Families of Congruent and Non-Congruent Numbers

Home , Congruent number

Families of Congruent and Non-congruent Numbers

Lindsey Kayla Reinholz

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

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

MASTER OF SCIENCE

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 congruent numbers have been studied for centuries, their complete classification is one of the central unresolved problems in the field of pure mathematics. 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 families 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 associated 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 offers 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 Chapter3 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. Chapter5 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 constructing 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 Chapter6 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 5.2 Proof of the Main Theorem...... 68

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 6.1 Proof of the Main Theorem...... 77 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 ∗2 Q Subgroup of squares in the multiplicative group of non-zero rational numbers ∗ ∗2 Q∗ = Q /Q Quotient group of square-free, non-zero rational numbers F2 Finite ﬁeld with two elements n Z Ordered n-tuples of integers Zn Cyclic group of order n

νi Cyclic group with prime-power order Zpi

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 Legendre symbol q 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 P Sum Q 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 potential 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 contributions 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 education 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

1 a2 + b2 = c2 and ab = n. 2

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 20 3 41 lengths 3 , 2 , and 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 rational 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 mathematicians, including Fibonacci, Fermat, and Euler have studied congruent numbers [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

3 2 2 2 An = #{(x, y, z) ∈ Z |n = 2x + y + 32z }, 3 2 2 2 Bn = #{(x, y, z) ∈ Z |n = 2x + y + 8z }, 3 2 2 2 Cn = #{(x, y, z) ∈ Z |n = 8x + 2y + 64z }, 3 2 2 2 Dn = #{(x, y, z) ∈ Z |n = 8x + 2y + 16z }. 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 1.1. Congruent and Non-congruent Numbers

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 numbers 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 Gérardin,Alter, Curtz, Kubota, Godwin, and Hunter worked to assem- ble a list containing all congruent numbers less than 1000 [Gér15, 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 p ≡ q ≡ r ≡ i (mod 8) (see Definition 1.24). In addition, pi is the i i i qj Legendre symbol (see Definition 1.28).

3 1.1. Congruent and Non-congruent Numbers

Table 1.1: Congruent Numbers

Heegner, 1952 [Hee52] −→ 2p3 and 2p7 and Birch, 1968 [Bir68]

Stephens, 1975 [Ste75] −→ p5 and p7

Monsky, 1990 [Mon90] −→ p3q7, p3q5, 2p3q5, and 2p5q7 −→ p q with p1 = −1 1 5 q5 −→ p q with p1 = −1 1 7 q7 −→ 2p q with p1 = −1 1 3 q3 −→ 2p q with p1 = −1 1 7 q7

Serf, 1991 [Ser91] −→ p3q3r5, p3q3r7, 2p3q3r7, 2p3q5r5, and 2p5q5r7 −→ p q r with p7 = − p7 = q7 7 7 7 q7 r7 r7 −→ 2p q r with p7 = − p7 = q7 7 7 7 q7 r7 r7

−→ p1q3r3s5 with p1 = p1 = p1 = +1 q3 r3 s5 or − p1 = − p1 = p1 = +1, q3 = r3 q3 r3 s5 s5 s5

−→ 2p1q3r5s5 with p1 = p1 = p1 = +1 q3 r5 s5 or p1 = − p1 = − p1 = +1, q3 = q3 q3 r5 s5 r5 s5

4 1.1. Congruent and Non-congruent Numbers

Table 1.2: Non-congruent Numbers

Genocchi, 1855 [Gen55] −→ p3, p3q3, 2p5, and 2p5q5 Lagrange, 1974 [Lag75] −→ p q with p1 = −1 1 3 q3 −→ p q with p5 = −1 5 7 q7

−→ 2p3q3 −→ 2p q with p1 = −1 1 5 q5 −→ 2p q with p3 = −1 3 7 q7 −→ p q r with p1 = − p1 1 3 3 q3 r3 −→ p q r with q5 = −1 3 5 7 r7 −→ p q r with p3 = − p3 = q7 3 7 7 q7 r7 r7 −→ 2p q r with p1 = − p1 1 3 3 q3 r3 −→ 2p q r with p1 = − p1 1 5 5 q5 r5 −→ 2p q r with p3 = − q5 3 5 7 r7 r7 −→ 2p q r with p5 = − p5 = q7 5 7 7 q7 r7 r7

Serf, 1991 [Ser91] −→ p5q5r7s7 with p5 = − p5 = − q5 = +1 r7 s7 r7 or − p5 = p5 = − q5 = +1 r7 s7 s7 or − p5 = − p5 = +1, q5 = − q5 r7 s7 r7 s7

−→ 2p1q1r3s3 with p1 = +1, p1 = − p1 , q1 = − q1 q1 r3 s3 r3 s3 or p1 = −1, p1 = p1 , q1 = − q1 q1 r3 s3 r3 s3 or p1 = −1, p1 = − p1 q1 r3 s3

5 1.1. Congruent and Non-congruent Numbers

In his paper “Non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8,” Iskra proved the existence of a new family of non-congruent numbers containing arbitrarily many prime factors, p , p , . . . , p satisfying p ≡ 3 (mod 8) for all 1 ≤ i ≤ t and pj = −1 for 1 2 t i pi j < i [Isk96]. A thorough discussion of Iskra’s results, including two different 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 Chapter3, 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 Chapter4, 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. Chapter5 offers an extension to Iskra’s work and applies the method presented in Chapter4 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 Chapter6, 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 Chapters3 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 Chapter2, 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 hG, ∗i, 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 hG, ∗i 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 a0 ∈ G such that a ∗ a0 = a0 ∗ a = e.

Deﬁnition 1.3. An abelian group hG, ∗i 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 hG, ∗i 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 hH, ∗i satisﬁes the three group axioms.

Deﬁnition 1.5. Let hG, ∗i and hG0, ∗0i be binary algebraic structures, where G and G0 are groups. A map φ of G into G0 is a homomorphism if

φ(x ∗ y) = φ(x) ∗0 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 1.2. Algebra and Number Theory Preliminaries

Deﬁnition 1.6. Let hG, ∗i and hG0, ∗0i be binary algebraic structures. An isomorphism, also known as a bijective homomorphism, of G with G0 is a one-to-one function φ mapping G onto G0 such that

φ(x ∗ y) = φ(x) ∗0 φ(y) for all x, y ∈ G. If such a map φ exists, then G and G0 are isomorphic binary structures, denoted by G =∼ G0.

Deﬁnition 1.7. Let G be a group and let a ∈ G. The element a generates n G and is a generator for G if G = {a |n ∈ Z} = hai. A group G is cyclic if there exists an element a in G that generates G.

Definition 1.8. A finitely generated abelian group hG, +i 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 = ν1 ⊕ ν2 ⊕ · · · ⊕ νs ⊕ ⊕ ⊕ · · · ⊕ , Zp1 Zp2 Zps Z Z Z where is an inﬁnite cyclic group and νi is a ﬁnite cyclic group with Z Zpi 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. Algebra and Number Theory Preliminaries

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 triangular 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 T j-th row of A. In other words, [A ]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 1.2. Algebra and Number Theory Preliminaries

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

0 0 det[a1, a2, . . . , bai+cai, . . . , an] = b·det[a1, a2, . . . , an]+c·det[a1, a2, . . . , ai, . . . , an].

0 Note that b and c are scalars and ai 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 summarized 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 column.

