nick Katz, to check his reasoning over thespring semester of 1993.

By mid-1993, Wiles was sufficiently confident of his results that he presented them in threelectures delivered on June 2123, 1993 at the Isaac Newton Institute for MathematicalSciences. Specifically, Wiles presented his proof of the TaniyamaShimura conjecture forsemistable elliptic curves; together with Ribet's proof of the epsilon conjecture, this impliedFermat's Last Theorem. However, it soon became apparent that Wiles' initial proof wasincorrect. A critical portion of the proof contained an error in a bound on the order of aparticular group. The error was caught by several mathematicians refereeing Wiles'manuscript, including Katz, who alerted Wiles on 23 August 1993.

Wiles and his former student Richard Taylor spent almost a year trying to repair the proof,without success. On 19 September 1994, Wiles had a flash of insight that the proof could besaved by returning to his original Horizontal Iwasawa theory approach, which he hadabandoned in favour of the KolyvaginFlach approach. On 24 October 1994, Wiles submittedtwo manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic

properties of certain Hecke algebras", the second of which was co-authored with Taylor. Thetwo papers were vetted and published as the entirety of the May 1995 issue of the Annals ofMathematics. These papers established the modularity theorem for semistable elliptic curves,the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.

Exponents other than positive integers

Rational exponents

All solutions of the Diophantine equation when n=1 were computed byLenstra in 1992. In the case in which the mth roots are required to be real and positive, allsolutions are given by

for positive integers r, s, t with s and t coprime.


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In 2004, for n>2, Bennett, Glass, and Szekely proved that if gcd(n,m)=1, then there areinteger solutions if and only if 6divides m, and a1 / m, b1 / m, and c-1 / mare different complex6th roots of the same real number.

Negative exponents

n = 1

All primitive (pairwise coprime) integer solutions to can be written as

for positive, coprime integers m, n.

n = 2

The case n = 2 also has an infinitude of solutions, and these have a geometricinterpretation in terms of right triangles with integer sides and an integer altitude to

the hypotenuse. All primitive solutions to are given by

for coprime integers u, vwith v > u. The geometric interpretation is that a and b arethe integer legs of a right triangle and d is the integer altitude to the hypotenuse.Then the hypotenuse itself is the integer

so (a, b, c) is a Pythagorean triple.

Integer n < 2

There are no solutions in integers for for integer n < 2. If there were,

the equation could be multiplied through by to obtain

, which is impossible by Fermat's Last Theorem.


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Did Fermat possess a general proof?

The mathematical techniques used in Fermat's "marvelous" proof are unknown. Only onedetailed proof of Fermat has survived, the above proof that no three coprime integers (x, y, z)satisfy the equation x4 y4 = z4.

Taylor and Wiles's proof relies on mathematical techniques developed in the twentieth century,which would be alien to mathematicians who had worked on Fermat's Last Theorem even acentury earlier. Fermat's alleged "marvellous proof", by comparison, would have had to beelementary, given mathematical knowledge of the time, and so could not have been the sameas Wiles' proof. Most mathematicians and science historians doubt that Fermat had a validproof of his theorem for all exponents n.

Harvey Friedman's grand conjecture implies that Fermat's last theorem can be proved inelementary arithmetic, a rather weak form of arithmetic with addition, multiplication,exponentiation, and a limited form of induction for formulas with bounded quantifiers. Anysuch proof would be elementary but possibly too long to write down.

Monetary prizes

In 1816 and again in 1850, the French Academy of Sciences offered a prize for a general proofof Fermat's Last Theorem. In 1857, the Academy awarded 3000 francs and a gold medal toKummer for his research on ideal numbers, although he had not submitted an entry for theprize. Another prize was offered in 1883 by the Academy of Brussels.

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed100,000 marks to the Gttingen Academy of Sciences to be offered as a prize for a completeproof of Fermat's Last Theorem. On 27 June 1908, the Academy published nine rules forawarding the prize. Among other things, these rules required that the proof be published in apeer-reviewed journal; the prize would not be awarded until two years after the publication;and that no prize would be given after 13 September 2007, roughly a century after thecompetition was begun. Wiles collected the Wolfskehl prize money, then worth $50,000, on 27

June 1997.Prior to Wiles' proof, thousands of incorrect proofs were submitted to the Wolfskehlcommittee, amounting to roughly 10 feet (3 meters) of correspondence. In the first year alone(19071908), 621 attempted proofs were submitted, although by the 1970s, the rate ofsubmission had decreased to roughly 34 attempted proofs per month. According to F.Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methodstaught in schools, and often submitted by "people with a technical education but a failedcareer". In the words of mathematical historian Howard Eves, "Fermat's Last Theorem has thepeculiar distinction of being the mathematical problem for which the greatest number ofincorrect proofs have been published."


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The 23 EnigmaWikipedia.org

The 23 enigma refers to the belief that most incidents and events aredirectly connected to the number 23, some modification of thenumber 23, or a number related to the number 23.

