Families of Congruent and Non-congruent Numbers

Families of Congruent andNon-congruent Numbers

by

Lindsey Kayla Reinholz

B.Sc., The University of British Columbia, 2011

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

in

The College of Graduate Studies

(Mathematics)

THE UNIVERSITY OF BRITISH COLUMBIA

(Okanagan)

August 2013

c© Lindsey Kayla Reinholz, 2013

Abstract

A positive integer n is a congruent number if it is equal to the area of a

right triangle with rational sides. Equivalently, the Mordell-Weil rank of

the elliptic curve

y2 = x(x2 − n2)

is positive. Otherwise n is a non-congruent number. Although congru-

ent numbers have been studied for centuries, their complete classification

is one of the central unresolved problems in the field of pure mathemat-

ics. However, by using algorithms such as the method of 2-descent, various

mathematicians have proven that numbers with prime factors of a specified

form that satisfy a certain pattern of Legendre symbols are either always

congruent or always non-congruent. In this thesis, we build upon these

results and not only prove the existence of new families of congruent and

non-congruent numbers, but also present a new method for generating fam-

ilies of non-congruent numbers. We begin by providing a technique for

constructing congruent numbers with three prime factors of the form 8k+3,

and then give a family of such numbers for which the rank of their asso-

ciated elliptic curves equals two, the maximal rank for congruent number

curves of this type. Following this, we offer an extension to work done by

Iskra and present our new method for generating families of non-congruent

numbers with arbitrarily many prime factors. This method employs Mon-

sky’s formula for the 2-Selmer rank. Unlike the method of 2-descent which

involves a series of lengthy and complex calculations, Monsky’s formula of-

fers an elegant approach for determining whether a given positive integer

is non-congruent. This theorem uses linear algebra, and through a series

of steps, allows one to compute the 2-Selmer rank of a congruent number

ii

Abstract

elliptic curve, which provides an upper bound for the curve’s Mordell-Weil

rank. By applying this method, we construct infinitely many distinct new

families of non-congruent numbers with arbitrarily many prime factors of

the form 8k + 3. In addition, by utilizing the aforementioned method once

again, we expand upon results by Lagrange to generate infinitely many new

families of non-congruent numbers that are a product of a single prime of

the form 8k + 1 and at least one prime of the form 8k + 3.

iii

Preface

The main results presented in my thesis are from collaborative research done

with Dr. Blair Spearman and Dr. Qiduan Yang. The contents of Chapter 3

were published in the journal Integers under the title “On congruent numbers

with three prime factors” [RSY11]. My colleagues and I contributed equally

to this article. Specifically, my research contributions to this publication

included conducting searches with the software program Magma to find

congruent numbers less than 10,000 with three prime factors of the form

8k + 3, applying Maple to solve torsors and find corresponding points on

congruent number elliptic curves, and utilizing Monsky’s formula for the

2-Selmer rank to verify that the maximal rank for our family of congruent

number elliptic curves is two. In addition, I was also an active participant

in the writing process. I was responsible for aiding in the organization of

the article’s content, for formatting the reference section, and for editing the

numerous drafts of the paper.

Chapter 5 is based on my paper “Families of non-congruent numbers

with arbitrarily many prime factors,” which was recently published in the

Journal of Number Theory [RSY13]. I played an important role in all as-

pects of the research process, from the formation of the initial hypothesis to

the submission of the completed article to The Journal of Number Theory.

When I began working on this project, I used the computer software program

Maple to carry out many numerical calculations. From these computations,

I noticed a pattern that enabled me to formulate a hypothesis. Further

numerical testing of my hypothesis allowed me to develop a method for con-

structing families of non-congruent numbers with arbitrarily many prime

factors. In addition to the research aspect of the project, I also contributed

to the writing of the paper and performed the necessary organizational and

iv

Preface

editing tasks. During the entire research process, I received assistance and

guidance from my collaborators, Dr. Spearman and Dr. Yang. As supervi-

sors of my research work, they introduced me to the topic of non-congruent

numbers and to Monsky’s formula for 2-Selmer rank. This, in turn, enabled

me to develop the hypothesis around which my paper is centred. Dr. Yang’s

linear algebra knowledge was indispensable to the proof of my hypothesis,

and Dr. Spearman’s extensive publication background was an asset to the

composition of our article.

It should also be noted that the results appearing in Chapter 6 are

intended for publication. I was responsible for developing and proving the

main theorem presented in this chapter. The supporting corollary, and its

proof were important additions suggested by Dr. Spearman.

v

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Congruent and Non-congruent Numbers . . . . . . . . . . . . 1

1.2 Algebra and Number Theory Preliminaries . . . . . . . . . . 7

1.2.1 Abstract Algebra . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Number Theory . . . . . . . . . . . . . . . . . . . . . 11

Chapter 2: Congruent Numbers and Elliptic Curves . . . . . 15

2.1 Introduction to Elliptic Curves . . . . . . . . . . . . . . . . . 15

2.2 The Group Law and Mordell’s Theorem . . . . . . . . . . . . 16

2.3 The Torsion Subgroup . . . . . . . . . . . . . . . . . . . . . . 20

vi

Table of Contents

2.4 The Method of 2-Descent . . . . . . . . . . . . . . . . . . . . 22

2.5 The Relationship Between Elliptic Curves and Congruent Num-

bers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 The Method of Complete 2-Descent . . . . . . . . . . . . . . 26

2.7 Monsky’s Formula for the 2-Selmer Rank . . . . . . . . . . . 30

Chapter 3: A Family of Congruent Numbers with Three Prime

Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 41

Chapter 4: Iskra’s Family of Non-congruent Numbers . . . . 42

4.1 The Proof of Iskra’s Theorem Using the Method of Complete

2-Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 The Proof of Iskra’s Theorem Using Monsky’s Formula. . . . . 55

Chapter 5: Families of Non-congruent Numbers with Arbi-

trarily Many Prime Factors of the Form 8k + 3 . . 60

5.1 Preliminary Results Involving the Generation of Non-congruent

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


Chapter 6: Families of Non-congruent Numbers with One

Prime Factor of the Form 8k + 1 and Arbitrar-

ily Many Prime Factors of the Form 8k + 3 . . . . 76


6.2 A Supporting Corollary . . . . . . . . . . . . . . . . . . . . . 87

Chapter 7: Conclusion and Future Work . . . . . . . . . . . . 89

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

vii

Table of Contents

Appendices

Chapter A: Magma Code . . . . . . . . . . . . . . . . . . . . . . 96

A.1 Elliptic Curve Calculations . . . . . . . . . . . . . . . . . . . 96

Chapter B: Maple Code . . . . . . . . . . . . . . . . . . . . . . . 99

B.1 Parametrization and 2-Selmer Rank Computations . . . . . . 99

viii

List of Tables

Table 1.1 Congruent Numbers . . . . . . . . . . . . . . . . . . . 4

Table 1.2 Non-congruent Numbers . . . . . . . . . . . . . . . . . 5

Table 3.1 Values of s(n) for n = p3q3r3 . . . . . . . . . . . . . . 40

ix

List of Figures

Figure 1.1 A rational right triangle with an area of 5. . . . . . . . 1

Figure 2.1 Elliptic curve with three real roots, y2 = (x − 1)(x −2)(x+ 1). . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2.2 Elliptic curve with one real root, y2 = (x+2)(x2−2x+3). 16

Figure 2.3 Cubic curve with a double root, y2 = x2(x+ 2). . . . . 16

Figure 2.4 Cubic curve with a triple root, y2 = (x+ 1)3. . . . . . 16

Figure 2.5 The chord and tangent method applied to distinct

points P and Q on the curve y2 = (x+ 2)(x2 − 2x+ 3). 17

Figure 2.6 The chord and tangent method applied to the point P

on the curve y2 = (x+ 2)(x2 − 2x+ 3). . . . . . . . . . 17

Figure 2.7 The group law applied to points P and Q on the curve

y2 = (x+ 2)(x2 − 2x+ 3). . . . . . . . . . . . . . . . . 19

x

List of Symbols

Z Set of integers

N+ Set of natural numbers excluding zero, {1, 2, 3, . . .}Q Set of rational numbers

Q∗ Multiplicative group of non-zero rational numbers

Q∗2 Subgroup of squares in the multiplicative group of

non-zero rational numbers

Q∗ = Q∗/Q∗2 Quotient group of square-free, non-zero rational numbers

F2 Finite field with two elements

Zn Ordered n-tuples of integers

Zn Cyclic group of order n

Zpνii Cyclic group with prime-power order

Zn1 ⊕ Zn2 ⊕ · · · ⊕ Zns Direct sum of cyclic groups

Z[x] Polynomial ring of integers in the variable x

Q(z) Ring of rational functions in the variable z

In or I Identity matrix of order n

0n or 0 Zero matrix of order n

AT Transpose of the matrix A

A−1 Inverse of the matrix A

rank(A) Rank of the matrix A

det(A) Determinant of the matrix A

gcd(a, b) Greatest common divisor of a and b

a|b a divides b

a - b a does not divide b

a ≡ b (mod m) a is congruent to b modulo m

a 6≡ b (mod m) a is incongruent to b modulo m

xi

List of Symbols

(p

q

)Legendre symbol

vp(n) p-adic valuation of n

∞ Infinite prime

MQ Set of all places of the field Q, {∞, 2, 3, . . .}O Point at infinity on an elliptic curve

x(2P ) x-coordinate of the point 2P

En Congruent number elliptic curve y2 = x(x2 − n2)E(Q) Group of rational points on the elliptic curve E

T Torsion part of E(Q)

F Free part of E(Q)

Γ (or Γ) Group of rational points on the elliptic curve E (or E)

α (or α) Homomorphism mapping Γ to Q∗ (or Γ to Q∗)r(n) Mordell-Weil rank of the elliptic curve En

s(n) 2-Selmer rank of the elliptic curve En

image(b) Image of the injective homomorphism b

M\S The set M excluding the elements in the set S∑Sum∏Product

| · | Cardinality

⊆ Subset

min{· · · } Minimum value of the elements in the set {· · · }

xii

Acknowledgements

First, and foremost, I would like to thank my supervisor, Dr. Blair Spear-

man, who has played an integral role in my educational journey. Over the

past four years, he has provided me with unfailing support, guidance, and

encouragement. Were it not for his recognition of and confidence in my po-

tential as a researcher, I never would have considered conducting research

or decided to pursue graduate studies. I am incredibly grateful for all of the

time and effort he has invested in my education, and feel truly fortunate to

have had the opportunity to work under his supervision.

I would like to thank my committee members, Dr. Qiduan Yang, Dr.

Sylvie Desjardins, and Dr. Shawn Wang. I greatly appreciate the contri-

butions that each of them has made to my education and the support and

guidance that they have given me throughout my academic studies. I am

thankful for the abundance of advice they have provided me with and the

numerous letters of reference they have written for me over the years.

I also wish to thank all of the donors who have contributed to my edu-

cation by providing me with financial support. In particular, I would like to

thank the Natural Sciences and Engineering Research Council of Canada.

Many of the ideas presented in this thesis were developed during two sum-

mers of undergraduate research funded by NSERC’s Undergraduate Student

Research Award Program. I credit this experience for sparking my interest

in research and for inspiring me to pursue graduate studies. In addition,

I would like to thank UBC Okanagan for their continued financial support

over my six years of studies. I am especially grateful that they provided me

with the opportunity to work as a teaching assistant.

Last, but certainly not least, I would like to thank my family and friends

for their endless reassurance, love, and encouragement throughout my edu-

xiii

Acknowledgements

cational journey. I appreciate the interest they have shown in my work, the

hours they have spent listening to my mathematical tirades, and the sanity-

preserving distractions they have provided. Most of all, I am thankful for

their unconditional support, which has made my accomplishments possible.

xiv

To my parents, who have instilled in me a strong work ethic

and have provided me with love, support, and encouragement along every

step of my educational journey.

xv

Chapter 1

Introduction

1.1 Congruent and Non-congruent Numbers

A positive integer n is a congruent number if it is equal to the area of a

right triangle with rational sides. In other words, there must exist rational

numbers a, b, and c such that

a2 + b2 = c2 and1

2ab = n.

Otherwise n is said to be a non-congruent number. For example, 5 is a

congruent number as it is equal to the area of a right triangle with side

lengths 203 ,

32 , and 41

6 [Cha98].

Figure 1.1: A rational right triangle with an area of 5.

In contrast, the integer 1 is non-congruent because no combination of ratio-

nal side lengths can be found to generate a right triangle with an area of 1

[Cha98, Joh09].

1

1.1. Congruent and Non-congruent Numbers

For centuries scholars have studied congruent numbers to find a solution to

a question known as the congruent number problem [Cha98, Hem06, Joh09]:

For a given positive integer n, is it possible to determine whether or

not n is a congruent number in a finite number of steps?

The first reference to this problem appears in an Arab manuscript written

in the tenth century [Alt80, Cha98]. Since then, many famous mathemati-

cians, including Fibonacci, Fermat, and Euler have studied congruent num-

bers [Alt80, Cha98, Joh09]. Fibonacci made a notable contribution to the

field by proving that both 5 and 7 are congruent numbers. He also conjec-

tured without proof that numbers that are perfect squares are not congruent

[Cha98]. This fact remained unproven until four centuries later when Fermat

developed the method of infinite descent. By applying this technique, Fer-

mat was able to prove that 1 is not a congruent number, which is equivalent

to showing that squares are not congruent [Cha98, Joh09]. In the twentieth

century, a link between congruent numbers and elliptic curves was estab-

lished [Kob93]. This significant discovery lead Tunnell to state and prove a

theorem that provides a simple criterion for determining whether or not a

given positive integer is a congruent number [Cha98, Kob93].

Theorem 1.1 (Tunnell’s Theorem). Let n be a square-free congruent

number and define

An = #{(x, y, z) ∈ Z3|n = 2x2 + y2 + 32z2},Bn = #{(x, y, z) ∈ Z3|n = 2x2 + y2 + 8z2},Cn = #{(x, y, z) ∈ Z3|n = 8x2 + 2y2 + 64z2},Dn = #{(x, y, z) ∈ Z3|n = 8x2 + 2y2 + 16z2}.

Then {2An = Bn if n is odd,

2Cn = Dn if n is even.

If the Birch and Swinnerton-Dyer conjecture holds for elliptic curves of the

form y2 = x3 − n2x then, conversely, these equalities imply that n is a

congruent number.

2


Proof. See Section 4 of Chapter IV in [Kob93].

Note that one direction of Tunnell’s theorem relies upon the Birch and

Swinnerton-Dyer conjecture, which has never been proven. This well-known

conjecture is widely believed to be true and is one of Clay Mathematics

Institute’s Millennium Prize Problems [Hem06, Joh09]. However, since the

results presented in this thesis do not require the use of the Birch and

Swinnerton-Dyer conjecture, additional details regarding it will be excluded

from the discussion.

Before Tunnell presented his ground-breaking theorem, classifying num-

bers as either congruent or non-congruent had been a difficult task. It took

until 1915 for all of the square-free congruent numbers less than 100 to be

discovered [Bas15, Alt80]. Following this, various mathematicians including

Gerardin, Alter, Curtz, Kubota, Godwin, and Hunter worked to assem-

ble a list containing all congruent numbers less than 1000 [Ger15, ACK72,

AC74, God78, Alt80]. However, it was not until 1983, when Tunnell proved

his theorem, that this list was officially completed [Joh09]. By 1993, this

list had been expanded to include all congruent numbers less than 10,000

[NW93]. Recently, computer software utilized in conjunction with Tunnell’s

theorem has enabled mathematicians to broaden their search and identify

all square-free congruent numbers less than one trillion [Joh09].

Due to the reliance of Tunnell’s theorem on the unproven Birch and

Swinnerton-Dyer conjecture, many scholars have chosen to avoid using this

theorem when studying the congruent number problem. Some of the results

that have been proven without the use of the Birch and Swinnerton-Dyer

conjecture include the ones listed in Tables 1.1 and 1.2. Note that pi, qi,

and ri denote distinct primes of the form 8k + i for k ∈ Z, or equivalently

pi ≡ qi ≡ ri ≡ i (mod 8) (see Definition 1.24). In addition,(piqj

)is the

Legendre symbol (see Definition 1.28).

3


Table 1.1: Congruent Numbers

Heegner, 1952 [Hee52] −→ 2p3 and 2p7and Birch, 1968 [Bir68]

Stephens, 1975 [Ste75] −→ p5 and p7

Monsky, 1990 [Mon90] −→ p3q7, p3q5, 2p3q5, and 2p5q7

−→ p1q5 with(p1q5

)= −1


)= −1

−→ 2p1q3 with(p1q3

)= −1


)= −1

Serf, 1991 [Ser91] −→ p3q3r5, p3q3r7, 2p3q3r7, 2p3q5r5, and 2p5q5r7

−→ p7q7r7 with(p7q7

)= −

(p7r7

)=(q7r7

)−→ 2p7q7r7 with

(p7q7

)= −

(p7r7

)=(q7r7

)−→ p1q3r3s5 with(

p1q3

)=(p1r3

)=(p1s5

)= +1

or −(p1q3

)= −

(p1r3

)=(p1s5

)= +1,

(q3s5

)=(r3s5

)−→ 2p1q3r5s5 with(

p1q3

)=(p1r5

)=(p1s5

)= +1

or(p1q3

)= −

(p1r5

)= −

(p1s5

)= +1,

(q3r5

)=(q3s5

)

4


Table 1.2: Non-congruent Numbers

Genocchi, 1855 [Gen55] −→ p3, p3q3, 2p5, and 2p5q5

Lagrange, 1974 [Lag75] −→ p1q3 with(p1q3

)= −1


)= −1

−→ 2p3q3


)= −1


)= −1


)= −

(p1r3

)−→ p3q5r7 with

(q5r7

)= −1


)= −

(p3r7

)=(q7r7


(p1q3

)= −

(p1r3


(p1q5

)= −

(p1r5


(p3r7

)= −

(q5r7


(p5q7

)= −

(p5r7

)=(q7r7

)Serf, 1991 [Ser91] −→ p5q5r7s7 with(

p5r7

)= −

(p5s7

)= −

(q5r7

)= +1

or −(p5r7

)=(p5s7

)= −

(q5s7

)= +1

or −(p5r7

)= −

(p5s7

)= +1,

(q5r7

)= −

(q5s7

)−→ 2p1q1r3s3 with(

p1q1

)= +1,

(p1r3

)= −

(p1s3

),(q1r3

)= −

(q1s3

)or

(p1q1

)= −1,

(p1r3

)=(p1s3

),(q1r3

)= −

(q1s3

)or

(p1q1

)= −1,

(p1r3

)= −

(p1s3

)

5


In his paper “Non-congruent numbers with arbitrarily many prime fac-

tors congruent to 3 modulo 8,” Iskra proved the existence of a new fam-

ily of non-congruent numbers containing arbitrarily many prime factors,

p1, p2, . . . , pt satisfying pi ≡ 3 (mod 8) for all 1 ≤ i ≤ t and(pjpi

)= −1 for

j < i [Isk96]. A thorough discussion of Iskra’s results, including two differ-

ent proofs of his main theorem (see Theorem 4.1), can be found in Chapter

4.