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 1.2. Algebra and Number Theory Preliminaries

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 " #! " #! AB A 0 det = det = det (A) det (D). 0 D CD

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

" #!  AB  det (A) det D − CA−1B, when A−1 exists, det = CD  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

T T −1 det B + CD = det(B) det(Ik + D B C).

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|b indicates 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 1.2. Algebra and Number Theory Preliminaries

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 (mod m1),

x ≡ a2 (mod m2), . .

x ≡ as (mod ms), 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 1.2. Algebra and Number Theory Preliminaries

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 a by p. The Legendre symbol p is deﬁned by  a  +1 if a is a quadratic residue of p, = 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 4 2 3 = = +1 and = = −1. 5 5 5 5

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 a b (i) if a ≡ b (mod p), then = . p p a b ab (ii) = . p p p a2 (iii) = +1. p 2. The Law of Quadratic Reciprocity. If p and q are distinct odd primes, then q p p−1 · q−1 = (−1) 2 2 . p q

3. The First and Second Supplements to the Law of Quadratic Reciprocity. If p is an odd prime, then

13 1.2. Algebra and Number Theory Preliminaries

 −1  +1 if p ≡ 1 (mod 4), (i) = p  −1 if p ≡ 3 (mod 4).  2  +1 if p ≡ 1, 7 (mod 8), (ii) = p  −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 = pαn0, where n0 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 information 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 curve Figure 2.2: Elliptic curve with three real roots, y2 = with one real root, y2 = (x − 1)(x − 2)(x + 1). (x + 2)(x2 − 2x + 3).

Figure 2.3: Cubic curve Figure 2.4: Cubic curve with a double root, y2 = with a triple root, y2 = x2(x + 2). (x + 1)3.

An important and well-known fact about elliptic curves is that the rational 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 2.2. The Group Law and Mordell’s Theorem 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 and Q [Hus04, Joh09, SZ03, ST92, Sil09]. Specifically, if P and Q 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 and Figure 2.6: The chord and tangent method applied to dis- tangent method applied to the tinct points P and Q on the point P on the curve y2 = curve y2 = (x+2)(x2 −2x+3). (x + 2)(x2 − 2x + 3).

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

17 2.2. The Group Law and Mordell’s Theorem 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 2.2. The Group Law and Mordell’s Theorem

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( ) = ν1 ⊕ ν2 ⊕ · · · ⊕ νs ⊕ ⊕ ⊕ · · · ⊕ , Q Zp1 Zp2 Zps Z Z Z

19 2.3. The Torsion Subgroup

where is an inﬁnite cyclic group and νi is a ﬁnite cyclic group with Z Zpi 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 = ν1 ⊕ ν2 ⊕ · · · ⊕ νs is the torsion part of E( ) and F = Zp1 Zp2 Zps Q 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 = O, | {z } m

0 0 0 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]:

x4 − 2bx2 − 8cx + b2 − 4ac x(2P ) = . 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: 2 3 2 y = x + ax + 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:y ¯2 =x ¯3 +a ¯x¯2 + ¯bx¯, wherea ¯ = −2a, ¯b = a2 − 4b, and (¯x, y¯) is a rational point. Let Γ be the group of rational points on E [Joh09, ST92]. Define ∗ ∗2 Q to be the multiplicative group of non-zero rational numbers, Q to be ∗ ∗ ∗2 the subgroup of squares in Q , and Q∗ = Q /Q 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  1 (mod ∗2) if P = O  Q ∗2 α(P ) = b (mod Q ) if P = (0, 0)  ∗2  x (mod Q ) if P = (x, y) with x 6= 0 and  1 (mod ∗2) if P¯ = O  Q ¯ ∗2 α(P ) = ¯b (mod Q ) if P¯ = (0, 0)  ∗2  x¯ (mod Q ) if P¯ = (¯x, y¯) withx ¯ 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

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

2 4 2 2 4 N = b1M + aM e + b2e , (2.1)

3 which is referred to as a torsor, has a solution (N, M, e) ∈ Z with M 6= 0, e 6= 0, and gcd(M, e) = gcd(N, e) = gcd(b1, e) = gcd(b2,M) = gcd(M,N) = ∗2 1, then b1 modulo Q is an element of α(Γ) [Joh09, ST92]. Each equation that we solve, produces a corresponding point on our elliptic curve of the form b M 2 b MN (x, y) = 1 , 1 (2.2) e2 e3

∗2 [ST92]. Similarly, working modulo Q , α(Γ)¯ contains 1, ¯b, and all divisors b1 of ¯b satisfying the torsor equation

2 4 2 2 4 N = b1M +aM ¯ e + b2e (2.3)

3 for some (N, M, e) ∈ Z with M 6= 0 and e 6= 0. Note thata ¯ = −2a, ¯ ¯ ∗2 b = b1 b2, and b1 6≡ 1, b (mod Q ). 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 3 (N, M, e) ∈ Z 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 specialization 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 S 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 n r rational points on our elliptic curve E is isomorphic to Z2 ⊕ Z2 ⊕ Z , 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 2 general Weierstrass equation y = (x−e1)(x−e2)(x−e3) with e1, e2, e3 ∈ K where 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 = 2 p1p2 ··· pk be a square-free positive integer with p1, p2, . . . , pk being primes that are not + n equal to two, ∈ {0, 1}, and k ∈ N . Let E be the elliptic curve over Q deﬁned 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

∗ ∗2 Q(S, 2) := {c ∈ Q /Q | vp(c) ≡ 0 (mod 2) ∀ p ∈ MQ\S}, where vp(c) is the p-adic valuation of c. Then there exists an injective homomorphism n n b: E (Q)/2E (Q) ,→ Q(S, 2) × Q(S, 2) deﬁned by

26 2.6. The Method of Complete 2-Descent

  (1, 1) if P = O   (−1, −n) if P = (0, 0) P = (x, y) 7→  (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:

2 2 b1z1 − b2z2 = n, (2.4)

2 2 b1z1 − b1b2z3 = −n. (2.5)

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

2 2 P = (b1z1, b1b2z1z2z3) = (b2z2 + n, b1b2z1z2z3).

n ∼ r n Recall that E (Q) = Z2 ⊕ Z2 ⊕ Z , so E (Q) can be described as a set of n (r + 2)-tuples. In addition, 2E (Q) is also a set of (r + 2)-tuples. Note that n an arbitrary (r + 2)-tuple in E (Q) can be written as (a1, a2, a3, . . . , ar+2), n where a1, a2 ∈ Z2 and ai ∈ Z for all 3 ≤ i ≤ (r + 2). In 2E (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, n n when we form the quotient group E (Q)/2E (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,

n n ∼ r+2 E (Q)/2E (Q) = (Z2) , 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 n Section 2.5 that the torsion subgroup of E (Q) contains four rational points of ﬁnite order. By applying the homomorphism deﬁned in Theorem 2.10,

27 2.6. The Method of Complete 2-Descent we know that these four points, P = O, (0, 0), (n, 0), and (−n, 0), are respectively 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 B + 4 ⇐⇒ 2r ≤ 4 B + 4 ⇐⇒ r ≤ log . (2.6) 2 4

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 equations 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 = 2 p1p2 ··· pk be a square-free positive integer with p1, p2, . . . , pk being primes that are not

28 2.6. The Method of Complete 2-Descent

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

1 2 k R := {±2 p1 p2 ··· pk |, 1, 2, . . . , k ∈ {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 Chapter4, the method of complete 2-descent is applied to prove a theorem 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 Chapter4. Unlike the method of complete 2-descent using quadratic equations which involves a series of lengthy and complex calculations, this new technique that I developed and describe in my paper [RSY13] oﬀers a simple and elegant approach 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 congruent 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

  l  0, if P = +1, d = i ii  1, if l = −1, Pi and   0, if Pj = +1, j 6= i,  Pi X aij = aii = aij. P  1, if j = −1, j 6= i, j:j6=i Pi Then ( 2t− rank (M ), if n = P P ··· P , s(n) = F2 o 1 2 t (2.7) 2t− rankF2 (Me), if n = 2P1P2 ··· Pt, where Mo and Me are the 2t × 2t matrices:     A + D2 D2 D2 A + D2 Mo = , Me = . (2.8)    T  D2 A + D−2 A + D2 D−1

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 generating 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 Chapter4, and to generate inﬁnitely many new families of non-congruent numbers in Chapters5 and6.