Origins

Robert Anton Wilson cites William S. Burroughs as being the firstperson to believe in the 23 enigma. Wilson, in an article in ForteanTimes, related the following story:

I first heard of the 23 enigma from William S Burroughs, author of Naked Lunch, NovaExpress, etc. According to Burroughs, he had known a certain Captain Clark, around 1960 inTangier, who once bragged that he had been sailing 23 years without an accident. That veryday, Clarks ship had an accident that killed him and everybody else aboard. Furthermore,while Burroughs was thinking about this crude example of the irony of the gods that evening,a bulletin on the radio announced the crash of an airliner in Florida, USA. The pilot wasanother captain Clark and the flight was Flight 23.

Burroughs wrote a short story in 1967 called "23 Skidoo." The term "23 skidoo" waspopularized in the early 1920s and means "it's time to leave while the getting is good." Itappeared in newspapers as early as 1906.

Discordianism

The Principia Discordia states that "All things happen in fives, or are divisible by or aremultiples of five, or are somehow directly or indirectly appropriate to 5"this is referred to asthe Law of Fives. The 23 Enigma is regarded as a corollary of this law. It can be seen inRobert Anton Wilson and Robert Shea's The Illuminatus! Trilogy (therein called the "23/17phenomenon"), Wilson's Cosmic Trigger I: The Final Secret of the Illuminati (therein called"The Law of fives" and "The 23 Enigma"), Arthur Koestler's Challenge of Chance, as well as thePrincipia Discordia. In these works, 23 is considered lucky, unlucky, sinister, strange, or

sacred to the goddess Eris or to the unholy gods of the Cthulhu Mythos.

As with most numerological claims, the enigma can be viewed as an example of apophenia,selection bias, and confirmation bias. In interviews, Wilson acknowledged the self-fulfillingnature of the enigma, implying that the real value of the Laws of Fives and Twenty-threes liesin their demonstration of the mind's power to perceive "truth" in nearly anything.

When you start looking for something you tend to find it. This wouldn't be like SimonNewcomb, the great astronomer, who wrote a mathematical proof that heavier than air flightwas impossible and published it a day before the Wright brothers took off. I'm talking aboutpeople who found a pattern in nature and wrote several scientific articles and got it acceptedby a large part of the scientific community before it was generally agreed that there was nosuch pattern, it was all just selective perception."

In the Illuminatus!Trilogy, he expresses the same view: that one can find a numerologicalsignificance to anything, provided "sufficient cleverness."


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13Wikipedia.org

13 (thirteen) is a natural number after 12 and before 14. It is the

smallest number with eight letters in its name spelled out in English.

Strikingly similar folkloric aspects of the number 13 have been notedin various cultures around the world: one theory is that this is due tothe cultures employing lunar-solar calendars (there are approximately12.41 lunations per solar year, and hence 12 "true months" plus asmaller, and often portentous, thirteenth month). This can bewitnessed, for example, in the "Twelve Days of Christmas" of WesternEuropean tradition.

In languages

Grammar

In all Germanic languages (such as English and German), 13 is the first compound number(in German dreizehn); the numbers 11 and 12 have their own names (in German elf andzwlf).

The Romance languages use different systems: In Italian, 11 is the first compound number(ndici), while in Spanish und Portuguese, the numbers up to and including 15 (Spanishquince, Portuguese quinze), in French up to and including 16 (seize) and in Romanian up toand including 19 have their own names.

Like in Italian, in many other languages, 11 is the first compound number, e.g. in Arabic,Chinese, Hungarian, Japanese, Swahili.

Like in Romanian, in Lithuanian and Slavic languages, the numbers from 11-19 have their

own names.

In Hindi-Urdu, nearly every number from 199 is irregular and needs to be memorized as aseparate numeral.

Spelling

In Germany, according to an old rule, 13 as the first compound number was the first numberto be written in digits; the numbers 0 through 12 were to be spelled out. The Duden (theGerman standard dictionary) now calls this rule outdated and no longer valid, but manywriters still follow it.

For the English language, different systems are used: Sometimes it is recommended to spell

out numbers up to and including nine or ten or twelve, like formerly in German, or evenninety-nine or one hundred. Another system spells out all numbers written in one or twowords (sixteen, twenty-seven, fifteen thousand, but 372 or 15,001 ).


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In religion

Roman Catholicism

The apparitions of the Virgin of Ftima in 1917 were claimed to occur on the 13th day of sixconsecutive months.

In Catholic devotional practice, the number thirteen is also associated with Saint Anthony ofPadua, since his feast day falls on June 13. A traditional devotion called the Thirteen Tuesdaysof St. Anthony involves praying to the saint every Tuesday over a period of thirteen weeks.Another devotion, St. Anthony's Chaplet, consists of thirteen decades of three beads each.