The results presented in this thesis broaden the current understanding

of congruent and non-congruent numbers by generating new families of both

types of these numbers. In Chapter 3, a method is provided for constructing

congruent numbers with three distinct prime factors of the form 8k + 3. A

family of such numbers is given for which the rank of their associated elliptic

curves equals two, the maximal rank for congruent number curves of this

type. These results were published in the journal Integers under the title

“On congruent numbers with three prime factors” [RSY11]. In Chapter 4,

Iskra’s work [Isk96] is discussed and a new, elegant method for generating

families of non-congruent numbers with arbitrarily many prime factors is

presented. This method is then applied to prove the existence of Iskra’s

family of non-congruent numbers. Chapter 5 offers an extension to Iskra’s

work and applies the method presented in Chapter 4 to construct infinitely

many distinct new families of non-congruent numbers with arbitrarily many

prime factors of the form 8k + 3. These results appeared in the paper

“Families of non-congruent numbers with arbitrarily many prime factors,”

which was recently published in the Journal of Number Theory [RSY13].

In Chapter 6, another collection of infinitely many new families of non-

congruent numbers is generated by utilizing the aforementioned method

once again. These numbers are a product of arbitrarily many primes, where

the first prime factor is of the form 8k + 1 and the remaining prime factors

are of the form 8k + 3. Before we prove the results of Chapters 3 through

6, a thorough discussion of the necessary background information must be

provided. In the next section, we recall various algebra and number theory

terminology. In Chapter 2, we describe the link between congruent numbers

and elliptic curves and present an introduction to the theory governing the

6

1.2. Algebra and Number Theory Preliminaries

properties of elliptic curves.

1.2 Algebra and Number Theory Preliminaries

1.2.1 Abstract Algebra

We begin by introducing some basic definitions involving binary algebraic

structures, denoted as 〈G, ∗〉, where G is a set and ∗ is a binary operation

on G. Note that the following definitions, taken from [Fra03], can be found

in most introductory abstract algebra textbooks.

Definition 1.2. A group 〈G, ∗〉 is a set G under a binary operation ∗ that

satisfies the following axioms:

1. (Associativity) For all a, b, c ∈ G, we have (a ∗ b) ∗ c = a ∗ (b ∗ c).

2. (Identity Element, e) There is an element e in G such that for all

g ∈ G, e ∗ g = g ∗ e = g.

3. (Inverse) For each a ∈ G, there exists an element a′ ∈ G such that

a ∗ a′ = a′ ∗ a = e.

Definition 1.3. An abelian group 〈G, ∗〉 is a group G with a commutative

binary operation ∗. This means that a ∗ b = b ∗ a for all a, b ∈ G.

Definition 1.4. Let 〈G, ∗〉 be a group and H be a non-empty subset of G.

Then H is called a subgroup of G if H is closed under the binary operation

∗ and 〈H, ∗〉 satisfies the three group axioms.

Definition 1.5. Let 〈G, ∗〉 and 〈G′, ∗′〉 be binary algebraic structures, where

G and G′ are groups. A map φ of G into G′ is a homomorphism if

φ(x ∗ y) = φ(x) ∗′ φ(y)

for all x, y ∈ G. A homomorphism that is one-to-one is called an injective

homomorphism, which is also known as a monomorphism.

7


Definition 1.6. Let 〈G, ∗〉 and 〈G′, ∗′〉 be binary algebraic structures. An

isomorphism, also known as a bijective homomorphism, of G with G′ is a

one-to-one function φ mapping G onto G′ such that

φ(x ∗ y) = φ(x) ∗′ φ(y)

for all x, y ∈ G. If such a map φ exists, then G and G′ are isomorphic binary

structures, denoted by G ∼= G′.

Definition 1.7. Let G be a group and let a ∈ G. The element a generates

G and is a generator for G if G = {an|n ∈ Z} = 〈a〉. A group G is cyclic if

there exists an element a in G that generates G.

Definition 1.8. A finitely generated abelian group 〈G,+〉 is an abelian

group for which there exist finitely many elements g1, g2, . . . , gn ∈ G such

that every g ∈ G can be written as

g = a1g1 + a2g2 + . . .+ angn,

where a1, a2, . . . , an ∈ Z.

An important theorem which provides complete structural information

about finitely generated abelian groups is the fundamental theorem of finitely

generated abelian groups [Fra03].

Theorem 1.9 (Fundamental Theorem of Finitely Generated Abelian

Groups). Every finitely generated abelian group G is isomorphic to a direct

sum of cyclic groups in the form

G ∼= Zpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνss ⊕ Z⊕ Z⊕ · · · ⊕ Z,

where Z is an infinite cyclic group and Zpνii is a finite cyclic group with

prime-power order for 1 ≤ i ≤ s with i, s ∈ Z. Note that the primes, pi, are

not necessarily distinct and that the νi are positive integers.

8


1.2.2 Linear Algebra

Next we recall some basic concepts and properties from linear algebra. This

information can be found in an introductory linear algebra textbook such

as [KH04].

Definition 1.10. A matrix A = [aij ] is called a square matrix of order n if

the number of rows and the number of columns are both equal to n.

Definition 1.11. Let A be a square matrix of order n.

1. The entries a11, a22, . . . , ann are called the diagonal entries of A.

2. A is said to be a diagonal matrix if all of the entries, except for the

diagonal entries, are zero.

3. A is an upper triangular matrix if all the entries below the diagonal

are zero. Similarly, A is a lower triangular matrix if all the entries

above the diagonal are zero.

Note that a product of upper triangular matrices is also an upper tri-

angular matrix. Similarly, any matrix that is a product of lower triangular

matrices is lower triangular.

Definition 1.12. The identity matrix of order n, denoted by In or I, is a

diagonal matrix whose diagonal entries are all equal to 1.

Definition 1.13. The zero matrix of order n, denoted by 0n or 0, is a

matrix whose entries are all equal to 0.

Definition 1.14. Let A = [aij ] be an m × n matrix. The transpose of A

is the n ×m matrix, denoted by AT , whose j-th column is taken from the

j-th row of A. In other words, [AT ]ij = [A]ji.

Definition 1.15. An n×n square matrix A is said to be invertible if there

exists a square matrix B of the same size such that

AB = In = BA.

The matrix B is called the inverse of A, and is denoted by A−1.

9


Definition 1.16. For an m×n matrix A, the rank of A is defined to be the

maximal number of linearly independent column vectors (or row vectors) of

A. The rank of A is denoted by rank(A).

Definition 1.17. The determinant of a square n×n matrix A, denoted by

det(A), is a real-valued function that satisfies the following three properties:

1. The value of the determinant changes sign if any two rows or columns

within the matrix A are interchanged.

2. The determinant is linear. This means that if A = [a1, a2, . . . , an],

where the aj are column vectors of length n, then

det[a1, a2, . . . , bai+ca′i, . . . , an] = b·det[a1, a2, . . . , an]+c·det[a1, a2, . . . , a

′i, . . . , an].

Note that b and c are scalars and a′i is a column vector of length n.

3. The determinant of the identity matrix is 1, so det(In) = 1.

The determinant satisfies some important properties that can be sum-

marized by the following theorem [KH04, Theorems 2.2, 2.3 & 3.26].

Theorem 1.18. Let A be an n × n square matrix. Then the determinant

satisfies the following properties:

1. If A has two identical rows (or columns), which means that the rows

(or columns) of A form a linearly dependent set, then det(A) = 0.

2. The determinant remains unchanged if a scalar multiple of one row

is added to another row. Similarly, the determinant’s value does not

change when a scalar multiple of one column is added to another col-

umn.

3. The determinant of a triangular matrix is equal to the product of the

diagonal entries.

4. The matrix A is invertible if and only if det(A) 6= 0.

5. det(AT ) = det(A).

10


6. det(A) 6= 0 if and only if rank(A) = n.

For a matrix subdivided into four separate blocks, the following identities

can be applied to compute its determinant [Mey00, p. 467, 475, and 483].

Proposition 1.19. If A and D are square matrices, then

det

([A B

0 D

])= det

([A 0

C D

])= det (A) det (D).

Proposition 1.20. If A and D are square matrices, then

det

([A B

C D

])=

det (A) det(D−CA−1B

), when A−1 exists,

det (D) det(A−BD−1C

), when D−1 exists.

Proposition 1.21. If B is an invertible n× n matrix, and if D and C are

n× k matrices, then

det(B + CDT

)= det(B) det(Ik + DTB−1C).

1.2.3 Number Theory

Now we introduce some terminology that is used in the field of number

theory. It is worthwhile to note that this information, taken from [Ros05],

can be found in most introductory number theory textbooks.

Definition 1.22. The greatest common divisor of two integers a and b,

which are not both 0, is the largest integer that divides both a and b.

Note that the greatest common divisor of a and b is written as gcd(a, b),

and that by definition gcd(0, 0) = 0. For integers a and b, the notation a|bindicates that a divides b and the notation a - b indicates that a does not

divide b.

Definition 1.23. The integers a1, a2, . . . , an are pairwise relatively prime

if, for each pair of integers ai and aj with i 6= j from the set, the greatest

common divisor of ai and aj is 1.

11


Definition 1.24. Let m be a positive integer. If a and b are integers, we

say that a is congruent to b modulo m if m divides (a− b).

If a is congruent to b modulo m, we write this as a ≡ b (mod m), whereas

if a and b are incongruent modulo m, we denote this by a 6≡ b (mod m).

Definition 1.25. A congruence class modulo m is a set of integers that are

mutually congruent modulo m.

For instance, there are three congruence classes modulo 3; one class

contains all integers congruent to 0 modulo 3, another class contains all

integers congruent to 1 modulo 3, and the third class contains all integers

congruent to 2 modulo 3.

An important theorem which provides a method for solving systems of

linear congruences is the Chinese remainder theorem [Ros05, Theorem 4.12].

Theorem 1.26 (Chinese Remainder Theorem). If m1,m2, . . . ,ms are

pairwise relatively prime positive integers, then the system of congruences

x ≡ a1 (modm1),

x ≡ a2 (modm2),

...

x ≡ as (modms),

has a unique solution modulo M = m1m2 · · ·ms.

Next, we discuss quadratic residues, quadratic nonresidues, and Legen-

dre symbols.

Definition 1.27. If m is a positive integer, we say that the integer b is a

quadratic residue of m if gcd(b,m) = 1 and the congruence x2 ≡ b (mod m)

has a solution. If this congruence does not have a solution, then we say that

b is a quadratic nonresidue of m.

As an example, consider the number 5 and try to determine its quadratic

residues. To do this, we must compute the squares of the integers 1, 2, 3, and

12


4. We find that 12 ≡ 42 ≡ 1 (mod 5) and 22 ≡ 32 ≡ 4 (mod 5). Therefore, 1

and 4 are quadratic residues of 5, whereas 2 and 3 are quadratic nonresidues

of 5.

Definition 1.28. Let p be an odd prime and a be an integer not divisible

by p. The Legendre symbol(ap

)is defined by

(a

p

)=

+1 if a is a quadratic residue of p,

−1 if a is a quadratic nonresidue of p.

For our previous example, the Legendre symbols for p = 5 and a = 1, 2,

3, and 4 are (1

5

)=

(4

5

)= +1 and

(2

5

)=

(3

5

)= −1.

Some important properties of Legendre symbols can be summarized by

the following theorem [Ros05, Theorems 11.4, 11.5, 11.6 & 11.7].

Theorem 1.29 (Properties of Legendre Symbols).

1. Let p be an odd prime and a and b be integers not divisible by p. Then

(i) if a ≡ b (mod p), then

(a

p

)=

(b

p

).

(ii)

(a

p

)(b

p

)=

(ab

p

).

(iii)

(a2

p

)= +1.

2. The Law of Quadratic Reciprocity.

If p and q are distinct odd primes, then(q

p

)=

(p

q

)(−1)

p−12· q−1

2 .

3. The First and Second Supplements to the Law of Quadratic

Reciprocity.

If p is an odd prime, then

13


(i)

(−1

p

)=

+1 if p ≡ 1 (mod 4),

−1 if p ≡ 3 (mod 4).

(ii)

(2

p

)=

+1 if p ≡ 1, 7 (mod 8),

−1 if p ≡ 3, 5 (mod 8).

Next, we state Dirichlet’s theorem on primes in arithmetic progression

[Ros05, Theorem 3.3].

Theorem 1.30 (Dirichlet’s Theorem on Primes in Arithmetic Pro-

gression). Suppose that a and b are relatively prime positive integers. Then

the arithmetic progression an + b where n ∈ N+ contains infinitely many

primes.

Finally, we define a couple of terms that are commonly discussed when

studying p-adic number theory [Isk98, Ogg09].

Definition 1.31. Let p be a prime number and n be a non-zero rational

number. If n = pαn′, where n′ is a rational number whose prime-power

factorization does not contain p, then the p-adic valuation, vp(n), of the

non-zero rational number n is

vp(n) = α.

Note that vp(0) is defined to be equal to ∞.

Definition 1.32. The set of all places of the field Q is the set of primes in

Q, denoted by MQ = {∞, 2, 3, . . .}.

Having gained an understanding of some of the basic background infor-

mation in the fields of algebra and number theory, we are now ready to

discuss elliptic curves and the theory that governs their properties.

14

Chapter 2

Congruent Numbers and

Elliptic Curves

2.1 Introduction to Elliptic Curves

An elliptic curve is an algebraic curve which has a cubic equation of the

form

y2 = x3 + ax2 + bx+ c,

where a, b, c ∈ Q [ST92]. For such a cubic curve to be considered an elliptic

curve, it must have distinct roots, or equivalently, its discriminant, given by

the equation

D = −4a3c+ a2b2 + 18abc− 4b3 − 27c2,

must be non-zero. It is important to note that the above elliptic curve is

written in Weierstrass normal form and that any cubic equation with a

rational point can be converted to this simple form [ST92].

Since elliptic curves are cubic polynomials with distinct roots, they must

have either three real roots, or one real root and a pair of complex conjugate

roots. If a cubic curve has a double or triple root, it is not an elliptic curve.

Figures 2.1 and 2.2 provide examples of elliptic curves, whereas Figures 2.3

and 2.4 depict cubic curves with double and triple roots that are not elliptic

curves. Note that these plots were created with the aid of MapleTM13.

15

2.2. The Group Law and Mordell’s Theorem

Figure 2.1: Elliptic curvewith three real roots, y2 =(x− 1)(x− 2)(x+ 1).

Figure 2.2: Elliptic curvewith one real root, y2 =(x+ 2)(x2 − 2x+ 3).

Figure 2.3: Cubic curvewith a double root, y2 =x2(x+ 2).

Figure 2.4: Cubic curvewith a triple root, y2 =(x+ 1)3.

An important and well-known fact about elliptic curves is that the ra-

tional points on a given curve E form an abelian group, denoted by E(Q).

To discuss the structure of E(Q), we must first define the group law that

governs the set of rational points on our elliptic curve.

2.2 The Group Law and Mordell’s Theorem

Before we describe the group law, we must develop an understanding of

how points on elliptic curves are related to one another. Given two points

16


on an elliptic curve, it is possible to find a third point on the curve by

applying a composition law known as the chord and tangent method [Hus04,

Joh09, SZ03, ST92, Sil09]. This technique allows us to map two points

P and Q on our elliptic curve to a third point P ∗ Q also on the curve.

Note that ∗ denotes the binary operator for the composition law. If P

and Q are distinct points then P ∗ Q is defined to be the third point of

intersection of the elliptic curve with the line passing through the points P

andQ [Hus04, Joh09, SZ03, ST92, Sil09]. Specifically, if P andQ are rational

points, then the line connecting the two points is a rational line. Therefore,

the point P ∗Q, which lies on the line, must also be rational [Hus04]. Figure

2.5 provides a geometric interpretation of this process for two distinct points

on the elliptic curve y2 = (x+ 2)(x2− 2x+ 3). If we only know one rational

point P on our elliptic curve, we can apply the same method to find another

rational point P ∗P on the curve [Hus04, Joh09, SZ03, ST92, Sil09]. In this

case, the line that passes through P lies tangent to the curve at that point;

this scenario is illustrated in Figure 2.6 for the curve y2 = (x+ 2)(x2− 2x+

3). Thus, the chord and tangent method provides us with a technique for

generating many rational points on a given elliptic curve.

Figure 2.5: The chord andtangent method applied to dis-tinct points P and Q on thecurve y2 = (x+2)(x2−2x+3).

Figure 2.6: The chord andtangent method applied to thepoint P on the curve y2 =(x+ 2)(x2 − 2x+ 3).

The set of rational points obtained by applying the chord and tangent

17


method can be made into a group by introducing the concept of the point at

infinity. This point, denoted by O, is a rational point on every elliptic curve

and is the identity element in our group. By definition, the point at infinity

is an inflection point on our elliptic curve, and the tangent line to the curve

at that point is the line at infinity [ST92]. It is a well-known fact that a

line meets an elliptic curve at exactly three points [SZ03, ST92]; therefore,

we know that the line at infinity intersects the curve at the point O three

times, a vertical line intersects the curve at two points in the xy-plane and

once at the point O, and a non-vertical line intersects the curve at three

points in the xy-plane [ST92]. Consider a specific vertical line that meets

the elliptic curve E at the point P . By definition, the vertical line must also

pass through the point at infinity, so the third point of intersection between

the curve E and the line must be P ∗O = O ∗P [Joh09, ST92, Sil09] . Thus,

P ∗ O is the reflection of P about the x -axis [Joh09, ST92].

Using this information, we can now define the group law associated with

rational points on elliptic curves. Let + be the binary operator for the group

law and let P and Q be rational points on the elliptic curve E. Consider

the line through the points P and Q and find the third intersection point,

P ∗Q, of the line with the curve E. Next, draw a vertical line through the

point P ∗ Q. This line also meets the curve at the point at infinity, so the

third point of intersection between E and the vertical line is O ∗ (P ∗ Q);

we define this point to be equal to P + Q [Joh09, ST92, Sil09]. Therefore,

P + Q is the reflection of P ∗ Q about the x -axis. Figure 2.7 provides a

visual depiction of the group law being applied to the distinct points P and

Q on the curve y2 = (x+ 2)(x2 − 2x+ 3).