31 Chapter 3

A Family of Congruent Numbers with Three Prime Factors

The purpose of this chapter is to provide a method for constructing congruent 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 Chapter1, 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 AppendixA 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):

2 4 2 2 4 N = b1M + aM e + b2e .

33 3.1. Preliminary Results

Notice that for our elliptic curve y2 = x3 − n2x, we have a = 0 and b = 2 −n = b1b2. Therefore, the torsor reduces to

2 4 4 N = b1M + b2e .

2 By choosing b = −v(v − 1)2 and b = −n = v(v + 1)2, the torsor becomes 1 2 b1

N 2 = −v(v − 1)2M 4 + v(v + 1)2e4.

Substituting M = 1 and e = 1 into the above equation and simplifying yields N 2 = 4v2.

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

b M 2 b MN (x, y) = 1 , 1 = −v(v − 1)2, −2v2(v − 1)2 (3.3) e2 e3 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

2 2 q3 = p3a − 16b , 2 2 r3 = p3a + 16b , for integers a and b. Then n = p3q3r3 is a congruent number.

34 3.1. Preliminary Results

2 2 Proof. Put v = p3a /16b into Equation (3.2) to give the congruent number

2 2 2 2 2 2 p3a /16b (p3a /16b − 1)(p3a /16b + 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 12 6 2 by 2 b /a 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). 2 For our congruent number elliptic curve, we have a = 0 and b = −n = b1b2, so the torsor reduces to

2 4 4 N = b1M + b2e .

2 We choose b = v(v − 1)(v + 1)2, so b = −n = −v(v − 1) and the torsor 1 2 b1 becomes N 2 = v(v − 1)(v + 1)2M 4 − v(v − 1)e4.

Substituting M = 1 and e = 1 into the above equation and simplifying yields N 2 = 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 3.1. Preliminary Results becomes a perfect square. By using Maple’s parametrization command, we −2−t2 deduce that v = 2t−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 AppendixB. Substituting this value for v into Equation (3.2) yields

−(2 + t2)(3 + t2 − 2t)(t + 1)2 v(v − 1)(v + 1) = . (3.5) (2t − 1)3

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 −3z2+1 in Equation (3.5) into a perfect square. Substituting t = 2 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 3z4 − 2z2 + 3 v = 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

4 2 2 2 4 2 2 2 (x1, y1) = −9(3 + 3z − 2z (z − 1) , 36(3 + 3z − 2z ) z(z − 1)) (3.6)

36 3.1. Preliminary Results and 3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2) (x , y ) = , 2 2 4z2

9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1) . (3.7) 8z3

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 3z4−2z2+3 v = 4z2 into the x-coordinate of the point and scaling it by a factor (z2−1)2 of 64z6 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,

4 2 2 2 4 2 2 2 (x1, y1) = −9(3 + 3z − 2z (z − 1) , 36(3 + 3z − 2z ) z(z − 1)) is a point on our curve. A second point on our curve can be found by using the solution to the 2 torsor that we solved at the beginning of the proof with b1 = v(v−1)(v+1) . Recall that each solvable torsor corresponds to a point on our elliptic curve. 2 This point is given by Equation (2.2) and for b1 = v(v − 1)(v + 1) , M = 1, and e = 1, its x-coordinate reduces to

v(v − 1)(v + 1)2.

3z4−2z2+3 (z2−1)2 Substituting v = 4z2 and scaling this coordinate by a factor of 64z6 yields 3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2) . 4z2

3(3+3z4+2z2)2(3+3z4−2z2) 2 3 2 Setting x = 4z2 and solving y = x − n x with n =

37 3.1. Preliminary Results

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

9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1) y = . 8z3

Thus, a second point on our elliptic curve is

3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2) (x , y ) = , 2 2 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 449049 289636605 (x , y ) = , . 2 2 16 64 The Magma code in AppendixA 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 3.1. Preliminary Results

4 2 2 4 2 for a rational number z 6= 0, ±1. If (3+3z −2z ) = p3c and (3+3z +2z ) = 2 q3d 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.

4 2 2 Proof. If we impose the restrictions that (3 + 3z − 2z ) = p3c and (3 + 4 2 2 3z + 2z ) = q3d 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. ∗2 Consider the x-coordinates of the points (x1, y1) and (x2, y2) modulo Q and notice that

2 2 2 ∗2 x1 = −9p3c (z − 1) ≡ −p3 (mod Q ) and 3(q d2)2p c2 x = 3 3 ≡ 3p (mod ∗2). 2 4z2 3 Q ∗2 Clearly −p3 and 3p3 are not congruent modulo Q , so they are two unique 4 2 2 4 2 2 elements in α(Γ). Hence, for (3+3z −2z ) = p3c and (3+3z +2z ) = q3d , 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 p3 , p3 , and q3 . We record the results for all eight cases in q3 r3 r3 Table 3.1 and provide the Maple code that was used to obtain the results in AppendixB.

39 3.1. Preliminary Results

Table 3.1: Values of s(n) for n = p3q3r3 p p q 3 3 3 s(n) q3 r3 r3 +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 m Y product Q(x) = fi(x) of polynomials fi(x) ∈ Z [x] has no fixed divisors, i=1 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 2 2 2 2 p3a − 16b and p3a + 16b (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 congruent 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 ∈ Z such 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 p ≡ 3 (mod 8) and pj = −1 for j < i. Then the product i pi 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 complete 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 2 = −1 and = −1. pi pi 2. Because pj = −1 for j < i, it follows that pi

p j = +1 if i < j. pi

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 n i = (−1)i−1. pi

Explanation: Since n = p1p2 ··· pt = nipi, we have ni = p1 ··· pi−1pi+1 ··· pt. This means that

n p ··· p p ··· p i = 1 i−1 i+1 t . pi pi

43 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

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:

p ··· p p ··· p p p p p p 1 i−1 i+1 t = 1 2 ··· i−1 i+1 ··· t . pi pi pi pi pi pi Recall that pj = −1 if j < i and that pj = +1 if i < j. As a pi pi result, the above equation reduces to

n i = (−1)(−1) ··· (−1)(+1) ··· (+1) = (−1)i−1. pi

4. Let b be a divisor of n, and deﬁne

 b  if pi|b, b0 = pi  b if pi - b.

Let k = |{j : pj|b and j < i}|. Then

b0 = (−1)k. pi

Explanation: Since b is a divisor of n = p1p2 ··· pt, we have b =

pa1 pa2 ··· paf where ah ∈ {1, 2, . . . , t} for all h = 1, 2, . . . , f, and the 0 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

0 b pa pa ··· pa p p pa = 1 2 f = a1 a2 ··· f . pi pi pi pi pi

p p We know that ah = −1 if a < i and that ah = +1 if a > i. pi h pi h 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 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

symbols have a value of +1. Thus,

b0 = (−1)k. pi

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), 1 2 k (n, 2), (−n, −2n)} with bi ∈ {±2 p1 p2 ··· pk |, 1, 2, . . . , k ∈ {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 - b2 Deﬁne

s = min{i : pi|b1 or pi|b2}.