Sikhism

According to famous Sakhi (Evidence) or story of Guru Nanak Dev Ji, when he was anaccountant at a town of Sultanpur Lodhi, he was distributing grocery to people and when hegave groceries to the 13th person he stopped there because in Gurmukhi and Hindi the word13 is called Terah, which means yours. And Guru Nanak Dev Ji kept on saying, "Yours, yours,yours..." remembering God. People reported to the emperor that Guru Nanak Dev Ji was

giving out free food to the people. When treasures were checked, there was more money thanbefore.

The Vaisakhi which commemorates the creation of "Khalsa" or pure Sikh was celebrated onApril 13 for many years.

Judaism

In Judaism, 13 signifies the age at which a boy matures and becomes a Bar Mitzvah,i.e., a full member of the Jewish faith (is qualified to be counted as a member ofMinyan).

The number of principles of Jewish faith according to Maimonides. According to Rabbinic commentary on the Torah, God has 13 Attributes of Mercy. The number of circles, or "nodes", that make up Metatron's Cube in Kaballistic

teachings.

Zoroastrianism

Evidently the number 13 had been considered sinister and wicked in ancient Iranian civilizationand Zoroastrianism; Since beginning of Nourooz tradition, the 13th day of each new Iranianyear is called Sizdah Be-dar and this tradition is still alive among Iranian people both insidemodern Iran and abroad. Since Sizdah Be-dar is the 13th day of the year, it is considered aday which evil's power might cause difficulties for people; Therefore people desert the citiesand urban areas for one day and camp in the countryside. Even in the current era after 1979Revolution and despite the wishes of Islamic government this day is officially holiday all overIran and its traditions are practiced by the majority of people.

Islam

In Shia Islam 13 signifies the 13th day of the month of Rajab (Lunar calendar) which is thebirth of Imam Ali. 13 also is a total of 1 Prophet and 12 Imams in the Shia school of thought.


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A repressed lunar cult

In ancient cultures, the number 13 represented femininity, because it corresponded to thenumber of lunar (menstrual) cycles in a year (13 x 28 = 364 days). The theory is that, as thesolar calendar triumphed over the lunar, the number thirteen became anathema.

Lucky 13

Several successful sports figures have worn the number 13. Park Ji-Sung, South-Koreanfootballer and midfielder for Queens Park Rangers wears number 13. Ozzie Guilln, managerof the 2005 World Series Champion Chicago White Sox, has worn the number throughout hisbaseball career. Alex Rodriguez began wearing it upon joining the New York Yankees (three,the number he had previously worn, is retired by the Bronx Bombers to honor Babe Ruth).Dan Marino, an American football player known for passing the 3rd most yards in NFL history,wore the number 13. Basketball great Wilt Chamberlain wore the number 13 on his jerseythroughout his NBA career. Also, FIBA rules require a player to wear the number ininternational competitions (only numbers from 4 to 15 could be worn, and as there are 12players, one must wear 13); Chris Mullin, who wore No. 20 in college and No. 17 in the NBA,wore No. 13 for both (1984 and 1992) of his Olympic appearances. Shaquille O'Neal wore No.13 in 1996; Tim Duncan wore No. 13 in 2004. Steve Nash wore it for most of his basketballcareer. Yao Ming wore it in the 2008 Olympics in Beijing. Chris Paul wore the number 13 forboth the 2008 and 2012 Olympics. Mats Sundin, Pavel Datsyuk, Bill Guerin, and MichaelCammalleri wear 13 in the NHL. One of Iceland's all time best handball players, SigururSveinsson, wore the number 13 when he played for the national team. In association football,both Gerd Mller and Michael Ballack have favoured the number 13, among others.

In Italy, 13 is also considered to be a lucky number, although in Campania the expression'tredici' (meaning 13) is said when one considers their luck to have turned for the worse.

Some people even have 13 tattooed onto them to represent the lucky number.

Music

American born Horror-Punk singer and musician Joseph Poole (Murderdolls) uses the nameWednesday 13 as his stage name, taking "Wednesday" from the girl Wednesday from theAddams Family and 13 from Friday the 13th.

American country-pop singer-songwriter Taylor Swift was born on December 13. Sheconsiders 13 her lucky number due to lucky events happening to her when the numberappears (her first album going gold in 13 weeks, being seated at awards shows in the 13thseat, row or section). She also wears the number written on her hand at her concerts so shehas it with her everywhere she goes.

There are 13 notes, by inclusive counting, in a full chromatic musical octave.


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Other

Colgate University also considers 13 to be a lucky number. They were founded in 1819 by 13men with 13 dollars, 13 prayers and 13 articles. (To this day, members of the Colgatecommunity consider the number 13 a good omen.) In fact, the campus address is 13 OakDrive in Hamilton, New York, and the male a cappella group is called the Colgate 13.

In the Mayan Tzolk'in calendar, trecenas mark cycles of 13 day periods. The pyramids are alsoset up in 9 steps divided into 7 days and 6 nights, 13 days total.