Let P , Q, and R be rational points on the elliptic curve E. The group

law operator + satisfies the following properties [Joh09, SZ03, ST92, Sil09]:

Commutative: P +Q = Q+ P for all P,Q ∈ E.

Closure: If P,Q ∈ E, then P +Q ∈ E.

Associative: (P +Q) +R = P + (Q+R) for all P,Q,R ∈ E.

Identity Element O: P + O = O + P = P for all P ∈ E.

18


Inverse: P + (−P ) = (−P ) + P = O for all P ∈ E.

Note that −P = O ∗P , which means that it is the reflection of P about the

x -axis. Together, these five properties imply that E(Q) is an abelian group

under the binary operation +. A proof of this fact can be found in [Sil09].

Figure 2.7: The group law applied to points P and Q on the curve y2 =(x+ 2)(x2 − 2x+ 3).

This leads us to an important theorem by Mordell that offers a detailed

description of the structure of the group of rational points [Hem06, Hus04,

Joh09, ST92, Sil09].

Theorem 2.1 (Mordell’s Theorem). Let E be an elliptic curve over the

field of rational numbers. The group of rational points, E(Q), is a finitely

generated abelian group.

Proof. See Chapter 6 of [Hus04], Chapter III of [ST92], or Chapter VIII of

[Sil09].

As a result of the Mordell’s theorem, we can apply the fundamental

theorem of finitely generated abelian groups (see Theorem 1.9) to the group

of rational points. This allows us to write it as the following direct sum of

cyclic groups:

E(Q) ∼= Zpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνss ⊕ Z⊕ Z⊕ · · · ⊕ Z,

19

2.3. The Torsion Subgroup

where Z is an infinite cyclic group and Zpνii is a finite cyclic group with

prime-power order for 1 ≤ i ≤ s with i, s ∈ Z [Hem06, Hus04, Joh09, SZ03,

ST92, Sil09]. An equivalent way of writing the group of rational points is

E(Q) ∼= T ⊕F ,

where T ∼= Zpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνss is the torsion part of E(Q) and F ∼=Z⊕Z⊕ · · · ⊕Z is the free part of E(Q). The number of copies of Z in F is

denoted by r and is called the rank, or the Mordell-Weil rank, of the elliptic

curve. We formally define the rank as follows [Joh09].

Definition 2.2. Let E(Q) be the group of rational points on the elliptic

curve E. The number of generators with infinite order in E(Q) is the rank

of the curve E denoted by r.

2.3 The Torsion Subgroup

In order to define the torsion subgroup, we must first explain what it means

for a point in E(Q) to have a particular order [ST92].

Definition 2.3. An element P = (x, y) in E(Q) is said to have order m if

mP = P + P + · · ·+ P︸︷︷︸m

= O,

but m′P 6= O for all 1 ≤ m′ < m, where m, m′ ∈ Z and O is the identity

element. If such an integer m exists, then point P is said to have finite

order ; otherwise P is said to have infinite order.

Note that the identity element, O, has order one and rational points,

(x, y), with y = 0 have order two [ST92]. The set of all points of finite order

in E(Q) forms a subgroup known as the torsion subgroup [Hus04, ST92,

Sil09].

Definition 2.4. The torsion subgroup, T , of E(Q) is the group consisting

of all the rational points of finite order on the elliptic curve E.

20

2.3. The Torsion Subgroup

Since the identity element, O, is a point on every elliptic curve, the

torsion subgroup always contains at least one rational point of finite order.

The rest of the points in the torsion subgroup can be found by applying the

following theorem [Joh09, ST92, Sil09].

Theorem 2.5 (Nagell-Lutz Theorem). Let y2 = f(x) = x3+ax2+bx+c

be an elliptic curve with a, b, c ∈ Z and let D be the the discriminant of f(x)

so

D = −4a3c+ a2b2 + 18abc− 4b3 − 27c3.

Let P = (x, y) be a rational point of finite order. Then x and y are integers

and either y = 0 or else y2 divides D.

Proof. See Chapter II of [ST92] or Chapter VIII of [Sil09].

Notice that this theorem is not an if-and-only-if statement. As a result,

it is possible to have points on the curve that are not of finite order, but

that do have integer coordinates with y2 dividing D [ST92]. To determine

whether a given point, P = (x, y) 6= O with y 6= 0, is finite, it is useful to

consider the duplication formula for the x-coordinate of P [ST92]:

x(2P ) =x4 − 2bx2 − 8cx+ b2 − 4ac

4y2.

If P = (x, y) 6= O is a rational point of finite order, then x and y are

integers and mP = O for some m ∈ Z. It follows that 2P also must have

finite order, so the x-coordinate of 2P , x(2P ), should have an integer value

too. Therefore, if we compute x(2P ) and find that it does not equal an

integer, we deduce that P is not a point of finite order [ST92]. Once all

of the points of finite order have been found, the following theorem can be

applied to determine the exact form of the torsion subgroup [Hem06, Hus04,

SZ03, ST92, Sil09].

Theorem 2.6 (Mazur’s Theorem). Let E be an elliptic curve defined over

Q. Then the torsion subgroup, T , of the group of rational points, E(Q), is

one of the following fifteen groups:

21

2.4. The Method of 2-Descent

1. A cyclic group of order N, ZN , with 1 ≤ N ≤ 10 or N = 12.

2. The product of a cyclic group of order two and a cyclic group of order

2N , Z2 ⊕ Z2N , with 1 ≤ N ≤ 4.

Proof. See [Maz77] or [Maz78].

2.4 The Method of 2-Descent

The method of 2-descent is an algorithm that is used for computing the

Mordell-Weil rank of an elliptic curve. Let us define the elliptic curve E:

y2 = x3 + ax2 + bx, where a, b ∈ Z and (x, y) is a rational point. In

addition, let Γ be the group of rational points on E. In order to compute

the rank of E, we must simultaneously consider another curve denoted by

E: y2 = x3 + ax2 + bx, where a = −2a, b = a2 − 4b, and (x, y) is a rational

point. Let Γ be the group of rational points on E [Joh09, ST92]. Define

Q∗ to be the multiplicative group of non-zero rational numbers, Q∗2 to be

the subgroup of squares in Q∗, and Q∗ = Q∗/Q∗2 to be the quotient group

consisting of square-free, non-zero rational numbers [ST92]. This allows the

following homomorphisms to be defined:

α : Γ −→ Q∗ and α : Γ −→ Q∗,

where

α(P ) =

1 (mod Q∗2) if P = O

b (mod Q∗2) if P = (0, 0)

x (mod Q∗2) if P = (x, y) with x 6= 0

and

α(P ) =

1 (mod Q∗2) if P = O

b (mod Q∗2) if P = (0, 0)

x (mod Q∗2) if P = (x, y) with x 6= 0,

for P = (x, y) ∈ Γ and P = (x, y) ∈ Γ [Joh09, ST92]. Furthermore, α(Γ) is a

subset of the square-free divisors of b, and α(Γ) is a subset of the square-free

22

2.4. The Method of 2-Descent

divisors of b [ST92]. The rank of E can be computed by using the equation

2r =|α(Γ)| |α(Γ)|

4,

where r is the rank of E, |α(Γ)| is the cardinality of α(Γ), and |α(Γ)| is

the cardinality of α(Γ) [Joh09, ST92]. Therefore, to calculate the rank,

we must first determine the elements in α(Γ) and α(Γ). Clearly, 1 and b

modulo Q∗2 are in α(Γ). To determine whether or not α(Γ) contains any

additional elements, we must consider all of the possible factors, b1, of b

with b1 6≡ 1, b (mod Q∗2), and b = b1b2. If the equation

N2 = b1M4 + aM2e2 + b2e

4, (2.1)

which is referred to as a torsor, has a solution (N,M, e) ∈ Z3 with M 6= 0,

e 6= 0, and gcd(M, e) = gcd(N, e) = gcd(b1, e) = gcd(b2,M) = gcd(M,N) =

1, then b1 modulo Q∗2 is an element of α(Γ) [Joh09, ST92]. Each equation

that we solve, produces a corresponding point on our elliptic curve of the

form

(x, y) =

(b1M

2

e2,b1MN

e3

)(2.2)

[ST92]. Similarly, working modulo Q∗2, α(Γ) contains 1, b, and all divisors

b1 of b satisfying the torsor equation

N2 = b1M4 + aM2e2 + b2e

4 (2.3)

for some (N,M, e) ∈ Z3 with M 6= 0 and e 6= 0. Note that a = −2a,

b = b1 b2, and b1 6≡ 1, b (mod Q∗2). In addition, we also require that the

gcd(M, e) = gcd(N, e) = gcd(b1, e) = gcd(b2,M) = gcd(M,N) = 1 [Joh09,

ST92].

Thus, the method of 2-descent provides us with a systematic procedure

for computing the Mordell-Weil rank of a given elliptic curve. However, if

b and b have many square-free divisors, then carrying out the method of 2-

descent can be a lengthy and tedious process. In addition, finding solutions

(N,M, e) ∈ Z3 that satisfy Equations (2.1) and (2.3) can be a challenging

23

2.5. The Relationship Between Elliptic Curves and Congruent Numbers

task, as there is no known method for determining whether equations of this

form are solvable [Joh09, ST92].

Nevertheless, if we can find a solution that satisfies a torsor equation,

we are guaranteed that there is corresponding rational point on our elliptic

curve. It is worthwhile to note that the rank of an elliptic curve is related

to the number of independent rational points of infinite order in the curve’s

group of rational points. An important theorem that can be used to show

that the points on an elliptic curve are independent is Silverman’s special-

ization theorem [Sil09, Theorem 20.3].

Theorem 2.7 (Silverman’s Specialization Theorem). Let C/K be a

curve and let E be an elliptic curve defined over the function field K(C)

such that j(E) 6∈ K, where j(E) is the j-invariant of the elliptic curve E.

Then the specialization map

σt : E(K(C))→ Et

is (well-defined) and injective for all but finitely many points t ∈ C(K).

(More generally, it is injective for all but finitely many points of⋃C(L),

where the union is over all fields L/K whose degree is bounded by a fixed

number.)

We can summarize this theorem as follows:

Suppose that there exists a family of elliptic curves, y2 = x3 − tx, in terms

of a parameter t and suppose that there is a finite set of points on these

curves also given in terms of t. If the points are independent in the group

of rational points for even a single value of t, then they are independent for

all rational values of t with at most finitely many possible exceptions.

2.5 The Relationship Between Elliptic Curves

and Congruent Numbers

Recall that in Chapter 1, we defined congruent numbers and non-congruent

numbers as follows:

24

2.5. The Relationship Between Elliptic Curves and Congruent Numbers

Definition 2.8. A positive integer n is a congruent number if it is equal to

the area of a right triangle with rational sides. Otherwise n is said to be a

non-congruent number.

Congruent numbers can be defined in an equivalent way by using elliptic

curves [DJS09, Hem06, Joh09, Kob93, NW93, RSY11, RSY13].

Lemma 2.9. A positive integer n is a congruent number if and only if the

rank of the elliptic curve

En: y2 = x(x2 − n2)

is positive. Otherwise, n is a non-congruent number. In other words, n is a

non-congruent number if and only if the rank of En is zero.

Note that the proof of the first if-and-only-if statement in Lemma 2.9 can

be found in Section 10 of Chapter 2 in [Hem06], or in Section 9 of Chapter

I in [Kob93].

By inspection, it is clear that the points (0, 0), (n, 0), and (−n, 0) are on

the curve y2 = x(x2−n2). These three points all have 0 as their y-coordinate,

so they are points of order two. Since O is a point of order one that lies on

every elliptic curve, we know that the torsion subgroup of En must contain

at least four elements. As it turns out, O, (0, 0), (n, 0), and (−n, 0) are the

only points of finite order on En; a proof of this fact can be found in Section

7 of Chapter 2 in [Hem06]. Since the torsion subgroup of En is composed

of one element of order one and three elements of order two, by Mazur’s

theorem (Theorem 2.6) we deduce that T ∼= Z2 ⊕ Z2. Thus, the group of

rational points on our elliptic curve En is isomorphic to Z2⊕Z2⊕Zr, where

r is the curve’s rank [Ser91]. Next we focus our attention on computing the

rank of elliptic curves En for various values of n, as the rank will allow us to

determine whether n is a congruent number or a non-congruent number. To

do this, we apply a technique known as the method of complete 2-descent.

25

2.6. The Method of Complete 2-Descent

2.6 The Method of Complete 2-Descent

The method of complete 2-descent is an algorithm that is used for computing

the rank of an elliptic curve. It considers pairs of quadratic equations and

determines whether or not they are solvable. For elliptic curves given by the

general Weierstrass equation y2 = (x−e1)(x−e2)(x−e3) with e1, e2, e3 ∈ Kwhere K is a number field, the process involved in carrying out the method of

complete 2-decent is described in Proposition 1.4 on page 315 of [Sil09]. For

curves of the form En : y2 = x(x2 − n2), the procedure can be summarized

by the following theorem [Ser91, Theorem 3.1].

Theorem 2.10 (Complete 2-Descent). Let

n = 2εp1p2 · · · pk

be a square-free positive integer with p1, p2, . . . , pk being primes that are not

equal to two, ε ∈ {0, 1}, and k ∈ N+. Let En be the elliptic curve over Qdefined by the equation

En : y2 = x(x2 − n2) = x(x− n)(x+ n),

and

S = {∞, 2, p1, . . . , pk}

be a finite subset of MQ, the set of all places of Q. In addition, define

Q(S, 2) := {c ∈ Q∗/Q∗2| vp(c) ≡ 0 (mod 2) ∀ p ∈MQ\S},

where vp(c) is the p-adic valuation of c. Then there exists an injective ho-

momorphism

b : En(Q)/2En(Q) ↪→ Q(S, 2)×Q(S, 2)

defined by

26


P = (x, y) 7→

(1, 1) if P = O

(−1,−n) if P = (0, 0)

(n, 2) if P = (n, 0)

(x, x− n) if P = (x, y) 6= O, (0, 0), (n, 0).

If (b1, b2) ∈ Q(S, 2)×Q(S, 2)\{(1, 1), (−1,−n), (n, 2)}, then (b1, b2) ∈ image(b)

if and only if there exist (z1, z2, z3) ∈ Q∗ × Q∗ × Q such that the following

two equations simultaneously hold:

b1z21 − b2z22 = n, (2.4)

b1z21 − b1b2z23 = −n. (2.5)

In this case, (b1, b2) = b(P ) for

P = (b1z21 , b1b2z1z2z3) = (b2z

22 + n, b1b2z1z2z3).

Recall that En(Q) ∼= Z2⊕Z2⊕Zr, so En(Q) can be described as a set of

(r+ 2)-tuples. In addition, 2En(Q) is also a set of (r+ 2)-tuples. Note that

an arbitrary (r + 2)-tuple in En(Q) can be written as (a1, a2, a3, . . . , ar+2),

where a1, a2 ∈ Z2 and ai ∈ Z for all 3 ≤ i ≤ (r + 2). In 2En(Q), this

(r+2)-tuple becomes (2a1, 2a2, 2a3, . . . , 2ar+2). However, since the first two

components of the (r + 2)-tuple are in Z2, they reduce to zero. Therefore,

when we form the quotient group En(Q)/2En(Q), the first two components

remain unchanged and are isomorphic to Z2. Each of the other r components

is isomorphic to Z/2Z, which is equivalent to Z2. Thus,

En(Q)/2En(Q) ∼= (Z2)r+2,

so since Z2 is a group of order two, the total order of the above quotient

group is 2r+2 [Ser91]. This means that there are 2r+2 rational points on our

elliptic curve En: y2 = x(x2 − n2).In order to compute the rank, r, of the curve En, we need to recall from

Section 2.5 that the torsion subgroup of En(Q) contains four rational points

of finite order. By applying the homomorphism defined in Theorem 2.10,

27


we know that these four points, P = O, (0, 0), (n, 0), and (−n, 0), are re-

spectively mapped to (1, 1), (−1,−n), (n, 2), and (−n,−2n). Now we must

consider the set of pairs (b1, b2) 6∈ {(1, 1), (−1,−n), (n, 2), (−n,−2n)} for

which Equations (2.4) and (2.5) simultaneously have a solution. Determin-

ing whether or not a given pair, (b1, b2), belongs to this set can sometimes

be a difficult task. Therefore, we define B to be the upper bound for the

number pairs (b1, b2) that simultaneously solve Equations (2.4) and (2.5).

This enables us to use the following inequality to bound the rank of En

[Ser91]:

2r+2 ≤ B + 4

⇐⇒ 2r ≤ B + 4

4

⇐⇒ r ≤ log2

(B + 4

4

). (2.6)

If we are able to conclusively determine the solvability of Equations (2.4)

and (2.5) for all pairs, then the above inequality becomes a strict equality.

Clearly, the more pairs, (b1, b2), we find for which our system of two equa-

tions has a solution, the higher the rank our elliptic curve En is guaranteed

to have.

Note that for n to be a non-congruent number, we require that r =

0. Therefore, we need the bound, B, for the number of pairs (b1, b2) to

be equal to zero as well. This means that we need to show that our

system of two equations does not have a solution for any pair (b1, b2) 6∈{(1, 1), (−1,−n), (n, 2), (−n,−2n)}. To do this, it is beneficial to make use

of the following theorem, since it reduces the number of cases that need to

be considered by providing a list of criteria for which Equations (2.4) and

(2.5) cannot simultaneously be solved [Ser91, Theorem 3.3].

Theorem 2.11 (Unsolvability Conditions). Let

n = 2εp1p2 · · · pk

be a square-free positive integer with p1, p2, . . . , pk being primes that are not

28


equal to two, ε ∈ {0, 1}, and k ∈ N+. Define

R := {±2εpε11 pε22 · · · p

εkk |ε, ε1, ε2, . . . , εk ∈ {0, 1}}

and let (b1, b2) ∈ R × R. The system of equations given by Equations (2.4)

and (2.5) has no solution (z1, z2, z3) ∈ Q∗ ×Q∗ ×Q in the following cases:

1. b1 · b2 < 0,

2. 2 - n and 2|b1.