If s exists, then ps|b1, or ps|b2, or both of these division statements hold. b0 b0 Consider 1 . By Property 4 in Remark 4.2, we know that 1 = (−1)k, ps ps 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

b0 1 = (−1)0 = +1. (4.1) ps

45 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

By using an analogous argument, we can also deduce that

b0 2 = +1. (4.2) ps

According to the deﬁnition of s, we now need to consider three separate subcases.

Subcase 1: ps|b1 and ps|b2

Consider Equation (2.5). Dividing both sides by ps yields

b1 2 b1 2 −n z1 − b2z3 = . ps ps ps

By using Properties 3 and 4 from Remark 4.2, we can replace n by n and ps s b1 by b0 . Therefore, the equation becomes ps 1

0 2 0 2 b1z1 − b1b2z3 = −ns.

Consider this equation modulo ps. Since b2 contains a factor of ps, we know 0 2 that b1b2z3 ≡ 0 (mod ps). As a result, our equation reduces to

0 2 b1z1 ≡ −ns (mod ps).

0 Multiplying both sides of this congruence by b1 yields

0 2 0 (b1z1) ≡ −nsb1 (mod ps).

0 0 Clearly ns and b1 are not divisible by ps, so nsb1 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

(b0 z )2 −n b0 1 1 = s 1 . ps ps

0 2 0 Since ps does not divide (b1z1) , ps does not divide (b1z1), so by 1(iii) The- (b0 z )2 orem 1.29, we know that 1 1 = +1. This allows us to conclude that ps

46 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

−n b0 s 1 = +1. (4.3) ps If we consider the Legendre symbol in Equation (4.3) and apply 1(ii) Theo- rem 1.29, we can write

−n b0 −1 n b0 s 1 = s 1 . ps ps ps ps Note that −1 = −1 by Property 1 in Remark 4.2, ns = (−1)s−1 by ps ps b0 Property 3 in Remark 4.2, and 1 = +1 by Equation (4.1). Thus, ps

−n b0 s 1 = (−1)(−1)s−1(+1) = (−1)s. (4.4) ps

Now consider the following equation:

2 2 b1b2z3 − b2z2 = 2n. (4.5)

2 2 This equation can be obtained by subtracting Equation (2.5), b1z1 −b1b2z3 = 2 2 −n, from Equation (2.4), b1z1 −b2z2 = n. If we divide both sides of Equation (4.5) by ps, we can write

b2 2 b2 2 n b1 z3 − z2 = 2 . ps ps ps

By applying Properties 3 and 4 from Remark 4.2, we can replace n by n ps s and b2 by b0 . Therefore, the equation becomes ps 2

0 2 0 2 b1b2z3 − b2z2 = 2ns.

Consider this equation modulo ps. Since b1 contains a factor of ps, we know 0 2 that b1b2z3 ≡ 0 (mod ps). As a result, our equation reduces to

0 2 −b2z2 ≡ 2ns (mod ps).

47 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

0 Multiplying both sides of this congruence by −b2 yields

0 2 0 (b2z2) ≡ −2nsb2 (mod ps).

0 0 Clearly 2, ns, and b2 are not divisible by ps, so 2nsb2 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

(b0 z )2 −2n b0 2 2 = s 2 . ps ps

0 2 0 Since ps does not divide (b2z2) , it follows that ps does not divide (b2z2), so (b0 z )2 by 1(iii) Theorem 1.29, we know that 2 2 = +1. This enables us to ps conclude that −2n b0 s 2 = +1. (4.6) ps If we consider the Legendre symbol in Equation (4.6) and apply 1(ii) Theo- rem 1.29, we can write

−2n b0 −1 2 n b0 s 2 = s 2 . ps ps ps ps ps Since −1 = −1 and 2 = −1 by Property 1 in Remark 4.2, ns = ps ps ps b0 (−1)s−1 by Property 3 in Remark 4.2, and 2 = +1 by Equation (4.2), ps the above product of Legendre symbols simpliﬁes to

−2n b0 s 2 = (−1)(−1)(−1)s−1(+1) = (−1)s−1. (4.7) ps

−n b0 Now we compare Equations (4.4) and (4.7). Because s 1 = (−1)s and ps −2n b0 s 2 = (−1)s−1, it follows that for every s, one of these two Legendre ps 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 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

Subcase 2: ps|b1 and ps - b2 Consider the following equation:

2 2 2 2b1z1 − b2z2 − b1b2z3 = 0. (4.8)

2 2 This equation can be obtained by adding Equation (2.4), b1z1 −b2z2 = n, to 2 2 Equation (2.5), b1z1 − b1b2z3 = −n. Dividing both sides of Equation (4.8) by ps yields 2 b1 2 b2z2 b1 2 2 z1 − − b2z3 = 0. ps ps ps By applying Property 4 from Remark 4.2, we can replace b1 by b0 . Therefore, ps 1 the equation becomes

2 0 2 b2z2 0 2 2b1z1 − − b1b2z3 = 0. ps

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, p must divide z . If z contains a factor of p , then b z z2 ≡ 0 (mod p ). s 2 2 s 2 2 ps s As a result, the above equation reduces to

0 2 0 2 2b1z1 − b1b2z3 ≡ 0 (mod ps).

0 Rearranging this congruence and multiplying both sides by 2b1 yields

0 2 2 0 2 2 (2b1) z1 ≡ 2(b1) b2z3 (mod ps).

2 Note that ps cannot divide both z1 and z2. If it did, then ps 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 2 n only contains a single factor of ps. Therefore, ps cannot divide n, so ps cannot divide both z1 and z2. We already deduced that ps is a factor of 0 z2, so it follows that ps cannot divide z1. Since ps does not divide 2, b1, 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 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

right-hand side either. We can now apply 1(i) Theorem 1.29 to write

(2b0 z )2 2b (b0 z )2 1 1 = 2 1 3 . ps 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

2b (b0 z )2 2b (b0 z )2 2b 2 1 3 = 2 1 3 = 2 (+1). ps ps ps ps

(2b0 z )2 Also, by 1(iii) Theorem 1.29, we know that 1 1 = +1. Utilizing these ps simpliﬁcations allows us to conclude that

2b 2 = +1. (4.9) ps

If we consider the Legendre symbol in Equation (4.9) and apply 1(ii) Theo- rem 1.29, we can write

2b 2 b 2 = 2 . ps ps ps

Since ps does not divide b2, by Property 4 in Remark 4.2, we can replace b2 by b0 . In addition, by Property 1 in Remark 4.2, we know that 2 = −1, 2 ps b0 and by Equation (4.2), we know that 2 = +1. Therefore, ps

2b 2 = (−1)(+1) = −1. ps

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 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

Subcase 3: ps - b1 and ps|b2 Consider Equation (4.8) and divide both sides of it by ps to get

2 2b1z1 b2 2 b2 2 − z2 − b1 z3 = 0. ps ps ps

By applying Property 4 from Remark 4.2, we can replace b2 by b0 , which ps 2 yields 2 2b1z1 0 2 0 2 − b2z2 − b1b2z3 = 0. ps 2 Since ps divides 2b1z1, 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 z . If z contains a factor of p , then 2b z z1 ≡ 0 (mod p ). As 1 1 s 1 1 ps s a result, the above equation reduces to

0 2 0 2 −b2z2 − b1b2z3 = 0 (mod ps).