In a tarot card deck, XIII is the card of Death, usually picturing the Pale horse with its rider.

Coperos

The number 13 in the Coperos religion (small culture in Brazil) is like a God number. Allcoperos must know that this number can save humankind.

History

The American flag has 13 stripes in honor of the first 13 colonies.

Apollo 13 was a NASA Moon mission famous for being a "successful failure" in that while thecrew were unable to land on the Moon as planned due to a technical malfunction, they werereturned safely home.

Age 13

In Judaism, 13 signifies the age at which a boy matures and becomes a Bar Mitzvah, i.e., a fullmember of the Jewish faith (is qualified to be counted as a member of Minyan).


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GoogolWikipedia.org

A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

The term was coined in 1938 by 9-year-old Milton Sirotta (19291981), nephew of Americanmathematician Edward Kasner. Kasner popularized the concept in his book Mathematics andthe Imagination (1940).

Other names for googol include ten duotrigintillion on the short scale, ten thousandsexdecillionon the long scale, or ten sexdecilliardon the Peletier long scale.

A googol has no particular significance in mathematics, but is useful when comparing withother very large quantities such as the number of subatomic particles in the visible universe orthe number of hypothetically possible chess moves. Edward Kasner used it to illustrate the

difference between an unimaginably large number and infinity, and in this role it is sometimesused in teaching mathematics.


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Magic SquareWikipedia.org

In recreational mathematics, a magic square of order n isan arrangement of n2 numbers, usually distinct integers,in a square, such that the n numbers in all rows, all

columns, and both diagonals sum to the same constant.A normal magic square contains the integers from 1 ton2. The term "magic square" is also sometimes used torefer to any of various types of word square.

Normal magic squares exist for all orders n 1 except n= 2, although the case n = 1 is trivial, consisting of asingle cell containing the number 1. The smallestnontrivial case, shown below, is of order 3.

The constant sum in every row, column and diagonal is called the magic constant or magicsum, M. The magic constant of a normal magic square depends only on n and has the value

For normal magic squares of order n = 3, 4, 5, ..., the magic constants are:

15, 34, 65, 111, 175, 260, ... (sequence A006003 in OEIS).

History

Left: Iron plate with an order 6 magic square in Arabic

numbers from China, dating to the Yuan Dynasty(1206-1368).

Magic squares were known to Chinese mathematicians,as early as 650 BCE and Arab mathematicians, possiblyas early as the 7th century, when the Arabs conquerednorthwestern parts of the Indian subcontinent andlearned Indian mathematics and astronomy, includingother aspects of combinatorial mathematics. The firstmagic squares of order 5 and 6 appear in anencyclopedia from Baghdad circa 983 CE, theEncyclopedia of the Brethren of Purity (Rasa'il Ihkwanal-Safa); simpler magic squares were known to several

earlier Arab mathematicians. Some of these squares were later used in conjunction with magic

letters as in (Shams Al-ma'arif) to assist Arab illusionists and magicians.


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Europe

Left: This page from Athanasius Kircher's OedipusAegyptiacus (1653) belongs to a treatise on magicsquares and shows the Sigillum Iovis associated with

JupiterIn 1300, building on the work of the Arab Al-Buni,Greek Byzantine scholar Manuel Moschopoulos wrote amathematical treatise on the subject of magic squares,leaving out the mysticism of his predecessors.Moschopoulos is thought to be the first Westerner tohave written on the subject. In the 1450s the ItalianLuca Pacioli studied magic squares and collected a largenumber of examples.

In about 1510 Heinrich Cornelius Agrippa wrote DeOcculta Philosophia, drawing on the Hermetic andmagical works of Marsilio Ficino and Pico della

Mirandola, and in it he expounded on the magicalvirtues of seven magical squares of orders 3 to 9, eachassociated with one of the astrological planets. Thisbook was very influential throughout Europe until the

counter-reformation, and Agrippa's magic squares, sometimes called Kameas, continue to beused within modern ceremonial magic in much the same way as he first prescribed.
http://en.wikipedia.org/wiki/File:Sigillum_Iovis.jpghttp://en.wikipedia.org/wiki/File:Sigillum_Iovis.jpg


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Left: The derivation of the sigil ofHagiel, the planetary intelligenceof Venus, drawn on the magicsquare of Venus. Each Hebrewletter provides a numerical value,giving the vertices of the sigil.