Once again, similar to the method of 2-descent, the method of complete

2-descent can be a long and tedious process to execute. In Section 4.1 of

Chapter 4, the method of complete 2-descent is applied to prove a theo-

rem of Iskra (see Theorem 4.1) that generates a family of non-congruent

numbers with arbitrarily many prime factors [Isk96]. A second approach

for proving Iskra’s theorem is also presented in Chapter 4. Unlike the

method of complete 2-descent using quadratic equations which involves a

series of lengthy and complex calculations, this new technique that I de-

veloped and describe in my paper [RSY13] offers a simple and elegant ap-

proach for determining whether a given square-free positive integer is non-

congruent. This method uses linear algebra in conjunction with a result of

Monsky [DJS09, HB94, Mon90] to compute the 2-Selmer rank of a congru-

ent number elliptic curve. Recall that the rank obtained by carrying out a

2-descent is called the Mordell-Weil rank. It is a known fact that an elliptic

curve’s Mordell-Weil rank must be less than or equal to its 2-Selmer rank

[DJS09, HB94, Mon90, Sil09]. Therefore, because congruent numbers have

elliptic curves with a positive Mordell-Weil rank, if a curve is found to have

a 2-Selmer rank of zero, it must correspond to a non-congruent number. For

a thorough explanation of the theory governing the relationship between the

Mordell-Weil rank and the 2-Selmer rank of an elliptic curve, see Chapter

X of [Sil09].

29

2.7. Monsky’s Formula for the 2-Selmer Rank

2.7 Monsky’s Formula for the 2-Selmer Rank

In order to bound the Mordell-Weil rank of the elliptic curve En : y2 =

x(x2−n2), we compute the curve’s 2-Selmer rank, s(n). To do this, we utilize

Monsky’s formula for the 2-Selmer rank given by Equation (2.7) [DJS09,

HB94, Mon90].

Let n be a square-free positive integer with odd prime factors P1, P2, . . . , Pt.

We define diagonal t×t matrices Dl = [di] for l ∈ {−2,−1, 2}, and the square

t× t matrix A = [aij ] by

dii =

0, if

(lPi

)= +1,

1, if(lPi

)= −1,

and

aij =

0, if

(PjPi

)= +1, j 6= i,

1, if(PjPi

)= −1, j 6= i,

aii =∑j:j 6=i

aij .

Then

s(n) =

{2t− rankF2(Mo), if n = P1P2 · · ·Pt,2t− rankF2(Me), if n = 2P1P2 · · ·Pt,

(2.7)

where Mo and Me are the 2t× 2t matrices:

Mo =

A + D2 D2

D2 A + D−2

, Me =

D2 A + D2

AT + D2 D−1

. (2.8)

We use the fundamental inequality

r(n) ≤ s(n), (2.9)

where r(n) is the Mordell-Weil rank of En.

Note that the inequality in Equation (2.9) is particularly useful for gen-

erating families of non-congruent numbers. For a given elliptic curve En, if

30

2.7. Monsky’s Formula for the 2-Selmer Rank

Monsky’s formula yields a 2-Selmer rank equal to zero, then the inequality

implies that the curve’s Mordell-Weil rank must also be zero. Hence, n is a

non-congruent number. We will use this technique to prove Iskra’s theorem

in Chapter 4, and to generate infinitely many new families of non-congruent

numbers in Chapters 5 and 6.

31

Chapter 3

A Family of Congruent

Numbers with Three Prime

Factors

The purpose of this chapter is to provide a method for constructing congru-

ent numbers with three prime factors of the form 8k + 3. A family of such

numbers is given for which the Mordell-Weil rank of their associated elliptic

curves equals two, the maximal rank for a congruent number curve of this

type [RSY11].

Recall that from Table 1.2 in Chapter 1, we know that p3 and p3q3

are non-congruent numbers. In addition to this, Kida noticed that 1419 =

3 · 11 · 43 is the only congruent number less that 4500 of the form p3q3r3

and that quite often a 2-descent shows that a number of the form p3q3r3 is

non-congruent [Kid93]. Other congruent numbers p3q3r3 less than 10, 000

include 4587 = 3·11·139, 4731 = 3·19·83, 6963 = 3·11·211, 7611 = 3·43·59,

and 9339 = 3 · 11 · 283 [RSY11]. The Magma code in Appendix A provides

verification that these five numbers are congruent. Our goal is to generate

a family of congruent numbers n = p3q3r3 for which we can prove that the

Mordell-Weil rank of

y2 = x(x2 − n2) (3.1)

is equal to two. We obtain this family by specializing a larger family used to

generate congruent numbers p3q3r3. Both of these families are conjecturally

infinite.

In Section 3.1, we state the main theorem of this chapter, give our

method of construction for congruent numbers p3q3r3, and provide the back-

32

3.1. Preliminary Results

ground material necessary for the proof of our theorem. In Section 3.2, we

prove our main theorem.

3.1 Preliminary Results

We begin by presenting the central theorem in this chapter.

Theorem 3.1. Suppose that the prime numbers q and r have the form

q = 3u4 + 3v4 − 2u2v2,

r = 3u4 + 3v4 + 2u2v2,

for non-zero integers u and v. Set n = 3qr. Then q ≡ r ≡ 3 (mod 8), n is a

congruent number, and the elliptic curve given by Equation (3.1) has a rank

of two.

Since the definition of a congruent integer can be immediately extended

to rational numbers, we can give the following lemma.

Lemma 3.2. Let v be a rational number with v /∈ (−∞,−1] ∪ [0, 1]. Then

v(v − 1)(v + 1) (3.2)

is a congruent number.

Proof. The restriction on v ensures that v(v − 1)(v + 1) is positive. If v is

an integer, then the congruent number v(v − 1)(v + 1) is a special case of

a formula in [Alt80]. It is sufficient to note that if n = v(v − 1)(v + 1) is a

rational number, then the elliptic curve given by (3.1) has the non-torsion

point

(x, y) =(−v(v − 1)2,−2v2(v − 1)2

).

This point is obtained by solving the torsor given by Equation (2.1):

N2 = b1M4 + aM2e2 + b2e

4.

33


Notice that for our elliptic curve y2 = x3 − n2x, we have a = 0 and b =

−n2 = b1b2. Therefore, the torsor reduces to

N2 = b1M4 + b2e

4.

By choosing b1 = −v(v − 1)2 and b2 = −n2

b1= v(v + 1)2, the torsor becomes

N2 = −v(v − 1)2M4 + v(v + 1)2e4.

Substituting M = 1 and e = 1 into the above equation and simplifying

yields

N2 = 4v2.

Hence N = ±2v, so by Equation (2.2), we know that the point

(x, y) =

(b1M

2

e2,b1MN

e3

)=(−v(v − 1)2,−2v2(v − 1)2

)(3.3)

is on our curve. Note that if we choose N = −2v, then we obtain the point(−v(v − 1)2, 2v2(v − 1)2

), which is simply the inverse of the point given by

Equation (3.3). Recall that elliptic curves of the form y2 = x3−n2x have the

following four torsion points: O, (0, 0), (n, 0), and (−n, 0). Clearly, the point

given by Equation (3.3) does not correspond to any of these four points, as

for v /∈ (−∞− 1] ∪ [0, 1], the y-coordinate of the point in Equation (3.3) is

not equal to zero. Thus,(−v(v − 1)2,−2v2(v − 1)2

)is a non-torsion point

on the curve (3.1). This indicates that the rank of the curve must be at

least one, so v(v − 1)(v + 1) is a congruent number.

Lemma 3.3. Suppose that the prime numbers p3, q3, and r3 satisfy

q3 = p3a2 − 16b2,

r3 = p3a2 + 16b2,

for integers a and b. Then n = p3q3r3 is a congruent number.

34


Proof. Put v = p3a2/16b2 into Equation (3.2) to give the congruent number

p3a2/16b2(p3a

2/16b2 − 1)(p3a2/16b2 + 1).

This number is positive if we impose the restrictions stated in Lemma 3.2.

Since congruent numbers scaled by squares are still congruent, we multiply

by 212b6/a2 to obtain the stated congruent number p3q3r3.

Lemma 3.4. If

n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) (3.4)

for a rational number z 6= 0,±1, then the rank of the elliptic curve given by

Equation (3.1) is at least two with at most finitely many exceptions.

Proof. To obtain the formula for n stated in (3.4), we begin by considering

Equation (3.2). We would like to solve the torsor given by Equation (2.1)

corresponding to the elliptic curve y2 = x(x2−n2) with n = v(v−1)(v+ 1).

For our congruent number elliptic curve, we have a = 0 and b = −n2 = b1b2,

so the torsor reduces to

N2 = b1M4 + b2e

4.

We choose b1 = v(v − 1)(v + 1)2, so b2 = −n2

b1= −v(v − 1) and the torsor

becomes

N2 = v(v − 1)(v + 1)2M4 − v(v − 1)e4.

Substituting M = 1 and e = 1 into the above equation and simplifying

yields

N2 = v2(v − 1)(v + 2).

Next we make a change of variable and set N = w ·v. This causes the above

equation to reduce to

w2 = (v − 1)(v + 2).

Now we would like to make a substitution for v such that (v − 1)(v + 2)

35


becomes a perfect square. By using Maple’s parametrization command, we

deduce that v =(−2−t22t−1

)transforms the right-hand of the above equation

into a square. The Maple code used to carry out this computation can be

found in Appendix B. Substituting this value for v into Equation (3.2) yields

v(v − 1)(v + 1) =−(2 + t2)(3 + t2 − 2t)(t+ 1)2

(2t− 1)3. (3.5)

We want this number to have the form 3q3r3. As a result, we must make a

substitution for t that produces a factor of 3 and changes the denominator

in Equation (3.5) into a perfect square. Substituting t = −3z2+12 into (3.5)

yields the desired result:

3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2)(z2 − 1)2

64z6.

Scaling this equation by squares gives us

3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2),

which is Equation (3.4). Thus, to summarize, we obtain Equation (3.4) by

substituting

v =3z4 − 2z2 + 3

4z2

into Equation (3.2) and then scaling by a factor of

(z2 − 1)2

64z6

to remove the squares. Note that the restriction z 6= 0,±1 ensures that

v > 1.

Our next step is to verify that n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) is

a congruent number. To do this, we will show that for this value of n, the

elliptic curve (3.1) over Q(z) possesses the two points

(x1, y1) =(−9(3 + 3z4 − 2z2

)(z2 − 1)2, 36(3 + 3z4 − 2z2)2z(z2 − 1)) (3.6)

36


and

(x2, y2) =

(3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2)

4z2,

9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1)

8z3

). (3.7)

By Lemma 3.2, we know that the point (x, y) =(−v(v − 1)2,−2v2(v − 1)2

)lies on the curve y2 = x3 − n2x with n = v(v − 1)(v + 1). Substituting

v = 3z4−2z2+34z2

into the x-coordinate of the point and scaling it by a factor

of (z2−1)264z6

yields

−9(3 + 3z4 − 2z2)(z2 − 1)2.

The corresponding y-coordinate, 36(3 + 3z4 − 2z2)2z(z2 − 1), can be found

by substituting x = −9(3 + 3z4 − 2z2(z2 − 1)2 into y2 = x3 − n2x where

n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) and then solving for y. Thus,

(x1, y1) =(−9(3 + 3z4 − 2z2

)(z2 − 1)2, 36(3 + 3z4 − 2z2)2z(z2 − 1))

is a point on our curve.

A second point on our curve can be found by using the solution to the

torsor that we solved at the beginning of the proof with b1 = v(v−1)(v+1)2.

Recall that each solvable torsor corresponds to a point on our elliptic curve.

This point is given by Equation (2.2) and for b1 = v(v − 1)(v + 1)2, M = 1,

and e = 1, its x-coordinate reduces to

v(v − 1)(v + 1)2.

Substituting v = 3z4−2z2+34z2

and scaling this coordinate by a factor of (z2−1)264z6

yields3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2)

4z2.

Setting x = 3(3+3z4+2z2)2(3+3z4−2z2)4z2

and solving y2 = x3 − n2x with n =

37


3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) for y gives

y =9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1)

8z3.

Thus, a second point on our elliptic curve is

(x2, y2) =

(3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2)

4z2,

9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1)

8z3

).

Finally, we must show that the two points, (x1, y1) and (x2, y2), are

independent. To do this, we apply Silverman’s specialization theorem (See

Theorem 2.7). If z = 2, then Equation (3.4) yields the congruent number

n = 7611 = 3 · 43 · 59, while (3.6) and (3.7) give two points on y2 =

x(x2 − 76112), namely

(x1, y1) = (−3483, 399384)

and

(x2, y2) =

(449049

16,289636605

64

).

The Magma code in Appendix A confirms that these two non-torsion points

are independent in the group of rational points on y2 = x(x2 − 76112). By

Silverman’s specialization theorem, the two points given by (3.6) and (3.7)

are independent over Q (z) and are therefore independent for all but finitely

many values of the rational number z. Thus, for n = 3(3 + 3z4 − 2z2)(3 +

3z4 +2z2), the rank of the curve given by Equation (3.1) is at least two with

at most finitely many exceptions.

Lemma 3.5. Let

n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2)

38


for a rational number z 6= 0,±1. If (3+3z4−2z2) = p3c2 and (3+3z4+2z2) =

q3d2 for distinct primes p3 and q3 different from 3, and rational numbers c

and d, then the rank of the congruent number curve given by Equation (3.1)

is at least two.

Proof. If we impose the restrictions that (3 + 3z4 − 2z2) = p3c2 and (3 +

3z4 +2z2) = q3d2 for distinct primes p3 and q3 different from 3, and rational

numbers z, c, and d, then an argument using the method of 2-descent shows

that the points given by Equations (3.6) and (3.7) are always independent.

Consider the x-coordinates of the points (x1, y1) and (x2, y2) modulo Q∗2

and notice that

x1 = −9p3c2(z2 − 1)2 ≡ −p3 (mod Q∗2)

and

x2 =3(q3d

2)2p3c2

4z2≡ 3p3 (mod Q∗2).

Clearly −p3 and 3p3 are not congruent modulo Q∗2, so they are two unique

elements in α(Γ). Hence, for (3+3z4−2z2) = p3c2 and (3+3z4+2z2) = q3d

2,

the points (x1, y1) and (x2, y2) are always independent.

In order to bound the rank, r(n), of the congruent curves in our theorem,

we need to make use of Monsky’s formula for s(n), the 2-Selmer rank, which

was introduced in Section 2.7.

Lemma 3.6. If n = p3q3r3, then s(n) ≤ 2.

Proof. We calculate s(n) using Equations (2.7) and (2.8) with P1 = p3,

P2 = q3 and P3 = r3 for all possible choices of values for the Legendre

symbols(p3q3

),(p3r3

), and

(q3r3

). We record the results for all eight cases in

Table 3.1 and provide the Maple code that was used to obtain the results in

Appendix B.

39


Table 3.1: Values of s(n) for n = p3q3r3(p3q3

) (p3r3

) (q3r3

)s(n)

+1 +1 +1 0

+1 +1 −1 0

+1 −1 +1 2

+1 −1 −1 0

−1 +1 +1 0

−1 +1 −1 2

−1 −1 +1 0

−1 −1 −1 0

Remark 3.7. In the proof of Lemma 3.6, the six cases where s(n) = 0

are related by permutation of the primes p3, q3, and r3. The cases where

s(n) = 2 are similarly related.

We recall Schinzel’s hypothesis H [SS58], which states that if a finite

product Q(x) =m∏i=1

fi(x) of polynomials fi(x) ∈ Z [x] has no fixed divisors,

then all of the fi(x) are simultaneously prime, for infinitely many integral

values of x. From this hypothesis we deduce that for any fixed prime p3 the

two forms

p3a2 − 16b2 and p3a

2 + 16b2 (3.8)

assume prime values infinitely often. To ensure that these two forms result

in q3 and r3 being prime numbers, a must be odd. By Lemma 3.3, the

number n = p3q3r3 is guaranteed to be congruent. All of the examples

of congruent numbers mentioned in the introduction have p3 = 3, but we

can generate examples for any fixed prime p3 by using Equation (3.8). For

40

3.2. Proof of the Main Theorem

example if p3 = 43 then using (3.8) with a = 9 and b = 1 yields the value

n = p3q3r3 = 43 · 3467 · 3499,

which by Lemma 3.3 is a congruent number.

3.2 Proof of the Main Theorem

We now provide the proof of Theorem 3.1.

Proof. If the formulas for q and r given in the statement of our theorem

assume prime values, then u and v must have opposite parity. Without loss

of generality, suppose that u = 2h+ 1 and v = 2j, with j, h ∈ Z and j 6= 0.

Then

q = 3u4 + 3v4 − 2u2v2

= 3(2h+ 1)4 + 3(2j)4 − 2(2h+ 1)2(2j)2

= 8(6h4 + 12h3 + 9h2 + 3h+ 6j4 − 4j2h2 − 4j2h− j2) + 3

≡ 3 (mod 8).

A similar argument shows that r is also congruent to 3 modulo 8. Thus,

q ≡ r ≡ 3 (mod 8). From Lemma 3.4, we know that the curve y2 = x(x2−n2)with n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) has rank at least two for all but

finitely many values of the rational number z. Hence, setting z = u/v and

scaling by v8 shows that n = 3qr is a congruent number. By Lemma 3.5,

the curve (3.1) with n = 3qr has rank at least two. However, Lemma 3.6

shows that s(n) ≤ 2, and since the rank is bounded above by s(n), the rank

is at most two. Thus, the rank equals two and the theorem is proved.

Example 3.8. A few smaller congruent numbers whose associated congru-

ent number curves have rank two and are generated by the formulas in our

theorem include 7611 = 3 · 43 · 59, 1021683291 = 3 · 13219 · 25763, and

2700420027 = 3 · 30203 · 29803.

41

Chapter 4

Iskra’s Family of

Non-congruent Numbers

This chapter focuses on a theorem proven by Iskra that describes a family of

non-congruent numbers with arbitrarily many prime factors. The theorem,

which appeared in Iskra’s paper “Non-congruent numbers with arbitrarily

many prime factors congruent to 3 modulo 8” [Isk96], provides an answer

to the following question posed by Kida [Kid93]:

Can we find an infinite set of primes of the form 8k + 3 with k ∈ Zsuch that for any product n of primes in the set, the elliptic curve

y2 = x(x2 − n2) has a rank of zero?

By applying the method of complete 2-descent, Iskra proved that the curve

y2 = x(x2 − n2) has a rank of zero for infinitely many values of n whose

prime factors are congruent to 3 modulo 8 and satisfy a certain pattern of

Legendre symbols. This family of non-congruent numbers is described by

Iskra’s theorem [Isk96].

Theorem 4.1 (Iskra’s Theorem). Let p1, p2, . . . , pt be distinct primes

such that pi ≡ 3 (mod 8) and(pjpi

)= −1 for j < i. Then the product

n = p1p2 · · · pt is a non-congruent number.

In this chapter, we prove Iskra’s theorem using two different approaches.