0 Rearranging this congruence and multiplying both sides by −b2 yields

02 2 02 2 b2 z2 = −b1b2 z3 (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 0 that ps does not divide z2. Because b2 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

(b0 z )2 −b (b0 z )2 2 2 = 1 2 3 . ps 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

−b (b0 z )2 −b (b0 z )2 −b 1 2 3 = 1 2 3 = 1 (+1). ps ps ps ps

51 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

(b0 z )2 In addition, by 1(iii) Theorem 1.29, we know that 2 2 = +1. Applying ps these simpliﬁcations allows us to conclude that

−b 1 = +1. (4.10) ps

Notice that if we apply 1(ii) Theorem 1.29 to the Legendre symbol in Equa- tion (4.10), we obtain

−b −1 b 1 = 1 . ps ps ps

Since ps does not divide b1, by Property 4 in Remark 4.2, we can replace b1 by b0 . In addition, by Property 1 in Remark 4.2, we know that −1 = −1, 1 ps b0 and by Equation (4.1), we know that 1 = +1. Therefore, ps

−b 1 = (−1)(+1) = −1. ps

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 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

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|b2

By 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 ∗ ∗2 ∗ ∗2 ﬁnite subgroup of Q /Q × Q /Q , by closure the pair (b1, b2) · (n, 2) = (nb1, 2b2) must also belong to the group. By assumption 2|b2, so we can ∗ ∗ ∗ ∗2 ∗ ∗ write b2 = 2b2, where b2 ∈ Q /Q and 2 - b2. If we set b1 = nb1, then we ∗ ∗ ∗ ∗ ∗ ∗2 ∗ ∗2 have (nb1, 2b2) = (b1, 4b2). Because (b1, 4b2) ∈ Q /Q × Q /Q , it follows ∗ ∗ ∗ ∗ ∗ that 4b2 is equivalent to b2. Notice that the pair (b1, b2) has b2 > 0 and ∗ ∗ ∗ 2 - b2. This means that the pair (b1, b2) 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, (b1, b2) 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 - b2 By 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 ∗ ∗2 ∗ ∗2 ﬁnite subgroup of Q /Q ×Q /Q , by closure the pair (b1, b2)·(−1, −n) = ∗ ∗ (−b1, −nb2) must also be an element of the group. Let b1 = −b1 and b2 = ∗ ∗ −nb2, so (−b1, −nb2) = (b1, b2). 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, b2 > 0 and 2 - b2, which means that the pair (b1, b2) has the same properties as the one described in Case 1. By the same argument as in Case ∗ ∗ 2, we conclude that (b1, b2) cannot belong to the group generated by the set

53 4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent

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|b2

By 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- ∗ ∗2 ∗ ∗2 erates a ﬁnite subgroup of Q /Q × Q /Q , by closure the pair (b1, b2) · (−n, −2n) = (−nb1, −2nb2) must also belong to the group. By assump- ~ ~ ∗ ∗2 ~ tion 2|b2, so we can write b2 = 2b2 , where b2 ∈ Q /Q and 2 - b2 . Be- ~ ∗ ∗2 ∗ ∗2 cause (−nb1, −2nb2) = (−nb1, −4nb2 ) ∈ Q /Q × Q /Q , it follows that ~ ~ ∗ ∗ ~ −4nb2 is equivalent to −nb2 . If we set b1 = −nb1 and b2 = −nb2 , we have ~ ∗ ∗ ~ (−nb1, −nb2 ) = (b1, b2). Clearly b2 < 0, since b2 < 0. By using this fact ~ ∗ and noting that n > 0, we deduce that −nb2 = b2 > 0. In addition, because ~ ∗ ∗ ∗ ∗ 2 - n and 2 - b2 , we know that 2 - b2. Therefore, the pair (b1, b2) has b2 > 0 ∗ and 2 - b2, 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 (b1, b2) 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), 1 2 k (−n, −2n)} with bi ∈ {±2 p1 p2 ··· pk |, 1, 2, . . . , k ∈ {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

0 + 4 r ≤ log = 0. 2 4

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 p ≡ 3 (mod 8) and pj = −1 for j < i, the elliptic curve y2 = x(x2 − n2) i pi 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 pj = −1 for j < i. pi

 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    Mo =  .  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 t t+2 1 and t, 2 and (t − 1), 3 and (t − 2),..., 2 and 2 . We then interchange 3t columns (t + 1) and 2t,(t + 2) and (2t − 1), (t + 3) and (2t − 2),..., 2 3t+2 and 2 . Following this, we exchange rows 1 and t, 2 and (t − 1), 3 and

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

t t+2 (t−2),..., 2 and 2 , and rows (t+1) and 2t,(t+2) and (2t−1), (t+3) and 3t 3t+2 (2t − 2),..., 2 and 2 . If t is odd, we exchange columns 1 and t, 2 and t−1 t+3 (t−1), 3 and (t−2),..., 2 and 2 , and columns (t+1) and 2t,(t+2) and 3t−1 3t+3 t+1 (2t − 1), (t + 3) and (2t − 2),..., 2 and 2 . However, columns 2 and 3t+1 2 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 t+1 3t+1 all of the rows except for rows 2 and 2 , which remain in their original positions. Upon carrying out these column and row operations, we obtain the following matrix:

 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  0   Mo =  .  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 0 interchanges to obtain the matrix Mo. By Property 1 in Deﬁnition 1.17, we obtain 2t 0 0 det(Mo) = (−1) det(Mo) = det(Mo).

Similarly, if t is odd, we must perform (t − 1) column exchanges and (t − 1)

56 4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . . row exchanges. By applying Property 1 in Deﬁnition 1.17, we obtain

2t−2 0 0 det(Mo) = (−1) det(Mo) = det(Mo).

0 Also, we can write the matrix Mo as " # 0 UIt Mo = , It U − It where It is the t × t identity matrix and U is a t × t matrix of the form   t 1 1 ...... 1 1    0 t − 1 1 ...... 1 1     .. . .   0 0 t − 2 . . .     ......  U =  ...... .    . . .   . . .. 3 1 1     . .   . . 0 2 1    0 0 ...... 0 0 1

0 We perform row interchanges on Mo t times to obtain the matrix " # I U − I N = t t . UIt

0 t By Property 1 in Deﬁnition 1.17, det(Mo) = (−1) det(N). Applying the formula for computing block determinants given in Proposition 1.20 yields

−1 det(N) = det(It)det(It − UIt (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 4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .

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,

0 det(Mo) = det(Mo) = (−1)tdet(N) t = (−1) det(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 pj = −1 for all 1 ≤ j < i ≤ t. For any value of t, the set H is pi t non-empty and, in fact, contains inﬁnitely many elements.

Proof. We need to use Dirichlet’s theorem on primes in arithmetic progression (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 progression, 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 pj = −1 for all 1 ≤ j < i < t. We would like to pi choose a prime pt satisfying   pt ≡ 3 (mod 8), (4.11)  pt ≡ 1 (mod pj) for each 1 ≤ j < t,

58 4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .

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 pj = −1 for all pt 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 infinitely many distinct new families of non-congruent numbers with arbitrarily 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 linear 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 theorem 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  p  −1 if 1 ≤ j < i and (j, i) 6= (1, m), j = (5.1) pi  +1 if 1 ≤ j < i and (j, i) = (1, m).

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 construction of non-congruent numbers.