The most common use for these Kameas is to provide a pattern upon which to construct thesigils of spirits, angels or demons; the letters of the entity's name are converted into numbers,and lines are traced through the pattern that these successive numbers make on the kamea.In a magical context, the term magic square is also applied to a variety of word squares or

number squares found in magical grimoires, including some that do not follow any obviouspattern, and even those with differing numbers of rows and columns. They are generallyintended for use as talismans. For instance the following squares are: The Sator square, oneof the most famous magic squares found in a number of grimoires including the Key ofSolomon; a square "to overcome envy", from The Book of Power; and two squares from TheBook of the Sacred Magic of Abramelin the Mage , the first to cause the illusion of a superbpalace to appear, and the second to be worn on the head of a child during an angelicinvocation:
http://en.wikipedia.org/wiki/File:Hagiel_sigil_derivation.svghttp://en.wikipedia.org/wiki/File:Hagiel_sigil_derivation.svg


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Albrecht Drer's magic square

The order-4 magic square in Albrecht Drer's engravingMelencolia Iis believed to be the first seen in Europeanart. It is very similar to Yang Hui's square, which wascreated in China about 250 years before Drer's time.

The sum 34 can be found in the rows, columns,diagonals, each of the quadrants, the center foursquares, and the corner squares(of the 4x4 as well asthe four contained 3x3 grids). This sum can also befound in the four outer numbers clockwise from thecorners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the twosolutions of the 4 queens puzzle), the two sets of foursymmetrical numbers (2+8+9+15 and 3+5+12+14),the sum of the middle two entries of the two outercolumns and rows (5+9+8+12 and 3+2+15+14), andin four kite or cross shaped quartets(3+5+11+15,

2+10+8+14, 3+9+7+15, and 2+6+12+14). The two numbers in the middle of the bottomrow give the date of the engraving: 1514. The numbers 1 and 4 at either side of the date

correspond to, in English, the letters 'A' and 'D' which are the initials of the artist.Drer's magic square can also be extended to a magic cube.

Drer's magic square and his Melencolia I both also played large roles in Dan Brown's 2009novel, The Lost Symbol.

Sagrada Famlia magic square

Left: A magic square on the Sagrada Famliachurch faade

The Passion faade of the Sagrada Famlia

church in Barcelona, designed by sculptorJosep Subirachs, features a 44 magicsquare:

The magic constant of the square is 33, theage of Jesus at the time of the Passion.Structurally, it is very similar to theMelancholia magic square, but it has had thenumbers in four of the cells reduced by 1.

While having the same pattern of summation, this is not a normal magicsquare as above, as two numbers (10 and 14) are duplicated and two

(12 and 16) are absent, failing the 1n2 rule.

Similarly to Drer's magic square, the Sagrada Familia's magic squarecan also be extended to a magic cube.
http://en.wikipedia.org/wiki/File:Ms_sf_2.jpghttp://en.wikipedia.org/wiki/File:Albrecht_D%C3%BCrer_-_Melencolia_I_(detail).jpghttp://en.wikipedia.org/wiki/File:Ms_sf_2.jpghttp://en.wikipedia.org/wiki/File:Albrecht_D%C3%BCrer_-_Melencolia_I_(detail).jpghttp://en.wikipedia.org/wiki/File:Ms_sf_2.jpghttp://en.wikipedia.org/wiki/File:Albrecht_D%C3%BCrer_-_Melencolia_I_(detail).jpg


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Types and construction

There are many ways to construct magic squares, but the standard (and most simple) way isto follow certain configurations/formulas which generate regular patterns. Magic squares existfor all values of n, with only one exception: it is impossible to construct a magic square oforder 2. Magic squares can be classified into three types: odd, doubly even (n divisible by

four) and singly even (n even, but not divisible by four). Odd and doubly even magic squaresare easy to generate; the construction of singly even magic squares is more difficult butseveral methods exist, including the LUX method for magic squares (due to John HortonConway) and the Strachey method for magic squares.

Group theory was also used for constructing new magic squares of a given order from one ofthem, please see.

The number of different nn magic squares for n from 1 to 5, not counting rotations andreflections:

1, 0, 1, 880, 275305224 (sequence A006052 in OEIS).

The number for n = 6 has been estimated to 1.77451019.

Method for constructing a magic square of odd order

Yang Hui's construction methodA method for constructing magic squares of odd order waspublished by the French diplomat de la Loubre in his book A new historical relation of thekingdom of Siam (Du Royaume de Siam, 1693), under the chapter entitled The problem of themagical square according to the Indians. The method operates as follows:

Starting from the central column of the first row with the number 1, the fundamentalmovement for filling the squares is diagonally up and right, one step at a time. If a filledsquare is encountered, one moves vertically down one square instead, then continuing asbefore. When a move would leave the square, it is wrapped around to the last row or firstcolumn, respectively.

Starting from other squares rather than the central column of the first row is possible, butthen only the row and column sums will be identical and result in a magic sum, whereas thediagonal sums will differ. The result will thus be a semimagic square and not a true magicsquare. Moving in directions other than north east can also result in magic squares.


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The following formulae help construct magic squares of odd order

Example:

The "Middle Number" is always in the diagonal bottom left to top right.The "Last Number" is always opposite the number 1 in an outside column or row.


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A method of constructing a magic square of doubly even order

Doubly even means that n is an even multiple of an even integer; or 4p (e.g. 4, 8, 12), wherep is an integer.