In Section 4.1 we prove the theorem by using the method of complete 2-

descent; this approach is based on Iskra’s proof [Isk96]. However, for the

sake of clarity, and to provide a thorough example of the method of com-

plete 2-descent, we include additional details. Following this, in Section 4.2,

we present a new technique for generating non-congruent numbers. This

42

4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

method uses linear algebra in conjunction with Monsky’s formula for the

2-Selmer rank to provide a simple and elegant proof of Iskra’s theorem. We

conclude this section by verifying that for any value of t, there always exists

a collection of primes satisfying the conditions of Theorem 4.1.

4.1 The Proof of Iskra’s Theorem Using the

Method of Complete 2-Descent

In order to apply the method of complete 2-descent to prove Theorem 4.1,

we need to use various properties of Legendre symbols and congruences.

Remark 4.2. Since all of the primes in Theorem 4.1 are congruent to 3

modulo 8, we can make the following useful simplifications [Isk96].

1. Since pi ≡ 3 (mod 8), by 3(i) and 3(ii) in Theorem 1.29 we deduce

that (−1

pi

)= −1 and

(2

pi

)= −1.

2. Because(pjpi

)= −1 for j < i, it follows that

(pjpi

)= +1 if i < j.

Explanation: Since pi and pj are distinct primes that are congruent

to 3 modulo 8, the above fact can easily be deduced from the law of

quadratic reciprocity that was stated in Theorem 1.29.

3. Let n = nipi. Then (nipi

)= (−1)i−1.

Explanation: Since n = p1p2 · · · pt = nipi, we have ni = p1 · · · pi−1pi+1 · · · pt.This means that (

nipi

)=

(p1 · · · pi−1pi+1 · · · pt

pi

).

43


By assumption, the primes are all distinct, so we can apply 1(ii) The-

orem 1.29 to rewrite the Legendre symbol on the right-hand side of the

above equation as a product of (t− 1) Legendre symbols:(p1 · · · pi−1pi+1 · · · pt

pi

)=

(p1pi

)(p2pi

)· · ·(pi−1pi

)(pi+1

pi

)· · ·(ptpi

).

Recall that(pjpi

)= −1 if j < i and that

(pjpi

)= +1 if i < j. As a

result, the above equation reduces to(nipi

)= (−1)(−1) · · · (−1)(+1) · · · (+1) = (−1)i−1.

4. Let b be a divisor of n, and define

b′ =

b

piif pi|b,

b if pi - b.

Let k = |{j : pj |b and j < i}|. Then(b′

pi

)= (−1)k.

Explanation: Since b is a divisor of n = p1p2 · · · pt, we have b =

pa1pa2 · · · paf where ah ∈ {1, 2, . . . , t} for all h = 1, 2, . . . , f , and the

ahs are all distinct. Because b′ does not have pi as a factor and the

primes pah are distinct, we can apply 1(ii) Theorem 1.29 to get(b′

pi

)=

(pa1pa2 · · · paf

pi

)=

(pa1pi

)(pa2pi

)· · ·(pafpi

).

We know that(pahpi

)= −1 if ah < i and that

(pahpi

)= +1 if ah > i.

By definition, there are k primes that divide b and satisfy the property

that ah < i, so k of the Legendre symbols on the right-hand side of the

above equation have a value of −1 and the remainder of the Legendre

44


symbols have a value of +1. Thus,(b′

pi

)= (−1)k.

We are now ready to apply the method of complete 2-descent to prove

Iskra’s theorem.

Proof. To prove that n = p1p2 · · · pt is a non-congruent number, we apply

Theorems 2.10 and 2.11 to show that for all pairs (b1, b2) 6∈ {(1, 1), (−1,−n),

(n, 2), (−n,−2n)} with bi ∈ {±2εpε11 pε22 · · · p

εkk |ε, ε1, ε2, . . . , εk ∈ {0, 1}}, Equa-

tions (2.4) and (2.5) cannot simultaneously be solved. By the unsolvability

conditions stated in Theorem 2.11, we know that Equations (2.4) and (2.5)

do not have a solution when b1 · b2 < 0 or when 2 - n and 2|b1. Since

n = p1p2 · · · pt with each of the pi ≡ 3 (mod 8), it follows that 2 - n. There-

fore, we only need to consider pairs (b1, b2) for which b1 · b2 > 0 and 2 - b1.We split these restrictions into four separate cases and verify that in each of

these cases, there does not exist a pair (b1, b2) that simultaneously satisfies

Equations (2.4) and (2.5).

Case 1: b2 > 0 and 2 - b2Define

s = min{i : pi|b1 or pi|b2}.

If s exists, then ps|b1, or ps|b2, or both of these division statements hold.

Consider(b′1ps

). By Property 4 in Remark 4.2, we know that

(b′1ps

)= (−1)k,

where k = |{j : pj |b1 and j < s}|. However, ps is by definition the prime

with the smallest subscript that divides b1 or b2. Therefore, there cannot

exist an integer j with j < s such that pj |b1. As a result, we conclude that

the set {j : pj |b1 and j < s} is empty and that k = 0. It follows that(b′1ps

)= (−1)0 = +1. (4.1)

45


By using an analogous argument, we can also deduce that(b′2ps

)= +1. (4.2)

According to the definition of s, we now need to consider three separate

subcases.

Subcase 1: ps|b1 and ps|b2Consider Equation (2.5). Dividing both sides by ps yields

b1psz21 −

b1psb2z

23 =−nps.

By using Properties 3 and 4 from Remark 4.2, we can replace nps

by ns andb1ps

by b′1. Therefore, the equation becomes

b′1z21 − b′1b2z23 = −ns.

Consider this equation modulo ps. Since b2 contains a factor of ps, we know

that b′1b2z23 ≡ 0 (mod ps). As a result, our equation reduces to

b′1z21 ≡ −ns (mod ps).

Multiplying both sides of this congruence by b′1 yields

(b′1z1)2 ≡ −nsb′1 (mod ps).

Clearly ns and b′1 are not divisible by ps, so nsb′1 is not divisible by ps.

Therefore, it follows that the left-hand side of the congruence is not divisible

by ps either. We can now apply 1(i) Theorem 1.29 to write((b′1z1)

2

ps

)=

(−nsb′1ps

).

Since ps does not divide (b′1z1)2, ps does not divide (b′1z1), so by 1(iii) The-

orem 1.29, we know that((b′1z1)

2

ps

)= +1. This allows us to conclude that

46


(−nsb′1ps

)= +1. (4.3)

If we consider the Legendre symbol in Equation (4.3) and apply 1(ii) Theo-

rem 1.29, we can write(−nsb′1ps

)=

(−1

ps

)(nsps

)(b′1ps

).

Note that(−1ps

)= −1 by Property 1 in Remark 4.2,

(nsps

)= (−1)s−1 by

Property 3 in Remark 4.2, and(b′1ps

)= +1 by Equation (4.1). Thus,

(−nsb′1ps

)= (−1)(−1)s−1(+1) = (−1)s. (4.4)

Now consider the following equation:

b1b2z23 − b2z22 = 2n. (4.5)

This equation can be obtained by subtracting Equation (2.5), b1z21−b1b2z23 =

−n, from Equation (2.4), b1z21−b2z22 = n. If we divide both sides of Equation

(4.5) by ps, we can write

b1b2psz23 −

b2psz22 = 2

n

ps.

By applying Properties 3 and 4 from Remark 4.2, we can replace nps

by ns

and b2ps

by b′2. Therefore, the equation becomes

b1b′2z

23 − b′2z22 = 2ns.

Consider this equation modulo ps. Since b1 contains a factor of ps, we know

that b1b′2z

23 ≡ 0 (mod ps). As a result, our equation reduces to

−b′2z22 ≡ 2ns (mod ps).

47


Multiplying both sides of this congruence by −b′2 yields

(b′2z2)2 ≡ −2nsb

′2 (mod ps).

Clearly 2, ns, and b′2 are not divisible by ps, so 2nsb′2 is not divisible by ps.

Therefore, the left-hand side of the congruence is not divisible by ps either.

We can now apply 1(i) Theorem 1.29 to write((b′2z2)

2

ps

)=

(−2nsb

′2

ps

).

Since ps does not divide (b′2z2)2, it follows that ps does not divide (b′2z2), so

by 1(iii) Theorem 1.29, we know that((b′2z2)

2

ps

)= +1. This enables us to

conclude that (−2nsb

′2

ps

)= +1. (4.6)


rem 1.29, we can write(−2nsb

′2

ps

)=

(−1

ps

)(2

ps

)(nsps

)(b′2ps

).

Since(−1ps

)= −1 and

(2ps

)= −1 by Property 1 in Remark 4.2,

(nsps

)=

(−1)s−1 by Property 3 in Remark 4.2, and(b′2ps

)= +1 by Equation (4.2),

the above product of Legendre symbols simplifies to(−2nsb

′2

ps

)= (−1)(−1)(−1)s−1(+1) = (−1)s−1. (4.7)

Now we compare Equations (4.4) and (4.7). Because(−nsb′1ps

)= (−1)s and(

−2nsb′2ps

)= (−1)s−1, it follows that for every s, one of these two Legendre

symbols must have a value of −1. This is a contradiction, since by Equations

(4.3) and (4.6), we know that both of these Legendre symbols have a value

of +1. Thus, we conclude that in the case where ps|b1 and ps|b2, there is no

solution that simultaneously solves Equations (2.4) and (2.5).

48


Subcase 2: ps|b1 and ps - b2Consider the following equation:

2b1z21 − b2z22 − b1b2z23 = 0. (4.8)

This equation can be obtained by adding Equation (2.4), b1z21−b2z22 = n, to

Equation (2.5), b1z21 − b1b2z23 = −n. Dividing both sides of Equation (4.8)

by ps yields

2b1psz21 −

b2z22

ps− b1psb2z

23 = 0.

By applying Property 4 from Remark 4.2, we can replace b1ps

by b′1. Therefore,

the equation becomes

2b′1z21 −

b2z22

ps− b′1b2z23 = 0.

Recall that if a prime divides a product, then it must divide at least one of

its factors. By assumption, we know that ps does not divide b2. Therefore,

ps must divide z2. If z2 contains a factor of ps, then b2z2z2ps≡ 0 (mod ps).

As a result, the above equation reduces to

2b′1z21 − b′1b2z23 ≡ 0 (mod ps).

Rearranging this congruence and multiplying both sides by 2b′1 yields

(2b′1)2z21 ≡ 2(b′1)

2b2z23 (mod ps).

Note that ps cannot divide both z1 and z2. If it did, then p2s would be

a common factor of the left-hand side of Equation (2.4), so it would also

be a common divisor of the right-hand side, n. However, by assumption

n only contains a single factor of ps. Therefore, p2s cannot divide n, so ps

cannot divide both z1 and z2. We already deduced that ps is a factor of

z2, so it follows that ps cannot divide z1. Since ps does not divide 2, b′1, or

z1, we conclude that ps is not a common factor of the left-hand side of the

above congruence. This means that ps must not be a common factor of the

49


right-hand side either. We can now apply 1(i) Theorem 1.29 to write((2b′1z1)

2

ps

)=

(2b2(b

′1z3)

2

ps

).

By using 1(ii) and 1(iii) from Theorem 1.29, we can rewrite the Legendre

symbol on the right-hand side of the equation as(2b2(b

′1z3)

2

ps

)=

(2b2ps

)((b′1z3)

2

ps

)=

(2b2ps

)(+1).

Also, by 1(iii) Theorem 1.29, we know that((2b′1z1)

2

ps

)= +1. Utilizing these

simplifications allows us to conclude that(2b2ps

)= +1. (4.9)


rem 1.29, we can write (2b2ps

)=

(2

ps

)(b2ps

).

Since ps does not divide b2, by Property 4 in Remark 4.2, we can replace b2

by b′2. In addition, by Property 1 in Remark 4.2, we know that(

2ps

)= −1,

and by Equation (4.2), we know that(b′2ps

)= +1. Therefore,

(2b2ps

)= (−1)(+1) = −1.

However, this result contradicts Equation (4.9). Thus, we conclude that

when ps|b1 and ps - b2, there does not exist a solution that simultaneously

satisfies Equations (2.4) and (2.5).

50


Subcase 3: ps - b1 and ps|b2Consider Equation (4.8) and divide both sides of it by ps to get

2b1z21

ps− b2psz22 − b1

b2psz23 = 0.

By applying Property 4 from Remark 4.2, we can replace b2ps

by b′2, which

yields2b1z

21

ps− b′2z22 − b1b′2z23 = 0.

Since ps divides 2b1z21 , it must divide at least one of its factors. By our

initial assumptions, we know that ps does not divide 2 or b1. Therefore, ps

must divide z1. If z1 contains a factor of ps, then 2b1z1z1ps≡ 0 (mod ps). As

a result, the above equation reduces to

−b′2z22 − b1b′2z23 = 0 (mod ps).

Rearranging this congruence and multiplying both sides by −b′2 yields

b′22 z22 = −b1b′22 z23 (mod ps).

By using the same argument as in Subcase 2, we conclude that ps cannot

divide both z1 and z2. Since we deduced that z1 is divisible by ps, it follows

that ps does not divide z2. Because b′2 and z2 are not divisible by ps, we

conclude that ps is not a factor of the left-hand side of the above congruence.

Consequently, we know that ps must not divide the right-hand side either.

As a result, we can apply 1(i) Theorem 1.29 to write((b′2z2)

2

ps

)=

(−b1(b′2z3)2

ps

).

By using 1(ii) and 1(iii) from Theorem 1.29, we can rewrite the Legendre

symbol on the right-hand side of the equation as(−b1(b′2z3)2

ps

)=

(−b1ps

)((b′2z3)

2

ps

)=

(−b1ps

)(+1).

51


In addition, by 1(iii) Theorem 1.29, we know that((b′2z2)

2

ps

)= +1. Applying

these simplifications allows us to conclude that(−b1ps

)= +1. (4.10)

Notice that if we apply 1(ii) Theorem 1.29 to the Legendre symbol in Equa-

tion (4.10), we obtain (−b1ps

)=

(−1

ps

)(b1ps

).

Since ps does not divide b1, by Property 4 in Remark 4.2, we can replace b1

by b′1. In addition, by Property 1 in Remark 4.2, we know that(−1ps

)= −1,

and by Equation (4.1), we know that(b′1ps

)= +1. Therefore,

(−b1ps

)= (−1)(+1) = −1.

However, this result contradicts Equation (4.10). Thus, we conclude that if

ps - b1 and ps|b2, there does not exist a solution that simultaneously solves

Equations (2.4) and (2.5).

None of the three subcases of Case 1 provide a pair, (b1, b2), for which

Equations (2.4) and (2.5) are simultaneously solvable. Therefore, we deduce

that s = min{i : pi|b1 or pi|b2} does not exist, which means that no prime

divides b1 or b2. As a result, since b1 · b2 > 0 and b2 > 0, we conclude

that (b1, b2) = (1, 1), which is a contradiction to our initial assumption that

(b1, b2) 6∈ {(1, 1), (−1,−n), (n, 2), (−n,−2n)}.

We now consider Case 2 where b2 > 0 and 2|b2, Case 3 where b2 < 0

and 2 - b2, and Case 4 where b2 < 0 and 2|b2. In his paper [Isk96], Iskra

utilized arguments similar to those presented in Case 1 to verify that the

remaining three cases do not yield pairs for which Equations (2.4) and (2.5)

are solvable. However, carrying out this process for each of the cases is

52


not only lengthy and tedious, but also unnecessary. By making use of the

properties of groups and our results from Case 1, we can easily verify that

that Cases 2, 3, and 4 do not yield any pairs either.

Case 2: b2 > 0 and 2|b2By way of contradiction, suppose there exists a pair (b1, b2) with b2 > 0 and

2|b2 for which Equations (2.4) and (2.5) are simultaneously solvable. Since

the set of points δ = {(1, 1), (−1,−n), (n, 2), (−n,−2n), (b1, b2)} generates a

finite subgroup of Q∗/Q∗2 × Q∗/Q∗2, by closure the pair (b1, b2) · (n, 2) =

(nb1, 2b2) must also belong to the group. By assumption 2|b2, so we can

write b2 = 2b∗2, where b∗2 ∈ Q∗/Q∗2 and 2 - b∗2. If we set b∗1 = nb1, then we

have (nb1, 2b2) = (b∗1, 4b∗2). Because (b∗1, 4b

∗2) ∈ Q∗/Q∗2 ×Q∗/Q∗2, it follows

that 4b∗2 is equivalent to b∗2. Notice that the pair (b∗1, b∗2) has b∗2 > 0 and

2 - b∗2. This means that the pair (b∗1, b∗2) has the same properties as the one

described in Case 1. However, from Case 1, we know that there does not

exist a solution that simultaneously solves Equations (2.4) and (2.5) for pairs

of this form. Therefore, (b∗1, b∗2) cannot belong to the group generated by the

set δ. This is a contradiction, so our initial assumption that (b1, b2) with

b2 > 0 and 2|b2 is an element of δ must be incorrect. Thus, we conclude that

there does not exist a pair (b1, b2) with b2 > 0 and 2|b2 for which Equations

(2.4) and (2.5) are solvable.

Case 3: b2 < 0 and 2 - b2By way of contradiction, assume there exists a pair (b1, b2) with b2 < 0 and

2 - b2 for which Equations (2.4) and (2.5) are simultaneously solvable. Since

the set of points δ = {(1, 1), (−1,−n), (n, 2), (−n,−2n), (b1, b2)} generates a

finite subgroup of Q∗/Q∗2×Q∗/Q∗2, by closure the pair (b1, b2) · (−1,−n) =

(−b1,−nb2) must also be an element of the group. Let b∗1 = −b1 and b∗2 =

−nb2, so (−b1,−nb2) = (b∗1, b∗2). Since 2 - n and 2 - b2, it follows that

2 - (−nb2). In addition, because b2 < 0 and n > 0, we know that −nb2 > 0.

Hence, b∗2 > 0 and 2 - b∗2, which means that the pair (b∗1, b∗2) has the same

properties as the one described in Case 1. By the same argument as in Case

2, we conclude that (b∗1, b∗2) cannot belong to the group generated by the set

53


δ. Hence, there does not exist a pair (b1, b2) with b2 < 0 and 2 - b2 for which

Equations (2.4) and (2.5) have a solution.

Case 4: b2 < 0 and 2|b2By way of contradiction suppose that there exists a pair (b1, b2) with b2 < 0

and 2|b2 for which Equations (2.4) and (2.5) are simultaneously solvable.