Deﬁnition 5.2. For a positive integer r, we deﬁne the matrices U, Q, and A by   r − 1 1 1 ··· 1 1    0 r − 2 1 ··· 1 1     ......   0 0 . . . .  U = Ur =  ,  . . ..   . . . 2 1 1     0 0 ··· 0 1 1    0 0 ··· 0 0 0

 1 0 0 ··· 0 1     0 0 0 ··· 0 0     0 0 0 ··· 0 0    Q = Qr =  . . . . . ,  . . . . .       0 0 0 ··· 0 0  1 0 0 ··· 0 1 and   r − 2 1 1 ··· 1 0    0 r − 2 1 ··· 1 1     ......   0 0 . . . .  A = Ar =  .  . . ..   . . . 2 1 1     0 0 ··· 0 1 1    1 0 ··· 0 0 1

61 5.1. Preliminary Results Involving the Generation of Non-congruent Numbers

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

2 Q = 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  r − 1 1 ··· 1     0  Ur =  . .  . U   . r−1  0 Using block multiplication we see that

 r − 1 1 ··· 1   r 1 ··· 1       0   0  Ur(U +Ir) =  .   . , r  . U   . U + I   . r−1   . r−1 r−1  0 0

62 5.1. Preliminary Results Involving the Generation of Non-congruent Numbers

which for some 1 × (r − 1) matrix W, simpliﬁes to

 r(r − 1) W   0 W       0   0   .  ≡  . (mod 2)  . U (U + I )   . 0   . r−1 r−1 r−1   . r−1  0 0

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 1 1 ··· 1 1     0 r − 2 1 ··· 1 1   . . . .   ......   0 0 . .  = r − 1 r − 1 ··· r − 1 + 1 1 ··· 1  . .   . . ..   . . . 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

 r 0 0 ··· 0 r     1 0 0 ··· 0 1     1 0 0 ··· 0 1    UQ =  . . . . . .  . . . . .       1 0 0 ··· 0 1  0 0 0 ··· 0 0

63 5.1. Preliminary Results Involving the Generation of Non-congruent Numbers

Lemma 5.6. With U and Q as given in Deﬁnition 5.2, we have

 r 1 1 ··· 1 0     0 0 0 ··· 0 0     0 0 0 ··· 0 0    Q(U + I) ≡  . . . . . (mod 2).  . . . . .       0 0 0 ··· 0 0  r 1 1 ··· 1 0

We now prove a lemma that establishes an identity involving A.

Lemma 5.7. With A as given in Deﬁnition 5.2, we have

 0 1 1 ··· 1 r     1 0 0 ··· 0 1     1 0 0 ··· 0 1    A(A + I) ≡  . . . . . (mod 2).  . . . . .       1 0 0 ··· 0 1  r 1 1 ··· 1 0

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 " # IA T = , A + II

we have det(T) ≡ 1 (mod 2).

64 5.1. Preliminary Results Involving the Generation of Non-congruent Numbers

Proof. Recalling Lemma 5.7, we have

 0 1 1 ··· 1 r   0 1 1 ··· 1 0       1 0 0 ··· 0 1   1 0 0 ··· 0 1       1 0 0 ··· 0 1   1 0 0 ··· 0 1      T A(A + I) ≡  . . . . .  ≡  . . . . .  ≡ CD (mod 2),  . . . . .   . . . . .           1 0 0 ··· 0 1   1 0 0 ··· 0 1  r 1 1 ··· 1 0 0 1 1 ··· 1 0

where   1 0    0 1  " #   0 1 ··· 1 0  . .  T C =  . .  and D = .   1 0 ··· 0 1  0 1    1 0 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 5.1. Preliminary Results Involving the Generation of Non-congruent Numbers

T det(T) ≡ det I2 − D C (mod 2)    1 0    " # " #  0 1   1 0 0 1 ··· 1 0     . .  ≡ det  −  . .  (mod 2)  0 1 1 0 ··· 0 1     0 1     1 0 " # " #! 1 0 0 r − 2 ≡ det − (mod 2) 0 1 2 0 " #! 1 0 ≡ det (mod 2) 0 1

≡ 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 " # U + II M = , IU with " # U U U = 11 12 . 0 U22

U11 is a (t − m) × (t − m) (possibly empty) matrix given by

 t − 1 1 1 ··· 1     0 t − 2 1 ··· 1   .   .. .. .  U11 =  0 0 . . . ,    ......   . . . . 1  0 0 ··· 0 m

66 5.1. Preliminary Results Involving the Generation of Non-congruent Numbers

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 " #! IU det 22 ≡ 1 (mod 2). (5.2) U22 + II

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 " # IU N = . U + II 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

−1 det(N) = det(It) det(It − UIt (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 5.2. Proof of the Main Theorem

" #" # U11 U12 U11 + It−m U12 It−U(U + It) = It− 0 U22 0 U22 + Im " # U11(U11 + It−m) ∗ ≡ It − (mod 2) 0 U22(U22 + Im) " # I − U (U + I ) ∗ ≡ t−m 11 11 t−m (mod 2). 0 Im − U22(U22 + Im) 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).

5.2 Proof of the Main Theorem

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 numbers. For the choice of prime factors with Legendre symbols as speciﬁed in

68 5.2. Proof of the Main Theorem

our theorem, the Monsky matrix given by Equation (2.8) becomes

 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  Mo =  ,  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

" # U + II M = , IU

69 5.2. Proof of the Main Theorem where " # U U U = 11 12 , 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 theorem (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 ∩ Nm0 , for even integers m and m0 with m0 > 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 ∈ Nm0 ,

70 5.2. Proof of the Main Theorem

where the qi are the prime factors of n and  q  −1 if 1 ≤ j < i and (j, i) 6= (1, m0), j = (5.6) qi  +1 if 1 ≤ j < i and (j, i) = (1, m0).

Let k denote the largest subscript for which p is not ﬁxed by the permuta- k q1 tion π. Clearly k ≥ 2. If k = 2 then q1 = p2 and q2 = p1 so that = +1, q2 0 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

q q q q 1 = +1, 1 = +1, 2 = +1, and 2 = +1, q2 q2 q3 q3 each of which contradicts (5.6) and the inequality m0 > 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 q 1 = −1 (5.8) qi for 2 ≤ i ≤ k and i 6= m0, we conclude by (5.7), (5.8), and the inequality p k ≥ 4 that the symbol k has a value of −1 for at least two values of ` p` 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 q j = −1, qk so that p k = −1, pi

71 5.2. Proof of the Main Theorem

which contradicts (5.1). Thus, the sets Nm and Nm0 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), (5.9)   pm ≡ 1 (mod pj) for each 1 < j < m.