Generic patternAll the numbers are written in order from left to right across each row in

turn, starting from the top left hand corner. The resulting square is also known as a mysticsquare. Numbers are then either retained in the same place or interchanged with theirdiametrically opposite numbers in a certain regular pattern. In the magic square of order four,the numbers in the four central squares and one square at each corner are retained in thesame place and the others are interchanged with their diametrically opposite numbers.

A construction of a magic square of order 4 Go left to right through the square fillingcounting and filling in on the diagonals only. Then continue by going left to right from the topleft of the table and fill in counting down from 16 to 1. As shown below.

An extension of the above example for Orders 8 and 12First generate a "truth" table,where a '1' indicates selecting from the square where the numbers are written in order 1 to n2(left-to-right, top-to-bottom), and a '0' indicates selecting from the square where the numbersare written in reverse order n2 to 1. For M = 4, the "truth" table is as shown below, (thirdmatrix from left.)

Note that a) there are equal number of '1's and '0's; b) each row and each column are"palindromic"; c) the left- and right-halves are mirror images; and d) the top- and bottom-halves are mirror images (c & d imply b.) The truth table can be denoted as (9, 6, 6, 9) forsimplicity (1-nibble per row, 4 rows.) Similarly, for M=8, two choices for the truth table are

(A5, 5A, A5, 5A, 5A, A5, 5A, A5) or (99, 66, 66, 99, 99, 66, 66, 99) (2-nibbles per row, 8rows.) For M=12, the truth table (E07, E07, E07, 1F8, 1F8, 1F8, 1F8, 1F8, 1F8, E07, E07,E07) yields a magic square (3-nibbles per row, 12 rows.) It is possible to count the number ofchoices one has based on the truth table, taking rotational symmetries into account.


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Medjig-method of constructing magic squares of even number of rows

This method is based on a 2006 published mathematical game called medjig (author: WillemBarink, editor: Philos-Spiele). The pieces of the medjig puzzle are squares divided in fourquadrants on which the numbers 0, 1, 2 and 3 are dotted in all sequences. There are 18squares, with each sequence occurring 3 times. The aim of the puzzle is to take 9 squares out

of the collection and arrange them in a 3 x 3 "medjig-square" in such a way that each row andcolumn formed by the quadrants sums to 9, along with the two long diagonals.

The medjig method of constructing a magic square of order 6 is as follows:

Construct any 3 x 3 medjig-square (ignoring the original game's limit on the number of timesthat a given sequence is used).Take the 3 x 3 magic square and divide each of its squares into four quadrants.Fill these quadrants with the four numbers from 1 to 36 that equal the original number modulo9, i.e. x+9y where x is the original number and y is a number from 0 to 3, following thepattern of the medjig-square.

Example:

Similarly, for any larger integer N, a magic square of order 2N can be constructed from any Nx N medjig-square with each row, column, and long diagonal summing to 3N, and any N x Nmagic square (using the four numbers from 1 to 4N^2 that equal the original number moduloN^2).


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Construction of panmagic squares

Any number p in the order-n square can be uniquely written in the form p = an + r, with rchosen from {1,...,n}. Note that due to this restriction, a and r are not the usual quotient andremainder of dividing p by n. Consequently the problem of constructing can be split in two

problems easier to solve. So, construct two matching square grids of order n satisfyingpanmagic properties, one for the a-numbers (0,..., n1), and one for the r-numbers (1,...,n).This requires a lot of puzzling, but can be done. When successful, combine them into onepanmagic square. Van den Essen and many others supposed this was also the way BenjaminFranklin (17061790) constructed his famous Franklin squares. Three panmagic squares areshown below. The first two squares have been constructed April 2007 by Barink, the third oneis some years older, and comes from Donald Morris, who used, as he supposes, the Franklinway of construction.

The order 8 square satisfies all panmagic properties, including the Franklin ones. It consists of4 perfectly panmagic 4x4 units. Note that both order 12 squares show the property that any

row or column can be divided in three parts having a sum of 290 (= 1/3 of the total sum of arow or column). This property compensates the absence of the more standard panmagicFranklin property that any 1/2 row or column shows the sum of 1/2 of the total. For the restthe order 12 squares differ a lot.The Barink 12x12 square is composed of 9 perfectly panmagic4x4 units, moreover any 4 consecutive numbers starting on any odd place in a row or columnshow a sum of 290. The Morris 12x12 square lacks these properties, but on the contraryshows constant Franklin diagonals. For a better understanding of the constructing decomposethe squares as described above, and see how it was done. And note the difference betweenthe Barink constructions on the one hand, and the Morris/Franklin construction on the otherhand.

In the book Mathematics in the Time-Life Science Library Series, magic squares by Euler andFranklin are shown. Franklin designed this one so that any four-square subset (any fourcontiguous squares that form a larger square, or any four squares equidistant from the center)

total 130. In Euler's square, the rows and columns each total 260, and halfway they total130and a chess knight, making its L-shaped moves on the square, can touch all 64 boxes inconsecutive numerical order.