Since the set of points δ = {(1, 1), (−1,−n), (n, 2), (−n,−2n), (b1, b2)} gen-

erates a finite subgroup of Q∗/Q∗2 × Q∗/Q∗2, by closure the pair (b1, b2) ·(−n,−2n) = (−nb1,−2nb2) must also belong to the group. By assump-

tion 2|b2, so we can write b2 = 2b~2 , where b~2 ∈ Q∗/Q∗2 and 2 - b~2 . Be-

cause (−nb1,−2nb2) = (−nb1,−4nb~2 ) ∈ Q∗/Q∗2 × Q∗/Q∗2, it follows that

−4nb~2 is equivalent to −nb~2 . If we set b∗1 = −nb1 and b∗2 = −nb~2 , we have

(−nb1,−nb~2 ) = (b∗1, b∗2). Clearly b~2 < 0, since b2 < 0. By using this fact

and noting that n > 0, we deduce that −nb~2 = b∗2 > 0. In addition, because

2 - n and 2 - b~2 , we know that 2 - b∗2. Therefore, the pair (b∗1, b∗2) has b∗2 > 0

and 2 - b∗2, which means that it has the same properties as the one described

in Case 1. By using the same argument as in Case 2, we deduce that (b∗1, b∗2)

cannot belong to the group generated by the set δ. Thus, we conclude that

there does not exist a pair (b1, b2) with b2 < 0 and 2|b2 for which Equations

(2.4) and (2.5) are solvable.

None of the four cases yields a pair (b1, b2) 6∈ {(1, 1), (−1,−n), (n, 2),

(−n,−2n)} with bi ∈ {±2εpε11 pε22 · · · p

εkk |ε, ε1, ε2, . . . , εk ∈ {0, 1}} for which

Equations (2.4) and (2.5) simultaneously have a solution. Therefore, our

bound, B, for the number of such pairs equals zero, so the inequality in

Equation (2.6) becomes

r ≤ log2

(0 + 4

4

)= 0.

Since the rank, r, must be a non-negative integer, the inequality implies that

r = 0. Hence, for n = p1p2 · · · pt, where p1, p2, . . . , pt are distinct primes with

pi ≡ 3 (mod 8) and(pjpi

)= −1 for j < i, the elliptic curve y2 = x(x2 − n2)

has a rank of zero. Thus, n is a non-congruent number.

54

4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .

4.2 The Proof of Iskra’s Theorem Using

Monsky’s Formula for the 2-Selmer Rank

We now provide a proof of Iskra’s theorem using Monsky’s formula for the

2-Selmer rank.

Proof. By applying Equation (2.8), we can define the Monsky matrix, Mo,

for numbers of the form n = p1p2 · · · pt with pi ≡ 3 (mod 8) for all i and(pjpi

)= −1 for j < i.

Mo =

1 0 0 . . . . . . . . . 0 1 0 0 . . . . . . . . . 0

1 2 0 . . . . . . . . . 0 0 1 0 . . . . . . . . . 0

1 1 3. . .

... 0 0 1. . .

......

.... . .

. . .. . .

......

.... . .

. . .. . .

......

.... . .

. . .. . .

......

.... . .

. . .. . .

......

.... . . t− 1 0

......

. . . 1 0

1 1 . . . . . . . . . 1 t 0 0 . . . . . . . . . 0 1

1 0 0 . . . . . . . . . 0 0 0 0 . . . . . . . . . 0

0 1 0 . . . . . . . . . 0 1 1 0 . . . . . . . . . 0

0 0 1. . .

... 1 1 2. . .

......

.... . .

. . .. . .

......

.... . .

. . .. . .

......

.... . .

. . .. . .

......

.... . .

. . .. . .

......

.... . . 1 0

......

. . . t− 2 0

0 0 . . . . . . . . . 0 1 1 1 . . . . . . . . . 1 t− 1

.

Note that Mo is a 2t × 2t matrix. We now execute a series of column and

row interchanges on this matrix. These operations vary slightly depending

on whether t is even or odd. If t is even, we begin by exchanging columns

1 and t, 2 and (t − 1), 3 and (t − 2), . . . , t2 and t+22 . We then interchange

columns (t + 1) and 2t, (t + 2) and (2t − 1), (t + 3) and (2t − 2), . . . , 3t2and 3t+2

2 . Following this, we exchange rows 1 and t, 2 and (t − 1), 3 and

55


(t−2), . . . , t2 and t+22 , and rows (t+1) and 2t, (t+2) and (2t−1), (t+3) and

(2t − 2), . . . , 3t2 and 3t+22 . If t is odd, we exchange columns 1 and t, 2 and

(t−1), 3 and (t−2), . . . , t−12 and t+32 , and columns (t+1) and 2t, (t+2) and

(2t− 1), (t+ 3) and (2t− 2), . . . , 3t−12 and 3t+32 . However, columns t+1

2 and3t+12 are left in their original positions. Similarly, in regards to the exchange

of the rows, we carry out the same procedure as for columns and interchange

all of the rows except for rows t+12 and 3t+1

2 , which remain in their original

positions. Upon carrying out these column and row operations, we obtain

the following matrix:

M′o =

t 1 1 . . . . . . 1 1 1 0 0 . . . . . . . . . 0

0 t− 1 1 . . . . . . 1 1 0 1 0 . . . . . . . . . 0

0 0 t− 2. . .

...... 0 0 1

. . ....

......

. . .. . .

. . ....

......

.... . .

. . .. . .

......

.... . . 3 1 1

......

. . .. . .

. . ....

...... 0 2 1

......

. . . 1 0

0 0 . . . . . . 0 0 1 0 0 . . . . . . . . . 0 1

1 0 0 . . . . . . . . . 0 t− 1 1 1 . . . . . . 1 1

0 1 0 . . . . . . . . . 0 0 t− 2 1 . . . . . . 1 1

0 0 1. . .

... 0 0 t− 3. . .

......

......

. . .. . .

. . ....

......

. . .. . .

. . ....

......

.... . .

. . .. . .

......

.... . . 2 1 1

......

. . . 1 0...

... 0 1 1

0 0 . . . . . . . . . 0 1 0 0 . . . . . . 0 0 0

.

Note that if t is even, we must carry out t column interchanges and t row

interchanges to obtain the matrix M′o. By Property 1 in Definition 1.17, we

obtain

det(Mo) = (−1)2tdet(M′o) = det(M′

o).

Similarly, if t is odd, we must perform (t− 1) column exchanges and (t− 1)

56


row exchanges. By applying Property 1 in Definition 1.17, we obtain

det(Mo) = (−1)2t−2det(M′o) = det(M′

o).

Also, we can write the matrix M′o as

M′o =

[U It

It U− It

],

where It is the t× t identity matrix and U is a t× t matrix of the form

U =

t 1 1 . . . . . . 1 1

0 t− 1 1 . . . . . . 1 1

0 0 t− 2. . .

......

......

. . .. . .

. . ....

......

.... . . 3 1 1

...... 0 2 1

0 0 . . . . . . 0 0 1

.

We perform row interchanges on M′o t times to obtain the matrix

N =

[It U− It

U It

].

By Property 1 in Definition 1.17, det(M′o) = (−1)tdet(N). Applying the

formula for computing block determinants given in Proposition 1.20 yields

det(N) = det(It)det(It −UI −1t (U− It))

= det(It −U(U− It)).

Notice that U(U− It) is a product of two upper triangular matrices, so

by the statement following Definition 1.11, we know that U(U−It) must be

an upper triangular matrix. Each diagonal entry in the matrix U(U − It)

is equal to the product of two consecutive integers, so the diagonal entries

must be even and hence congruent to 0 modulo 2. Therefore, It −U(U −

57


It) is an upper triangular matrix with diagonal entries all congruent to 1

modulo 2. By Property 3 in Theorem 1.18, it follows that the determinant

of It −U(U− It) is congruent to 1 modulo 2. Hence,

det(Mo) = det(M′o)

= (−1)tdet(N)

= (−1)tdet(It −U(U− It))

≡ 1 (mod 2).

Since det(Mo) 6≡ 0 (mod 2), by Property 6 in Theorem 1.18, we know

that rankF2(Mo) = 2t. Therefore, by Equation (2.7), we deduce that s(n) =

0. Thus, the inequality in Equation (2.9) implies that r(n) = 0, so n is a

non-congruent number.

Finally, we verify that for any value of t, there always exists a collection

of primes satisfying the conditions of Theorem 4.1.

Corollary 4.3. Let Ht denote the collection of positive integers with prime

factorization p1p2 · · · pt, where the pi are distinct primes of the form 8k + 3

satisfying(pjpi

)= −1 for all 1 ≤ j < i ≤ t. For any value of t, the set Ht is

non-empty and, in fact, contains infinitely many elements.

Proof. We need to use Dirichlet’s theorem on primes in arithmetic progres-

sion (See Theorem 1.30) to verify that this is true. The case where t = 1 is

obviously true, since by Dirichlet’s theorem on primes in arithmetic progres-

sion, there are infinitely many primes of the form 8k + 3. We use induction

on t, and assume that the result is true up to (t − 1) for t > 1. Now we

must show that the set Ht is infinite. By the induction hypothesis, we know

that there exist integers p1p2 · · · pt−1, where the pi are distinct primes of the

form 8k + 3 satisfying(pjpi

)= −1 for all 1 ≤ j < i < t. We would like to

choose a prime pt satisfying pt ≡ 3 (mod 8),

pt ≡ 1 (mod pj) for each 1 ≤ j < t,(4.11)

58


and append this prime onto the end of the product p1p2 · · · pt−1. The Chi-

nese remainder theorem (See Theorem 1.26) guarantees that the system of

congruences given by (4.11) has a solution. By applying this theorem in

conjunction with Dirichlet’s theorem on primes in arithmetic progression,

we are able to conclude that there exist infinitely many primes pt satisfying

the system of congruences in (4.11). Note that since pt ≡ 1 (mod pj), we

know that pt is a quadratic residue modulo pj for each 1 ≤ j < t. Hence,

by applying Property 2 from Remark 4.2, it follows that(pjpt

)= −1 for all

1 ≤ j < t. Thus, the set Ht contains infinitely many elements.

59

Chapter 5

Families of Non-congruent

Numbers with Arbitrarily

Many Prime Factors of the

Form 8k + 3

In this chapter, we provide an extension to Iskra’s work and generate in-

finitely many distinct new families of non-congruent numbers with arbitrar-

ily many prime factors of the form 8k+3. In order to construct these families,

we utilize the technique involving Monsky’s formula for the 2-Selmer rank

that was presented in Section 4.2.

In Section 5.1, we present the main theorem of this chapter and use lin-

ear algebra to establish necessary conditions to construct the new families

of non-congruent numbers. Following this in Section 5.2, we prove our main

theorem and then conclude with a couple of supplementary corollaries. The

first corollary provides verification that the sets described by our main the-

orem are non-empty, and the second modifies our main theorem in such a

way that it yields congruent numbers.

5.1 Preliminary Results Involving the

Generation of Non-congruent Numbers

We begin by stating the main result of this chapter.

Theorem 5.1. Let m be a fixed non-negative even integer and let t be any

positive integer satisfying t ≥ m. Let Nm denote the set of positive integers

60

5.1. Preliminary Results Involving the Generation of Non-congruent Numbers

with prime factorization p1p2 · · · pt, where p1, p2, . . . , pt are distinct primes

of the form 8k + 3 such that

(pjpi

)=

−1 if 1 ≤ j < i and (j, i) 6= (1,m),

+1 if 1 ≤ j < i and (j, i) = (1,m).(5.1)

If n ∈ Nm, then n is non-congruent. Moreover for m > 0, the sets Nm are

pairwise disjoint.

For convenience we define three matrices that will be used in our con-

struction of non-congruent numbers.

Definition 5.2. For a positive integer r, we define the matrices U, Q, and

A by

U = Ur =

r − 1 1 1 · · · 1 1

0 r − 2 1 · · · 1 1

0 0. . .

. . ....

......

.... . . 2 1 1

0 0 · · · 0 1 1

0 0 · · · 0 0 0

,

Q = Qr =

1 0 0 · · · 0 1

0 0 0 · · · 0 0

0 0 0 · · · 0 0...

......

......

0 0 0 · · · 0 0

1 0 0 · · · 0 1

,

and

A = Ar =

r − 2 1 1 · · · 1 0

0 r − 2 1 · · · 1 1

0 0. . .

. . ....

......

.... . . 2 1 1

0 0 · · · 0 1 1

1 0 · · · 0 0 1

.

61


As usual, I = Ir denotes the r × r identity matrix and 0 = 0r denotes the

r × r zero matrix.

Our first lemma is a direct calculation.

Lemma 5.3. With Q defined as in Definition 5.2, we have

Q2 = 2Q ≡ 0r (mod 2).

The next lemma establishes an identity involving U.

Lemma 5.4. With U defined as in Definition 5.2, we have

U(U + I) ≡ 0r (mod 2).

Proof. We apply mathematical induction on r. The lemma is true when

r = 1 since U =[

0]

and U + I =[

1]. Now assume that

Ur−1 (Ur−1 + Ir−1) ≡ 0r−1 (mod 2).

We can write

Ur =

r − 1 1 · · · 1

0...

0

Ur−1

.Using block multiplication we see that

Ur(Ur+Ir) =

r − 1 1 · · · 1

0...

0

Ur−1

r 1 · · · 10...

0

Ur−1 + Ir−1

,

62


which for some 1× (r − 1) matrix W, simplifies tor(r − 1) W

0...

0

Ur−1 (Ur−1 + Ir−1)

≡

0 W

0...

0

0r−1

(mod 2)

by the induction hypothesis. It remains to calculate W. We see that

W = [r − 1][

1 1 · · · 1]

+[

1 1 · · · 1]

[Ur−1 + Ir−1]

=[r − 1 r − 1 · · · r − 1

]+[

1 1 · · · 1]

r − 1 1 1 · · · 1 1

0 r − 2 1 · · · 1 1

0 0. . .

. . ....

......

.... . . 3 1 1

...... · · · 0 2 1

0 0 · · · 0 0 1

=[r − 1 r − 1 · · · r − 1

]+[r − 1 r − 1 · · · r − 1

]≡[

0 0 · · · 0](mod 2).

The proofs of the next two lemmas use direct calculation.

Lemma 5.5. With U and Q as given in Definition 5.2, we have

UQ =

r 0 0 · · · 0 r

1 0 0 · · · 0 1

1 0 0 · · · 0 1...

......

......

1 0 0 · · · 0 1

0 0 0 · · · 0 0

.

63


Lemma 5.6. With U and Q as given in Definition 5.2, we have

Q(U + I) ≡

r 1 1 · · · 1 0

0 0 0 · · · 0 0

0 0 0 · · · 0 0...

......

......

0 0 0 · · · 0 0

r 1 1 · · · 1 0

(mod 2).

We now prove a lemma that establishes an identity involving A.

Lemma 5.7. With A as given in Definition 5.2, we have

A(A + I) ≡

0 1 1 · · · 1 r

1 0 0 · · · 0 1

1 0 0 · · · 0 1...

......

......

1 0 0 · · · 0 1

r 1 1 · · · 1 0

(mod 2).

Proof. From Definition 5.2 we have

A(A + I) ≡ (U + Q)(U + I + Q) ≡ U(U + I) + UQ + Q(U + I) + Q2(mod 2).

Applying Lemmas 5.3, 5.4, 5.5, and 5.6 yields the desired result.

The next lemma provides the starting point for our families of non-

congruent numbers.

Lemma 5.8. With A = Ar as given in Definition 5.2, r even, and T defined

by

T =

[I A

A + I I

],

we have det(T) ≡ 1 (mod 2).

64


Proof. Recalling Lemma 5.7, we have

A(A + I) ≡

0 1 1 · · · 1 r

1 0 0 · · · 0 1

1 0 0 · · · 0 1...

......

......

1 0 0 · · · 0 1

r 1 1 · · · 1 0

≡

0 1 1 · · · 1 0

1 0 0 · · · 0 1

1 0 0 · · · 0 1...

......

......

1 0 0 · · · 0 1

0 1 1 · · · 1 0

≡ CDT (mod 2),

where

C =

1 0

0 1...

...

0 1

1 0

and DT =

[0 1 · · · 1 0

1 0 · · · 0 1

].

By using this fact and applying Proposition 1.20, we deduce that

det(T) = det (I−A(A + I))

≡ det(I−CDT

)(mod 2).

Furthermore, by applying Proposition 1.21 with B = Ir for r even, we are

able to determine that

65


det(T) ≡ det(I2 −DTC

)(mod 2)

≡ det

[

1 0

0 1

]−

[0 1 · · · 1 0

1 0 · · · 0 1

]

1 0

0 1...

...

0 1

1 0

(mod 2)

≡ det

([1 0

0 1

]−

[0 r − 2

2 0

])(mod 2)

≡ det

([1 0

0 1

])(mod 2)

≡ 1 (mod 2).

Our final lemma is a crucial step in producing families of non-congruent

numbers with arbitrarily many prime factors.

Lemma 5.9. Let m be a fixed non-negative even integer and let t be any

positive integer satisfying t ≥ m. Suppose that the matrix M = M2t is given

by

M =

[U + I I

I U

],

with

U =

[U11 U12

0 U22

].

U11 is a (t−m)× (t−m) (possibly empty) matrix given by

U11 =

t− 1 1 1 · · · 1

0 t− 2 1 · · · 1

0 0. . .

. . ....

......

. . .. . . 1

0 0 · · · 0 m

,

66


U12 is a (t −m) × m (possibly empty) matrix with all of its entries equal

to 1, and U22 is a (possibly empty) m×m matrix of integers with

det

([I U22

U22 + I I

])≡ 1 (mod 2). (5.2)

Then det(M) ≡ 1 (mod 2). We note that by convention the empty matrix

has determinant 1 and if U22 is empty then U22 + I0 is equal to the empty

matrix.

Proof. After performing t row interchanges on M, we obtain the matrix N

given by

N =

[I U

U + I I

].

By Property 1 in Definition 1.17, it follows that

det(M) = (−1)t det(N). (5.3)

Applying the formula for block determinants given in Proposition 1.20 yields

det(N) = det(It) det(It −UI−1t (U + It)) = det(It −U(U + It)). (5.4)

Meanwhile, since U11 and (U11 + It−m) are upper triangular matrices, it

follows that U11(U11 + It−m) is an upper triangular matrix. Each of the

diagonal entries in U11(U11+It−m) is equal to the product of two consecutive

integers, so by Property 3 in Theorem 1.18 we have

det (U11(U11 + It−m)) ≡ 0 (mod 2). (5.5)

Therefore,

67


It−U(U + It) = It−

[U11 U12

0 U22

][U11 + It−m U12

0 U22 + Im

]

≡ It −

[U11(U11 + It−m) ∗

0 U22(U22 + Im)

](mod 2)

≡

[It−m −U11(U11 + It−m) ∗

0 Im −U22(U22 + Im)

](mod 2).