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 congruences in (5.9). Since pm ≡ −1 (mod p1), it follows that pm is a quadratic nonresidue modulo p . Therefore, p1 = +1. Conversely, pj = −1 for 1 pm pm 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 5.2. Proof of the Main Theorem

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 pj = −1 if 1 ≤ j < i. Deﬁne the integer b by b = pi p2p3 ··· pm−1. Since b is a product of an odd number of primes of the form bx2 8k + 3, it follows that b ≡ 3 (mod 8). Let v = for positive integers x 16y2 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 Y fi(x) of polynomials fi(x) ∈ Z [x] has no fixed divisors, then all of the i=1 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 5.2. Proof of the Main Theorem three equations hold:

bx2 − 16y2 bx2 − 16y2 p = −1, = +1, and = −1. p bx2 + 16y2 bx2 + 16y2

bx2−16y2 We begin by considering the Legendre symbol p ﬁrst. Since p|b, we know that bx2 ≡ 0 (mod p). Therefore,

bx2 − 16y2 −16y2 = . p p

2 2 Note that p - (−16y ). By way of contradiction, suppose that p|(−16y ). 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 2 2 2 (bx + 16y ) is prime. Therefore, p - y so p - (−16y ). As a result, we can use 1(ii) Theorem 1.29 to write

−16y2 −1 (4y)2 = . p p p

Finally, we apply 1(iii) and 3(i) from Theorem 1.29 to conclude that

bx2 − 16y2 = −1. p

bx2−16y2 2 2 2 Next, we verify that bx2+16y2 = +1. Clearly, bx ≡ −16y (mod (bx + 16y2)). Therefore,

bx2 − 16y2 −16y2 − 16y2 −2(16y2) = = . bx2 + 16y2 bx2 + 16y2 bx2 + 16y2

2 2 5 2 2 2 Note that (bx + 16y ) - (−2 y ), because if it did, the prime (bx + 16y ) 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 5.2. Proof of the Main Theorem equation:

−2(16y2) −1 2 (4y)2 = . bx2 + 16y2 bx2 + 16y2 bx2 + 16y2 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 = +1. bx2 + 16y2

p The third and ﬁnal Legendre symbol that we need to consider is bx2+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 16y2 − = − . p p

16y2 By 1(iii) Theorem 1.29, it is clear that p = +1. This enables us to conclude that p = −1. bx2 + 16y2

2 2 2 2 Therefore, (bx −16y )b(bx +16y ) 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

Chapter6 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 p p 1.2 and have the form n = pq with q = −1, or n = pqr with q = p − r , 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 Chapter5. 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 6.1. Proof of the Main Theorem

6.1 Proof of the Main Theorem

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 factorization 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 −1 if i = m, = qi +1 if i 6= m, and q j = −1 if j < i. qi

If n ∈ Sm, then n is non-congruent. Moreover for diﬀerent m, the sets Sm are pairwise disjoint.

77 6.1. Proof of the Main Theorem

Proof. Going directly to the Monsky matrix we have

 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  Mo =  .  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 0 subtracting row t + (m + 1) from row (t + 1). We obtain a matrix Mo given

78 6.1. Proof of the Main Theorem

below.

 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  M0 =  . o  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 " # 0 UIt Mo = , It V

79 6.1. Proof of the Main Theorem

where

 1 0 0 ······ 0 1 0 ············ 0   . .   0 1 0 ········· 0 . .     . . . .   0 1 2 .. . . .     ......   ......     ......   ......     . .   0 1 ······ 1 m − 1 0 . .  " #   U11 U12 U =  1 1 ········· 1 m + 1 0 ············ 0  = ,   U U   21 22  0 1 ············ 1 m + 1 0 ········· 0     ......   . . . 1 m + 2 . .     ......   ......     ......   ......     ......   . . . . . t − 2 0  0 1 ············ 1 1 ········· 1 t − 1

and   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 . .  " #   V V   11 12 V =  1 − m 1 ········· 1 m 0 ············ 0  = .   V21 V22  −1 1 ············ 1 m 0 ········· 0     . . . . .   . . . 1 m + 1 .. .     ......   ......     ......   ......     . . . . .   ...... t − 3 0    −1 1 ············ 1 1 ········· 1 t − 2

80 6.1. Proof of the Main Theorem

0 The matrix resulting from performing t row interchanges on Mo is " # I V N = t . UIt

Note that by Property 1 in Deﬁnition 1.17 and Property 2 in Theorem 1.18,

0 t det(Mo) = det(Mo) = (−1) det(N). (6.1)

In addition, by the formula for block determinants given in Proposition 1.20,

" #! It V −1 det(N) = det = det(It) det(It−UIt V) = det(It−UV). (6.2) UIt

Consider     1 0 0 ······ 0 1 m −1 −1 · · · · · · −1 1 − m     0 1 0 ········· 0   0 0 0 ········· 0       .. .   .  0 1 2 . .   0 1 1 0 .      ......   .. .  U11V11 = ......   0 1 1 2 . .      ......   ......  ......   ......      0 1 ······ 1 m − 1 0   0 1 ······ 1 m − 2 0      1 1 ········· 1 m + 1 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 6.1. Proof of the Main Theorem

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   1 0 ··· 0 1        0 0 ··· 0 0     . . . .   . . . .   . . . .  U11V11 ≡   (mod 2).  . . . .   . . . .       . .   0 . . 0      −m2 + m + 1 0 ··· 0 m2 + 1

Further reduction modulo 2 yields      1 0 ··· 0 1   0 0 0     . . .  . .. . (mod 2) if m is even,        0 0 0     1 0 ··· 0 1 U11V11 ≡     1 0 ··· 0 1   0 0 0     . . .  . .. . (mod 2) if m is odd.        0 0 0     1 0 ··· 0 0

82 6.1. Proof of the Main Theorem

Returning to the matrices U and V, we notice that all of the entries in U12 and V12 are equal to zero. In addition,     m + 1 0 ········· 0 m 0 ········· 0  . .   . .   1 m + 2 .. .   1 m + 1 .. .       . . . . .   . . . . .   ......   ......      U22V22 =  . . . . .   . . . . . .  ......   ......       . .   . .   . .. t − 2 0   . .. t − 3 0      1 ········· 1 t − 1 1 ········· 1 t − 2

Since U22V22 is a product of two lower triangular matrices, it is lower triangular. Each diagonal entry in the matrix U22V22 is equal to the product of two consecutive integers, hence is congruent to 0 modulo 2. Therefore,

    U11 U12 V11 V12     It − UV = It −     U21 U22 V21 V22   U11V11 0   = It −  . ∗ U22V22

83 6.1. Proof of the Main Theorem

If m is even, then

 0 0 0 ··· 0 1 0 ...... 0   . .   0 1 0 ······ 0 . .     . . . .   0 0 1 .. . . .     ......   ......     . . .   0 .. 1 0 . .       1 0 ······ 0 0 0 ...... 0  It − UV ≡   (mod 2).  1 0 ········· 0     . . .   .. .. .   ∗ .   . . . . .   ......   . .   . . . . .   ......   ∗ . .   . . .   . .. ..   . 0  ∗··· ··· ··· ∗ 1

By using Equations (6.1) and (6.2), and Proposition 1.19, we deduce that

t t det(Mo) = (−1) det(N) = (−1) det(It − UV) 0 0 0 ··· 0 1   0 1 0 ······ 0  . . 0 0 1 .. . t   ≡ (−1) det . . (mod 2). ......  . . . . .    ..  0 . 1 0 1 0 ······ 0 0

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 6.1. Proof of the Main Theorem we have   1 0 ······ 0  . . 0 1 .. .   . . . . . det(Mo) ≡ − det ......  ≡ 1 (mod 2).   . .  . .. 1 0   0 ······ 0 1

If m is odd, then

 0 0 0 ··· 0 1 0 ...... 0   . .   0 1 0 ······ 0 . .     . . . .   0 0 1 .. . . .     ......   ......     . . .   0 .. 1 0 . .       1 0 ······ 0 1 0 ...... 0  It − UV ≡   (mod 2).  1 0 ········· 0     . . .   .. .. .   ∗ .   . . . . .   ......   . .   . . . . .   ......   ∗ . .   . . .   . .. ..   . 0  ∗··· ··· ··· ∗ 1

By combining Equations (6.1) and (6.2), and using Proposition 1.19, we

85 6.1. Proof of the Main Theorem deduce that

Continuing by subtracting the ﬁrst row of this matrix from the last row yields

0 0 0 ··· 0 1   0 1 0 ······ 0  . . 0 0 1 .. . t   det(Mo) ≡ (−1) det . . (mod 2). ......  . . . . .    ..  0 . 1 0 1 0 ······ 0 0

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 Sm0 satisfy

n ∈ Sm ∩ Sm0 ,

86 6.2. A Supporting Corollary where we may assume that m > m0 ≥ 1. Let π denote a permutation of the prime factors qi of n and suppose that

0 0 0 pq1q2 ··· qt ∈ Sm and pπ(q1)π(q2) ··· π(qt) = pq1q2 ··· qt ∈ Sm0 .