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Construction similar to the Kronecker Product

There is a method reminiscent of the Kronecker product of two matrices, that builds an nm xnm magic square from an n x n magic square and an m x m magic square.

The construction of a magic square using genetic algorithms

A magic square can be constructed using genetic algorithms. In this process an initialpopulation of magic squares with random values are generated. The fitness scores of theseindividual magic squares are calculated based on the degree of deviation in the sums of therows, columns, and diagonals. The population of magic squares reproduce by exchangingvalues, together with some random mutations. Those squares with a higher fitness score aremore likely to reproduce. The next generation of the magic square population is againcalculated for their fitness, and this process continues until a solution has been found or atime limit has been reached.

Generalizations

Extra constraints

Certain extra restrictions can be imposed on magic squares. If not only the main diagonals butalso the broken diagonals sum to the magic constant, the result is a panmagic square. Ifraising each number to certain powers yields another magic square, the result is a bimagic, atrimagic, or, in general, a multimagic square.

Different constraints

Sometimes the rules for magic squares are relaxed, so that only the rows and columns but notnecessarily the diagonals sum to the magic constant (this is usually called a semimagicsquare).

In heterosquares and antimagic squares, the 2n + 2 sums must all be different.


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Multiplicative magic squares

Instead of adding the numbers in each row, column and diagonal, one can apply some otheroperation. For example, a multiplicative magic square has a constant product of numbers. Amultiplicative magic square can be derived from an additive magic square by raising 2 (or anyother integer) to the power of each element. For example, the original Lo-Shu magic square

becomes:

Other examples of multiplicative magic squares include:

Ali Skalli's non iterative method of construction is also applicable to multiplicative magicsquares. On the 7x7 example below, the products of each line, each column and each diagonalis 6,227,020,800.


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Multiplicative magic squares of complex numbers

Still using Ali Skalli's non iterative method, it is possible to produce an infinity of multiplicativemagic squares of complex numbers belonging to set. On the example below, the real andimaginary parts are integer numbers, but they can also belong to the entire set of realnumbers . The product is: 352,507,340,640 400,599,719,520 i.

Other magic shapes

Other shapes than squares can be considered. The general case is to consider a design with Nparts to be magic if the N parts are labeled with the numbers 1 through N and a number ofidentical sub-designs give the same sum. Examples include magic dodecahedrons, magictriangles magic stars, and magic hexagons. Going up in dimension results in magic cubes,magic tesseracts and other magic hypercubes.

Edward Shineman has developed yet another design in the shape of magic diamonds.

Possible magic shapes are constrained by the number of equal-sized, equal-sum subsets of thechosen set of labels. For example, if one proposes to form a magic shape labeling the partswith {1, 2, 3, 4}, the sub-designs will have to be labeled with {1,4} and {2,3}.

Other component elements

Magic squares may be constructed which contain geometric shapes rather than numbers, as inthe "geomagic squares" introduced by Lee Sallows.

Combined extensions

One can combine two or more of the above extensions, resulting in such objects asmultiplicative multimagic hypercubes. Little seems to be known about this subject.


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Related problems

Over the years, many mathematicians, including Euler, Cayley and Benjamin Franklin haveworked on magic squares, and discovered fascinating relations.

Magic square of primes

Rudolf Ondrejka (19282001) discovered the following 3x3 magic square of primes, in thiscase nine Chen primes:

The GreenTao theorem implies that there are arbitrarily large magic squares consisting ofprimes.

Using Ali Skalli's non-iterative method of magic squares construction, it is easy to createmagic squares of primes of any dimension. In the example below, many symmetries appear(including all sorts of crosses), as well as the horizontal and vertical translations of all those.The magic constant is 13665.

It is believed that an infinite number of Skalli's magic squares of prime exist, but nodemonstration exists to date. However, it is possible to easily produce a considerable numberof them, not calculable in the absence of demonstration.

n-Queens problem

In 1992, Demirrs, Rafraf, and Tanik published a method for converting some magic squaresinto n-queens solutions, and vice versa.


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*Square Root of -1Wikipedia.org

[text ]


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*InfinityWikipedia.org

[text ]


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Mbius StripWikipedia.org

A Mbius strip made with a piece of paper and

tape. If an ant were to crawl along the lengthof this strip, it would return to its startingpoint having traversed every part of the strip(on both sides of the original paper) withoutever crossing an edge.

The Mbius strip or Mbius band (pronounced UK: /m:bis/ or US: /mobis/ in English,[m:bis] in German) (alternatively written Mobius or Moebius in English) is a surface with

only one side and only one boundary component. The Mbius strip has the mathematicalproperty of being non-orientable. It can be realized as a ruled surface. It was discoveredindependently by the German mathematicians August Ferdinand Mbius and Johann BenedictListing in 1858.