Finally, by applying Equations (5.3) and (5.4), and Proposition 1.19, we

deduce that

det(M) = (−1)t det (N)

≡ det (It−U(U + It)) (mod 2)

≡ det(It−m −U11(U11 + It−m)) det(Im −U22(U22 + Im)) (mod 2).

Equation (5.5) implies that det(It−m −U11(U11 + It−m)) ≡ 1 (mod 2) and

by Equation (5.2), we know that det(Im − U22(U22 + Im)) ≡ 1 (mod 2).

Thus, we conclude that

det(M) ≡ 1 (mod 2).


We are now prepared to provide the proof of Theorem 5.1.

Proof. We apply Lemma 5.9 to generate our families of non-congruent num-

bers. For the choice of prime factors with Legendre symbols as specified in

68


our theorem, the Monsky matrix given by Equation (2.8) becomes

Mo =

2 0 · · · · · · · · · 0 1 0 · · · · · · 0 1 0 0 · · · · · · · · · · · · · · · · · · · · · 0

1 2 0 · · · · · · · · · · · · · · · · · · · · · 0 0 1 0 · · · · · · · · · · · · · · · · · · · · · 0

1 1 3 0 · · · · · · · · · · · · · · · · · · 0 0 0 1. . .

...

1 1 1 4 0...

. . .. . .

. . ....

......

.... . .

......

. . .. . .

. . ....

1 1 1 · · · 1 m− 1 0 · · · · · · · · · 0...

. . .. . .

. . ....

0 1 1 · · · · · · 1 m− 1 0 · · · · · · 0...

. . .. . .

. . ....

1 1 1 · · · · · · · · · 1 m+ 1 0 · · · 0...

. . .. . .

. . ....

......

.... . .

......

. . .. . .

. . ....

......

... t− 1 0...

. . .. . . 0

1 1 1 · · · · · · · · · · · · · · · · · · 1 t 0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1

1 0 0 · · · · · · · · · · · · · · · · · · · · · 0 1 0 · · · · · · · · · 0 1 0 · · · · · · 0

0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 1 1 0 · · · · · · · · · · · · · · · · · · · · · 0

0 0 1. . .

... 1 1 2 0 · · · · · · · · · · · · · · · · · · 0...

. . .. . .

. . .... 1 1 1 3 0

.... . .

. . .. . .

......

......

. . ....

.... . .

. . .. . .

... 1 1 1 · · · 1 m− 2 0 · · · · · · · · · 0...

. . .. . .

. . .... 0 1 1 · · · · · · 1 m− 2 0 · · · · · · 0

.... . .

. . .. . .

... 1 1 1 · · · · · · · · · 1 m 0 · · · 0...

. . .. . .

. . .......

......

. . ....

.... . .

. . . 0...

...... t− 2 0

0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1 1 1 1 · · · · · · · · · · · · · · · · · · 1 t− 1

,

Note that Mo is a 2t × 2t matrix. We now apply a series of column and

row interchanges to the matrix Mo. These column and row exchanges are

described in the proof of Iskra’s theorem presented in Section 4.2. Upon

executing these operations, we obtain a matrix of the form

M =

[U + I I

I U

],

69


where

U =

[U11 U12

0 U22

],

with U11 and U12 as given in Lemma 5.9. The matrix U22 is the empty

matrix if m = 0, while U22 is equal to Am if m > 0. Lemma 5.8 shows

that the conditions of Lemma 5.9 are fulfilled with these choices. Applying

Lemma 5.9, we deduce that

det(M) ≡ 1(mod 2).

In addition, we know by the explanation presented in the proof of Iskra’s

theorem in Section 4.2 that for all values of t satisfying the requirements of

Theorem 5.1

det(Mo) = det(M) ≡ 1(mod 2).

Thus, by Property 6 in Theorem 1.18, the rank of Mo is equal to 2t. It

follows from (2.7) that s(n) = 0 if n ∈ Nm and by inequality (2.9) that the

rank of (3.1) is equal to zero. Hence n is non-congruent.

We note that N0 is the family of non-congruent numbers in Iskra’s theo-

rem (see Theorem 4.1) and that N2 ⊆ N0 by permuting the first two primes

in any n ∈ N2. We now prove that all other sets Nm are new. Assume that

the positive integer n satisfies

n ∈ Nm ∩Nm′ ,

for even integers m and m′ with m′ > m ≥ 4. Suppose that the prime

factorization of the integer n, satisfying (5.1) is given by

n = p1p2 · · · pt ∈ Nm

and that a permutation π of the prime factors pi of n results in

n = q1q2 · · · qt ∈ Nm′ ,

70


where the qi are the prime factors of n and

(qjqi

)=

−1 if 1 ≤ j < i and (j, i) 6= (1,m′),

+1 if 1 ≤ j < i and (j, i) = (1,m′).(5.6)

Let k denote the largest subscript for which pk is not fixed by the permuta-

tion π. Clearly k ≥ 2. If k = 2 then q1 = p2 and q2 = p1 so that

(q1q2

)= +1,

contradicting (5.6) as m′ > m ≥ 4. If k = 3, then the ordered set {q1, q2, q3}is one of the ordered sets {p3, p1, p2}, {p3, p2, p1}, {p1, p3, p2}, or {p2, p3, p1}.Considering these choices in order leads to the Legendre symbol values(

q1q2

)= +1,

(q1q2

)= +1,

(q2q3

)= +1, and

(q2q3

)= +1,

each of which contradicts (5.6) and the inequality m′ > m ≥ 4. Therefore,

k ≥ 4. By the definition of k we know that pk = qj for some j satisfying

1 ≤ j < k. If pk = q1 then as

{p1, p2, . . . , pk−1} = {q2, q3, . . . , qk} (5.7)

and (q1qi

)= −1 (5.8)

for 2 ≤ i ≤ k and i 6= m′, we conclude by (5.7), (5.8), and the inequality

k ≥ 4 that the symbol

(pkp`

)has a value of −1 for at least two values of `

satisfying 1 ≤ ` ≤ k− 1. This contradicts (5.1). If qk = p1 then we obtain a

contradiction in a similar manner. Therefore, pk = qj for some j satisfying

2 ≤ j ≤ k− 1. We also have that qk = pi for some i satisfying 2 ≤ i ≤ k− 1.

From (5.6) we must have (qjqk

)= −1,

so that (pkpi

)= −1,

71


which contradicts (5.1). Thus, the sets Nm and Nm′ are distinct.

A similar argument shows that for m ≥ 4, the integers in the sets Nm

are different from those in Iskra’s family of non-congruent numbers, which

was described in Theorem 4.1.

Next, we prove a corollary that provides verification that the sets Nm

described in Theorem 5.1 are non-empty.

Corollary 5.10. Let Nm denote the set of positive integers defined in the

statement of Theorem 5.1. For any value of m, the set Nm is non-empty

and, in fact, contains infinitely many elements.

Proof. Recall Corollary 4.3 and consider a positive integer of the form

p1p2 · · · pm−1 whose prime factors fulfill the conditions of Theorem 4.1. Ap-

pend a prime pm onto the end of this product that satisfies the following

system of congruences:pm ≡ 3 (mod 8),

pm ≡ −1 (mod p1),

pm ≡ 1 (mod pj) for each 1 < j < m.

(5.9)

By applying the Chinese remainder theorem and Dirichlet’s theorem on

primes in arithmetic progression (See Theorems 1.26 and 1.30), we deduce

that there exist infinitely many primes pm that satisfy the system of con-

gruences in (5.9). Since pm ≡ −1 (mod p1), it follows that pm is a quadratic

nonresidue modulo p1. Therefore,(p1pm

)= +1. Conversely,

(pjpm

)= −1 for

each 1 < j < m, because the third congruence in (5.9) indicates that pm is

a quadratic residue modulo pj . Thus, (5.1) is satisfied, so we conclude that

the sets Nm contain infinitely many elements.

It is worthwhile to note that we proved this corollary without imposing

the restrictions t ≥ m and m is even from the statement of Theorem 5.1.

72


Finally, we conclude this chapter by presenting a corollary that offers

evidence that congruent numbers whose prime factors are of the form 8k+3

and satisfy Equation (5.1) exist whenever m is odd and m ≥ 3.

Corollary 5.11. Let m ≥ 3 be a fixed odd integer and let t be any positive

integer satisfying t ≥ m. Let Nm denote the set of positive integers with

prime factorization p1p2 · · · pt, where p1, p2, . . . , pt are distinct primes of the

form 8k + 3 satisfying Equation (5.1). If n ∈ Nm, then n is congruent.

Proof. We recall Lemma 3.2 which states that for any rational number v /∈(−∞,−1] ∪ [0, 1], the form v(v − 1)(v + 1), properly scaled to an integer

by squares of rational numbers, produces a congruent number. Let m ≥ 3

be odd, and assume that p2, p3, . . . , pm−1 are distinct prime numbers of the

form 8k + 3 satisfying(pjpi

)= −1 if 1 ≤ j < i. Define the integer b by b =

p2p3 · · · pm−1. Since b is a product of an odd number of primes of the form

8k + 3, it follows that b ≡ 3 (mod 8). Let v =bx2

16y2for positive integers x

and y. Scaling by squares yields the congruent number

(bx2 − 16y2)b(bx2 + 16y2).

Schinzel’s hypothesis H [SS58] states that if a finite product Q(x) =m∏i=1

fi(x) of polynomials fi(x) ∈ Z [x] has no fixed divisors, then all of the

fi(x) are simultaneously prime, for infinitely many integral values of x. From

this hypothesis we deduce that the two forms

bx2 − 16y2 and bx2 + 16y2

assume prime values infinitely often. Notice that bx2− 16y2 and bx2 + 16y2

only attain prime values if x is odd. Since b ≡ 3 (mod 8), it is easy to verify

that bx2− 16y2 and bx2 + 16y2 are primes of the form 8k+ 3 by using basic

properties of congruences.

Furthermore, we can prove that the product (bx2 − 16y2)b(bx2 + 16y2)

where b = p2p3 · · · pm−1 satisfies the conditions given by (5.1) in Theorem

5.1. To do this, we must show that for any prime divisor p of b, the following

73


three equations hold:(bx2 − 16y2

p

)= −1,

(bx2 − 16y2

bx2 + 16y2

)= +1, and

(p

bx2 + 16y2

)= −1.

We begin by considering the Legendre symbol(bx2−16y2

p

)first. Since p|b,

we know that bx2 ≡ 0 (mod p). Therefore,(bx2 − 16y2

p

)=

(−16y2

p

).

Note that p - (−16y2). By way of contradiction, suppose that p|(−16y2).

Clearly, p cannot divide 2 as p is a prime of the form 8k + 3. Therefore,

we must have p|y. This means that p is a common factor of both b and

y, so it follows that p|(bx2 + 16y2). However, since p 6= (bx2 + 16y2), we

cannot have p|(bx2 + 16y2) as this is a contradiction to our assumption that

(bx2 + 16y2) is prime. Therefore, p - y so p - (−16y2). As a result, we can

use 1(ii) Theorem 1.29 to write(−16y2

p

)=

(−1

p

)((4y)2

p

).

Finally, we apply 1(iii) and 3(i) from Theorem 1.29 to conclude that(bx2 − 16y2

p

)= −1.

Next, we verify that(bx2−16y2bx2+16y2

)= +1. Clearly, bx2 ≡ −16y2 (mod (bx2+

16y2)). Therefore,(bx2 − 16y2

bx2 + 16y2

)=

(−16y2 − 16y2

bx2 + 16y2

)=

(−2(16y2)

bx2 + 16y2

).

Note that (bx2 + 16y2) - (−25y2), because if it did, the prime (bx2 + 16y2)

would have to divide 2 or y. However, since 2 and y are both less than

(bx2+16y2), it is impossible for (bx2+16y2) to divide −2(16y2). By making

use of this fact and applying 1(ii) Theorem 1.29, we obtain the following

74


equation:(−2(16y2)

bx2 + 16y2

)=

(−1

bx2 + 16y2

)(2

bx2 + 16y2

)((4y)2

bx2 + 16y2

).

Since (bx2 + 16y2) is a prime of the form 8k+ 3, by applying 1(iii), 3(i), and

3(ii) from Theorem 1.29, we can conclude that(bx2 − 16y2

bx2 + 16y2

)= +1.

The third and final Legendre symbol that we need to consider is(

pbx2+16y2

).

Since p and (bx2 + 16y2) are primes of the form 8k + 3, it follows from the

law of quadratic reciprocity that(p

bx2 + 16y2

)= −

(bx2 + 16y2

p

).

In addition, because p|b we know that bx2 ≡ 0 (mod p). Therefore,

−(bx2 + 16y2

p

)= −

(16y2

p

).

By 1(iii) Theorem 1.29, it is clear that(16y2

p

)= +1. This enables us to

conclude that (p

bx2 + 16y2

)= −1.

Therefore, (bx2−16y2)b(bx2+16y2) with b = p2p3 · · · pm−1 is a congruent

number that satisfies the conditions given by (5.1) in Theorem 5.1. Thus,

when m is odd and m ≥ 3, we cannot generate families of non-congruent

numbers.

75

Chapter 6

Families of Non-congruent

Numbers with One Prime

Factor of the Form 8k + 1 and

Arbitrarily Many Prime

Factors of the Form 8k + 3

Chapter 6 focuses on the generation of families of non-congruent numbers

with arbitrarily many prime factors. However, unlike the non-congruent

numbers presented in the previous two chapters whose prime divisors are

only of the form 8k+3, the numbers described in this chapter are a product

of primes belonging to two different congruence classes modulo 8; the non-

congruent numbers in these new families contain a single prime factor of the

form 8k+ 1 and at least one prime factor of the form 8k+ 3. It is important

to note that these families of non-congruent numbers are an extension of

work done by Lagrange involving non-congruent numbers with two or three

prime factors [Lag75]. Lagrange’s non-congruent numbers are listed in Table

1.2 and have the form n = pq with(pq

)= −1, or n = pqr with

(pq

)=

−(pr

), where p ≡ 1 (mod 8) and q ≡ r ≡ 3 (mod 8). To construct these

new families, we utilize the technique introduced in Section 4.2 and used to

prove our main theorem in Chapter 5.

In Section 6.1 we state and prove the main theorem for this chapter, and

in Section 6.2 we discuss and prove a supporting corollary.

76



We begin by stating the main theorem of this chapter.

Theorem 6.1. Let m be a fixed positive integer and let t be any integer

satisfying t ≥ m. Let Sm denote the set of positive integers with prime fac-

torization pq1q2 · · · qt, where p is a prime of the form 8k+1 and q1, q2, . . . , qt

are distinct primes of the form 8k + 3 such that(p

qi

)=

{−1 if i = m,

+1 if i 6= m,

and (qjqi

)= −1 if j < i.

If n ∈ Sm, then n is non-congruent. Moreover for different m, the sets Sm

are pairwise disjoint.

77


Proof. Going directly to the Monsky matrix we have

Mo =

1 0 · · · · · · · · · 0 1 0 · · · · · · 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · 0

0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 1 0 · · · · · · · · · · · · · · · · · · · · · 0

0 1 2 0 · · · · · · · · · · · · · · · · · · 0 0 0 1. . .

...

0 1 1 3 0...

. . .. . .

. . ....

......

.... . .

......

. . .. . .

. . ....

0 1 1 · · · 1 m− 1 0 · · · · · · · · · 0...

. . .. . .

. . ....

1 1 1 · · · · · · 1 m+ 1 0 · · · · · · 0...

. . .. . .

. . ....

0 1 1 · · · · · · · · · 1 m+ 1 0 · · · 0...

. . .. . .

. . ....

......

.... . .

......

. . .. . .

. . ....

......

... t− 2 0...

. . .. . . 0

0 1 1 · · · · · · · · · · · · · · · · · · 1 t− 1 0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1

0 0 0 · · · · · · · · · · · · · · · · · · · · · 0 1 0 · · · · · · · · · 0 1 0 · · · · · · 0

0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · 0

0 0 1. . .

... 0 1 1 0 · · · · · · · · · · · · · · · · · · 0...

. . .. . .

. . .... 0 1 1 2 0

.... . .

. . .. . .

......

......

. . ....

.... . .

. . .. . .

... 0 1 1 · · · 1 m− 2 0 · · · · · · · · · 0...

. . .. . .

. . .... 1 1 1 · · · · · · 1 m 0 · · · · · · 0

.... . .

. . .. . .

... 0 1 1 · · · · · · · · · 1 m 0 · · · 0...

. . .. . .

. . ....

......

.... . .

......

. . .. . . 0

......

... t− 3 0

0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1 0 1 1 · · · · · · · · · · · · · · · · · · 1 t− 2

.

Note that each block in Mo is a t × t matrix. We start by applying a

sequence of elementary row and column operations on Mo. Specifically, we

add column 1 to column (t+ 1) and then subtract column t+ (m+ 1) from

column (t + 1). This is followed by adding row 1 to row (t + 1) and then

subtracting row t+ (m+ 1) from row (t+ 1). We obtain a matrix M′o given

78


below.

M′o =

1 0 · · · · · · · · · 0 1 0 · · · · · · 0 1 0 0 · · · · · · · · · · · · · · · · · · · · · 0

0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 1 0 · · · · · · · · · · · · · · · · · · · · · 0

0 1 2 0 · · · · · · · · · · · · · · · · · · 0 0 0 1. . .

...

0 1 1 3 0...

. . .. . .

. . ....

......

.... . .

......

. . .. . .

. . ....

0 1 1 · · · 1 m− 1 0 · · · · · · · · · 0...

. . .. . .

. . ....

1 1 1 · · · · · · 1 m+ 1 0 · · · · · · 0...

. . .. . .

. . ....

0 1 1 · · · · · · · · · 1 m+ 1 0 · · · 0...

. . .. . .

. . ....

......

.... . .

......

. . .. . .

. . ....

......

... t− 2 0...

. . .. . . 0

0 1 1 · · · · · · · · · · · · · · · · · · 1 t− 1 0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1

1 0 0 · · · · · · · · · · · · · · · · · · · · · 0 m −1 · · · · · · · · · −1 1−m 0 · · · · · · 0

0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · 0

0 0 1. . .

... 0 1 1 0 · · · · · · · · · · · · · · · · · · 0

.... . .

. . .. . .

... 0 1 1 2 0

.... . .

. . .. . .

......

......

. . ....

.... . .

. . .. . .

... 0 1 1 · · · 1 m− 2 0 · · · · · · · · · 0

.... . .

. . .. . .

... 1−m 1 1 · · · · · · 1 m 0 · · · · · · 0

.... . .

. . .. . .

... −1 1 1 · · · · · · · · · 1 m 0 · · · 0

.... . .

. . .. . .

......

......

. . ....

.... . .

. . . 0...

...... t− 3 0

0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1 −1 1 1 · · · · · · · · · · · · · · · · · · 1 t− 2

.

We write this matrix in the form

M′o =

[U It

It V

],

79


where

U =

1 0 0 · · · · · · 0 1 0 · · · · · · · · · · · · 0

0 1 0 · · · · · · · · · 0...

...

0 1 2. . .

......

......

.... . .

. . .. . .

......

......

.... . .

. . .. . .

......

...

0 1 · · · · · · 1 m− 1 0...

...

1 1 · · · · · · · · · 1 m+ 1 0 · · · · · · · · · · · · 0

0 1 · · · · · · · · · · · · 1 m+ 1 0 · · · · · · · · · 0...

...... 1 m+ 2

. . ....

......

......

. . .. . .

. . ....

......

......

. . .. . .

. . ....

......

......

. . . t− 2 0

0 1 · · · · · · · · · · · · 1 1 · · · · · · · · · 1 t− 1

=

[U11 U12

U21 U22

],

and

V =

m −1 −1 · · · · · · −1 1−m 0 · · · · · · · · · · · · 0

0 0 0 · · · · · · · · · 0...

...

0 1 1 0...

......

0 1 1 2. . .

......

......

.... . .

. . .. . .

......

...

0 1 · · · · · · 1 m− 2 0...

...

1−m 1 · · · · · · · · · 1 m 0 · · · · · · · · · · · · 0

−1 1 · · · · · · · · · · · · 1 m 0 · · · · · · · · · 0...

...... 1 m+ 1

. . ....

......

......

. . .. . .

. . ....

......

......

. . .. . .

. . ....

......

......

. . . t− 3 0

−1 1 · · · · · · · · · · · · 1 1 · · · · · · · · · 1 t− 2

=

[V11 V12

V21 V22

].

80


The matrix resulting from performing t row interchanges on M′o is

N =

[It V

U It

].

Note that by Property 1 in Definition 1.17 and Property 2 in Theorem 1.18,

det(Mo) = det(M′o) = (−1)t det(N). (6.1)

In addition, by the formula for block determinants given in Proposition 1.20,

det(N) = det

([It V

U It

])= det(It) det(It−UI−1t V) = det(It−UV). (6.2)

Consider

U11V11 =

1 0 0 · · · · · · 0 1

0 1 0 · · · · · · · · · 0

0 1 2. . .

......

.... . .

. . .. . .

......

.... . .

. . .. . .

...

0 1 · · · · · · 1 m− 1 0

1 1 · · · · · · · · · 1 m+ 1

m −1 −1 · · · · · · −1 1−m0 0 0 · · · · · · · · · 0

0 1 1 0...

0 1 1 2. . .

......

.... . .

. . .. . .

...

0 1 · · · · · · 1 m− 2 0

1−m 1 · · · · · · · · · 1 m

=

m+ (1−m) 0 0 · · · · · · 0 (1−m) +m

0 0 0 · · · · · · 0 0

0 2 2. . .

...

... 4 4 6. . .

...

.... . .

. . ....

0. . . 0

m+ (m+ 1)(1−m) ∗ · · · · · · · · · ∗ (1−m) +m(m+ 1)

.

81


Notice that all of the diagonal entries in the matrix U11V11 except for the

two corner ones are equal to the product of two consecutive integers, so they

are congruent to 0 modulo 2. Moreover all of the entries of U11V11 except

for the corner entries in the first and last row are even, which means that

they are congruent to 0 modulo 2. We note that the entries denoted by ∗are of the form

−1 + (m− 2) + (m+ 1),

hence are even. We reduce U11V11 modulo 2 to obtain

U11V11 ≡

1 0 · · · 0 1

0 0 · · · 0 0

......

......

......

......

0...

... 0

−m2 +m+ 1 0 · · · 0 m2 + 1

(mod 2).

Further reduction modulo 2 yields

U11V11 ≡

1 0 · · · 0 1

0 0 0...

. . ....

0 0 0

1 0 · · · 0 1

(mod 2) if m is even,

1 0 · · · 0 1

0 0 0...

. . ....

0 0 0

1 0 · · · 0 0

(mod 2) if m is odd.

82


Returning to the matrices U and V, we notice that all of the entries in U12

and V12 are equal to zero. In addition,

U22V22 =

m+ 1 0 · · · · · · · · · 0

1 m+ 2. . .

......

. . .. . .

. . ....

.... . .

. . .. . .

......

. . . t− 2 0

1 · · · · · · · · · 1 t− 1

m 0 · · · · · · · · · 0

1 m+ 1. . .

......

. . .. . .

. . ....

.... . .

. . .. . .

......

. . . t− 3 0

1 · · · · · · · · · 1 t− 2

.

Since U22V22 is a product of two lower triangular matrices, it is lower tri-

angular. Each diagonal entry in the matrix U22V22 is equal to the product

of two consecutive integers, hence is congruent to 0 modulo 2. Therefore,

It −UV = It −

U11 U12

U21 U22

V11 V12

V21 V22

= It −

U11V11 0

∗ U22V22

.

83


If m is even, then

It −UV ≡

0 0 0 · · · 0 1 0 . . . . . . . . . . . . 0

0 1 0 · · · · · · 0...

...

0 0 1. . .

......

......

. . .. . .

. . ....

......

0. . . 1 0

......

1 0 · · · · · · 0 0 0 . . . . . . . . . . . . 0

1 0 · · · · · · · · · 0

∗ . . .. . .

......

. . .. . .

. . ....

∗...

. . .. . .

. . ....

.... . .

. . . 0

∗ · · · · · · · · · ∗ 1

(mod 2).

By using Equations (6.1) and (6.2), and Proposition 1.19, we deduce that

det(Mo) = (−1)t det(N) = (−1)t det(It −UV)

≡ (−1)t det

0 0 0 · · · 0 1

0 1 0 · · · · · · 0

0 0 1. . .

......

. . .. . .

. . ....

0. . . 1 0

1 0 · · · · · · 0 0

(mod 2).

Finally, by applying Property 1 of Definition 1.17 and exchanging the first

and last rows of the matrix whose determinant we are trying to compute,

84


we have

det(Mo) ≡ −det

1 0 · · · · · · 0

0 1. . .

......

. . .. . .

. . ....

.... . . 1 0

0 · · · · · · 0 1

≡ 1 (mod 2).

If m is odd, then

It −UV ≡

0 0 0 · · · 0 1 0 . . . . . . . . . . . . 0

0 1 0 · · · · · · 0...

...

0 0 1. . .

......

......

. . .. . .

. . ....

......

0. . . 1 0

......

1 0 · · · · · · 0 1 0 . . . . . . . . . . . . 0

1 0 · · · · · · · · · 0

∗ . . .. . .

......

. . .. . .

. . ....

∗...

. . .. . .

. . ....

.... . .

. . . 0

∗ · · · · · · · · · ∗ 1

(mod 2).

By combining Equations (6.1) and (6.2), and using Proposition 1.19, we

85


deduce that

det(Mo) = (−1)t det(N) = (−1)t det(It −UV)

≡ (−1)t det

0 0 0 · · · 0 1

0 1 0 · · · · · · 0

0 0 1. . .

......

. . .. . .

. . ....

0. . . 1 0

1 0 · · · · · · 0 1

(mod 2).

Continuing by subtracting the first row of this matrix from the last row

yields

det(Mo) ≡ (−1)t det

0 0 0 · · · 0 1

0 1 0 · · · · · · 0

0 0 1. . .

......

. . .. . .

. . ....

0. . . 1 0

1 0 · · · · · · 0 0

(mod 2).

This matrix is the same as the one we obtained in the case where m was

even. As a result, we deduce that

det(Mo) ≡ 1 (mod 2)

when m is odd. Thus, the matrix Mo has full rank for all m. We apply

Equations (2.7) and (2.9) to conclude that n is a non-congruent number.

Next we show that for different m, the sets Sm are pairwise disjoint.

Suppose that for some positive integer n the two sets Sm and Sm′ satisfy

n ∈ Sm ∩ Sm′ ,

86

6.2. A Supporting Corollary

where we may assume that m > m′ ≥ 1. Let π denote a permutation of the

prime factors qi of n and suppose that

pq1q2 · · · qt ∈ Sm and pπ(q1)π(q2) · · ·π(qt) = pq′1q′2 · · · q′t ∈ Sm′ .

By definition of the sets Sm and Sm′ , we deduce that

q′m′ = qm.

As m > m′ ≥ 1, we conclude that

{q1, q2, . . . , qm−1} ⊆{q′1, q

′2, . . . , q

′m′−1

}is impossible. Therefore, for some integer j with 1 ≤ j ≤ m− 1, we have

qj ∈{q′m′+1, q

′m′+2, . . . , q

′t

}.

It follows that (q′m′

qj

)= −1,

or (qmqj

)= −1,

contradicting the definition of Sm. Thus, the sets Sm and Sm′ are distinct.

This completes the proof of the theorem.

6.2 A Supporting Corollary

By applying Dirichlet’s theorem on primes in arithmetic progression and

using a similar argument to the one presented in Corollary 5.10, we can

deduce that the sets Sm are non-empty and can verify that it is possible to

form sequences

p, q1, q2, . . .

87

6.2. A Supporting Corollary

of prime numbers satisfying the hypotheses of Theorem 6.1. In addition, re-

call that a sequence of primes {pi} that satisfies the conditions of Theorem

4.1 has the additional property that any product of primes chosen from this

sequence is non-congruent. The families of non-congruent numbers gener-

ated by Theorem 6.1 have a property similar to this, as they also give rise

to a sequence of integers such that any product of them is non-congruent.

This leads to the following corollary.

Corollary 6.2. Let {p, q1, q2, . . . , qm, qm+1, . . .} be a sequence of prime num-

bers satisfying the hypotheses of Theorem 6.1. Any product of integers from

the set

{pq1q2 · · · qm, qm+1, qm+2, . . .}

is non-congruent.

Proof. Let w be a product of integers belonging to the set

{pq1q2 · · · qm, qm+1, qm+2, . . .}.

If this product does not contain the integer factor pq1q2 · · · qm, then it is non-

congruent by Theorem 4.1. If it does contain pq1q2 · · · qm, then Theorem 6.1

implies that w is a non-congruent number.

88

Chapter 7

Conclusion and Future Work

7.1 Conclusion

The focus of this thesis was the construction of new families of congruent

and non-congruent numbers. In order to generate these families of num-

bers, techniques that did not rely upon the currently unproven Birch and

Swinnerton-Dyer conjecture were used.

In Chapter 3, a method was provided for constructing congruent numbers

with three prime factors of the form 8k + 3. A family of such numbers

was given for which the rank of their associated elliptic curves equals two,

the maximal rank for congruent number curves of this type. In order to

compute the rank, both the method of 2-descent and Monsky’s formula for

the 2-Selmer rank were applied. We showed that the rank of the curves

was at least two by solving torsors to find two independent points on the

corresponding elliptic curves. Furthermore, we applied Monsky’s formula to

deduce that the upper bound for the rank of the elliptic curves corresponding

to numbers with three prime factors of the form 8k + 3 was two. Together,

these results proved the existence of a family of congruent numbers with

associated elliptic curves of rank two.

Chapter 4 focused on an important result by Iskra that describes a fam-

ily of non-congruent numbers with arbitrarily many prime factors of the

form 8k + 3. Since a new method for generating non-congruent numbers

with arbitrarily many prime factors is presented in Section 4.2, this chap-

ter arguably contains the most valuable information in the thesis. This

new method utilizes linear algebra and employs Monsky’s formula for the

2-Selmer rank. Unlike the method of 2-descent which uses quadratic equa-

tions and involves a series of lengthy and complex calculations, Monsky’s

89

7.2. Future Work

formula offers a simple and elegant approach for determining whether a given

square-free positive integer is non-congruent. We demonstrated the beauty

of this method by applying it to prove Iskra’s theorem in Section 4.2. If

we compare Iskra’s original proof contained in Section 4.1 to the proof we

provided in Section 4.2, it becomes obvious that the new method is not only

less mathematically complex than the method of complete 2-descent, but it

is also more efficient at generating families of non-congruent numbers.

In Chapters 5 and 6 the method described in Section 4.2 was used to

generate new families of non-congruent numbers. Specifically, Chapter 5

provided an important extension to Iskra’s work by proving the existence

of infinitely many distinct new families of non-congruent numbers with ar-

bitrarily many prime factors of the form 8k + 3. Chapter 6 expanded upon

results by Lagrange to generate families of non-congruent numbers whose

prime factors belonged to two different congruence classes modulo 8; these

integers are a product of a single prime of the form 8k + 1 and at least one

prime of the form 8k+3, and have distinct prime factors satisfying a specific

pattern of Legendre symbols.

7.2 Future Work

The field of study involving congruent and non-congruent numbers has con-

siderable potential for future research work. The open problems in this area

of mathematics are diverse and vary in their level of difficulty. Of utmost

importance would be the discovery of a proof that verifies the validity of the

Birch and Swinnerton-Dyer conjecture. Proving this significant conjecture

would have extensive implications, including the verification of Tunnell’s

theorem. This, in turn, would provide an answer to the congruent number

problem by establishing a complete classification of congruent numbers. Al-

though the Birch and Swinnerton-Dyer conjecture is widely believed to be

true, finding a proof for this conjecture is an immensely difficult and poten-

tially even impossible task. Therefore, it is important to look for alternative

solutions to the congruent number problem.

Of particular interest is the search for families of congruent and non-

90

7.2. Future Work

congruent numbers with arbitrarily many prime factors. I believe that the

method for generating non-congruent numbers introduced in Section 4.2

could be applied to prove the existence of infinitely many more families of

non-congruent numbers with arbitrarily many prime factors. Specifically,

it may be possible to extend some of the results stated in Table 1.2 by

following the approach used to extend Lagrange’s results in Chapter 6. It is

worthwhile to note that since Monsky’s formula for the 2-Selmer rank only

provides an upper bound for the Mordell-Weil rank, the method described

in Section 4.2 cannot be used to verify the existence of families of congruent

numbers. Nevertheless, as illustrated in Chapter 3, the bound on the rank

provided by Monsky’s formula can be a useful tool in determining the precise

value of the rank for a specific family of congruent numbers. Additional

research could be done on this topic to generate other families of congruent

numbers with a specified rank. As Johnstone notes in her thesis [Joh09],

congruent number elliptic curves with a provable rank greater than two are

quite rare. Therefore, another topic of interest is the search for families of

congruent number elliptic curves with moderate or high rank. Thus, the

field of congruent and non-congruent numbers has an immense potential for

future research work and exciting new mathematical discoveries.

91

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95

Appendix A

Magma Code

The calculations in this section were carried out by using Magma Version

2.19-4, which can be found online at

http://magma.maths.usyd.edu.au/calc/.

A.1 Elliptic Curve Calculations

To verify that the numbers of the form p3q3r3 mentioned at the beginning

of Chapter 3 are congruent, we use Magma to compute the rank of their

corresponding elliptic curves. Recall that we know that the number n is

congruent if the elliptic curve En: y2 = x(x2−n2) has a rank that is greater

than or equal to one.

First consider the number n = 4587 = 3 ·11 ·139. Notice that the elliptic

curve y2 = x3 − 45872x has a rank of two, which means that 4587 is a

congruent number.

Input:

E:=EllipticCurve([-4587^2,0]);

Rank(E);

AnalyticRank(E);

Output:

Warning: rank computed (1) is only a lower bound

(It may still be correct, though)

1

2 47.276

96

http://magma.maths.usyd.edu.au/calc/

A.1. Elliptic Curve Calculations

Similarly, the number n = 4731 = 3 · 19 · 83 is congruent, as its corre-

sponding elliptic curve also has a rank of two.

Input:


Rank(E);

AnalyticRank(E);

Output:



1

2 42.726

In addition, the elliptic curve y2 = x3− 69632x has a rank of two, which

indicates that n = 6963 = 3 · 11 · 211 is a congruent number.

Input:


Rank(E);

AnalyticRank(E);

Output:



1

2 52.336

The number n = 7611 = 3 · 43 · 59 is also congruent, as the elliptic curve

y2 = x3 − 76112x has a positive rank equal to two.

Input:


Rank(E);

AnalyticRank(E);

Output:

97

A.1. Elliptic Curve Calculations

2

2 7.8098

Finally, n = 9339 = 3 · 11 · 283 is a congruent number, because its

corresponding elliptic curve has a rank equal to two.

Input:


Rank(E);

AnalyticRank(E);

Output:



1

2 42.069

We use the following Magma code to provide verification that the two

points, (x1, y1) and (x2, y2) on the curve y2 = x(x2 − 76112) in the proof of

Lemma 3.4 are linearly independent.

Input:


P:=E![-3483,399384];

Q:=E![449049/16,289636605/64];

S:=[P,Q];

IsLinearlyIndependent(S);

Output:

true

98

Appendix B

Maple Code

The calculations in this section were carried out with MapleTM13. Note that

the symbol > indicates Maple input and the text centred under each line of

input code is Maple output.

B.1 Parametrization and 2-Selmer Rank

Computations

We used the following Maple code to determine the parametrization for v

in terms of t in the proof of Lemma 3.4.

The Maple code used to determine the values for the 2-Selmer rank listed

in Table 3.1 is given below.

Recall that for primes p3 and q3 that are congruent to 3 modulo 8, the

law of quadratic reciprocity implies that(p3q3

)= −

(q3p3

). In the following,

the block matrices D2 and D−2 within the matrix Mo described in Equation

(2.8) are denoted by D2 and Dneg2, respectively.

99

B.1. Parametrization and 2-Selmer Rank Computations

Case 1:(p3q3

)= +1,

(p3r3

)= +1, and

(q3r3

)= +1

We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to find that

s(n) = 0.

100


Case 2:(p3q3

)= +1,

(p3r3

)= +1, and

(q3r3

)= −1

We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to deduce that

s(n) = 0.

101


Case 3:(p3q3

)= +1,

(p3r3

)= −1, and

(q3r3

)= +1


s(n) = 2.

102


Case 4:(p3q3

)= +1,

(p3r3

)= −1, and

(q3r3

)= −1


s(n) = 0.

103


Case 5:(p3q3

)= −1,

(p3r3

)= +1, and

(q3r3

)= +1


s(n) = 0.

104


Case 6:(p3q3

)= −1,

(p3r3

)= +1, and

(q3r3

)= −1


s(n) = 2.

105


Case 7:(p3q3

)= −1,

(p3r3

)= −1, and

(q3r3

)= +1


s(n) = 0.

106


Case 8:(p3q3

)= −1,

(p3r3

)= −1, and

(q3r3

)= −1


s(n) = 0.

107

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