A model can easily be created by taking a paper strip and giving it a half-twist, and thenjoining the ends of the strip together to form a loop. In Euclidean space there are in fact twotypes of Mbius strips depending on the direction of the half-twist: clockwise andcounterclockwise. That is to say, it is a chiral object with "handedness" (right-handed or left-handed).

It is straightforward to find algebraic equations the solutions of which have the topology of aMbius strip, but in general these equations do not describe the same geometric shape thatone gets from the twisted paper model described above. In particular, the twisted paper modelis a developable surface (it has zero Gaussian curvature). A system of differential-algebraicequations that describes models of this type was published in 2007 together with its numericalsolution.

The Euler characteristic of the Mbius strip is zero.


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Mbius band with flat edge

The edge of a Mbius strip is topologically equivalent to the circle. Under the usualembeddings of the strip in Euclidean space, as above, this edge is not an ordinary (flat) circle.It is possible to embed a Mbius strip in three dimensions so that the edge is a circle. One wayto think of this is to begin with a minimal Klein bottle immersed in the 3-sphere and take half

of it, which is an embedded Mbius band in 4-space; this figure M has been called the"Sudanese Mbius Band". (The name comes from a combination of the names of twotopologists, Sue Goodman and Daniel Asimov). Applying stereographic projection to M puts itin 3-dimensional space, as can be seen here as well as in the pictures below. (Some haveincorrectly labeled the stereographic image in 3-space "Sudanese", but this is rather an imageof the actual Sudanese one, which has a high degree of symmetry as a Riemannian surface:its isometry group contains SO(2). A well-known parametrization of it follows.)

To see this, first consider such an embedding into the 3-sphere S3 regarded as a subset of R4.A parametrization for this embedding is given by {(z1(,), z2(,))}, where

Here we have used complex notation and regarded R4 as C2. The parameter runs from 0 to and runs from 0 to 2. Since | z1|2 + |z2|2 = 1 the embedded surface lies entirely onS3. The boundary of the strip is given by | z2| = 1 (corresponding to = 0, ), which isclearly a circle on the 3-sphere.

To obtain an embedding of the Mbius strip in R3 one maps S3 to R3 via a stereographicprojection. The projection point can be any point on S3 which does not lie on the embeddedMbius strip (this rules out all the usual projection points). Stereographic projections mapcircles to circles and will preserve the circular boundary of the strip. The result is a smoothembedding of the Mbius strip into R3 with a circular edge and no self-intersections.
http://en.wikipedia.org/wiki/File:MobiusSnail2B.pnghttp://en.wikipedia.org/wiki/File:MobiusSnail2B.png


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Related objects

A closely related 'strange' geometrical object is the Klein bottle. A Klein bottle can be producedby gluing two Mbius strips together along their edges; this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections.

Another closely related manifold is the real projective plane. If a circular disk is cut out of thereal projective plane, what is left is a Mbius strip. Going in the other direction, if one glues adisk to a Mbius strip by identifying their boundaries, the result is the projective plane. Inorder to visualize this, it is helpful to deform the Mbius strip so that its boundary is anordinary circle (see above). The real projective plane, like the Klein bottle, cannot beembedded in three-dimensions without self-intersections.

In graph theory, the Mbius ladder is a cubic graph closely related to the Mbius strip.

Applications

There have been several technical applications for the Mbius strip. Giant Mbius strips have

been used as conveyor belts that last longer because the entire surface area of the belt getsthe same amount of wear, and as continuous-loop recording tapes (to double the playingtime). Mbius strips are common in the manufacture of fabric computer printer and typewriterribbons, as they allow the ribbon to be twice as wide as the print head while using both half-edges evenly.

A device called a Mbius resistor is an electronic circuit element that has the property ofcanceling its own inductive reactance. Nikola Tesla patented similar technology in the early1900s: "Coil for Electro Magnets" was intended for use with his system of global transmissionof electricity without wires.

The Mbius strip is the configuration space of two unordered points on a circle. Consequently,in music theory, the space of all two note chords, known as dyads, takes the shape of aMbius strip; this and generalizations to more points is a significant application of orbifolds to

music theory.In physics/electro-technology:

as compact resonator with the resonance frequency which is half that of identicallyconstructed linear coils

as inductionless resistance as superconductors with high transition temperature

In chemistry/nano-technology:

as molecular knots with special characteristics (Knotane , Chirality) as molecular engines as graphene volume (nano-graphite) with new electronic characteristics, like helical

magnetism in a special type of aromaticity: Mbius aromaticity charged particles, which were caught in the magnetic field of the earth, can move on a

Mbius band the cyclotide (cyclic protein) Kalata B1, active substance of the plant Oldenlandia

affinis, contains Mbius topology for the peptide backbone.

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Man and Mystery Vol 10 - Math Wonders [Rev06]

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