By deﬁnition of the sets Sm and Sm0 , we deduce that

0 qm0 = qm.

As m > m0 ≥ 1, we conclude that

0 0 0 {q1, q2, . . . , qm−1} ⊆ q1, q2, . . . , qm0−1 is impossible. Therefore, for some integer j with 1 ≤ j ≤ m − 1, we have

0 0 0 qj ∈ qm0+1, qm0+2, . . . , qt .

It follows that q0 m0 = −1, qj or q m = −1, qj contradicting the deﬁnition of Sm. Thus, the sets Sm and Sm0 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, recall 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 generated 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 numbers 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 numbers, techniques that did not rely upon the currently unproven Birch and Swinnerton-Dyer conjecture were used. In Chapter3, 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. Chapter4 focused on an important result by Iskra that describes a family 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 chapter 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 equations 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 Chapters5 and6 the method described in Section 4.2 was used to generate new families of non-congruent numbers. Specifically, Chapter5 provided an important extension to Iskra’s work by proving the existence of infinitely many distinct new families of non-congruent numbers with arbitrarily 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 Chapter6. 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 Chapter3, 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 Bibliography

[AC74] R. Alter and T. B. Curtz. A note on congruent numbers. Math. Comp., 28(125):303–305, 1974. → page3.

[ACK72] R. Alter, T. B. Curtz, and K. K. Kubota. Remarks and results on congruent numbers. Proc. Third Southeastern Conf. on Combina- torics, Graph Theory and Computing, pages 27–35, 1972. → page 3.

[Alt80] R. Alter. The congruent number problem. The American Mathe- matical Monthly, 87(1):43–45, 1980. → pages2,3, 33.

[Bas15] L. Bastien. Nombres congruents. Interm´ediaire des Math., 22:231– 232, 1915. → page3.

[Bir68] B. J. Birch. Diophantine analysis and modular functions. Internat. Colloq. on Algebraic Geometry. Tata Inst. Studies in Math., 4:35– 42, 1968. → page4.

[Cha98] V. Chandrasekar. The congruent number problem. Resonance, 3(8):33–45, 1998. → pages1,2.

[DJS09] A. Dujella, A. S. Janfada, and S. Salami. A search for high rank congruent number elliptic curves. J. Integer Seq., 12(5), 2009. Article 09.5.8. → pages 25, 29, 30.

[Fra03] J. B. Fraleigh. A First Course in Abstract Algebra. Addison Wes- ley, seventh edition, 2003. → pages7,8.

[Gen55] A. Genocchi. Note analitiche sopra tre scritti. Annali di Scienze Matematiche e Fisiche, 6:273–317, 1855. → page5.

92 Bibliography

[Gér15] A. Gérardin. Nombres congruents. Intermédiaire des Math., 22:52–53, 1915. → page3.

[God78] H. J. Godwin. A note on congruent numbers. Math. Comp., 32(141):293–295, 1978. → page3.

[HB94] D. R. Heath-Brown. The size of Selmer groups for the congruent number problem, II. Invent. Math., 118(2):331–370, 1994. → pages 29, 30.

[Hee52] K. Heegner. Diophantische analysis und modulfunktionen. Math Z., 56(3):227–253, 1952. → page4.

[Hem06] B. Hemenway. On Recognizing Congruent Primes. Master’s thesis, Simon Fraser University, 2006. → pages2,3, 19, 20, 21, 25.

[Hus04] D. Husem¨oller. Elliptic Curves. Springer-Verlag, New York, second edition, 2004. → pages 17, 19, 20, 21.

[Isk96] B. Iskra. Non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8. Proc. Japan Acad. Ser. A Math. Sci., 72(7):168–169, 1996. → pages6, 29, 42, 43, 52.

[Isk98] B. Iskra. Families of Rank Zero Twists of Ellitpic Curves. PhD thesis, University of Illinois, 1998. → page 14.

[Joh09] J. Johnstone. Congruent Numbers and Elliptic Curves. Master’s thesis, The University of British Columbia, 2009. → pages1,2,3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 91.

[KH04] J. H. Kwak and S. Hong. Linear Algebra. Springer, New York, second edition, 2004. → pages9, 10.

[Kid93] M. Kida. On the rank of an elliptic curve in elementary 2- extensions. Proc. Japan Acad. Ser. A Math. Sci., 69(10):422–425, 1993. → pages 32, 42.

93 Bibliography

[Kob93] N. Koblitz. Introduction to Elliptic Curves and Modular Forms. Springer-Verlag, New York, second edition, 1993. → pages2,3, 25.

[Lag75] J. Lagrange. Nombres congruents et courbes elliptiques. Séminaire Delange-Pisot-Poitou, Théorie des nombres, 16e année(16), 1974/1975. → pages5, 76.

[Maz77] B. Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Etudes´ Sci. Publ. Math., 47:33–186, 1977. → page 22.

[Maz78] B. Mazur. Rational isogenies of prime degree. Invent. Math., 44(2):129–162, 1978. → page 22.

[Mey00] C. D. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, 2000. → page 11.

[Mon90] P. Monsky. Mock Heegner points and congruent numbers. Math. Z., 204(1):45–68, 1990. → pages4, 29, 30.

[NW93] K. Noda and H. Wada. All congruent numbers less than 10000. Proc. Japan Acad. Ser. A Math. Sci., 69(6):175–178, 1993. → pages3, 25.

[Ogg09] F. Oggier. Introduction to algebraic number theory. https://docs.google.com/viewer? a=v&q=cache:8nwN_oGpKCQJ:www1.spms.ntu.edu. sg/~frederique/ANT10.pdf+&hl=en&gl=ca&pid= bl&srcid=ADGEESjPU6VHUybv26npwfOe5BxI9Th8KAnF72R- imosYkrhU0HDE9gPfvtTOz3VsetpXS5UGn0riPXbW2-nYZK- 5eJDZA_7RjCuO1kUr1FmtiW2xI2pZCMBsf7fxSO98jcixI8repVH&sig= AHIEtbTxVTpraA5v8M3zLmatV6iowsZLqg, 2009. → page 14.

[Ros05] K. H. Rosen. Elementary Number Theory and its Applications. Addison Wesley, Boston, ﬁfth edition, 2005. → pages 11, 12, 13, 14.

94 Bibliography

[RSY11] L. Reinholz, B. K. Spearman, and Q. Yang. On congruent numbers with three prime factors. Integers, 11(A20), 2011. → pages iv,6, 25, 32.

[RSY13] L. Reinholz, B. K. Spearman, and Q. Yang. Families of non- congruent numbers with arbitrarily many prime factors. J. Num- ber Theory, 133(1):318–327, 2013. → pages iv,6, 25, 29.

[Ser91] P. Serf. Congruent numbers and elliptic curves. In A. Peth¨o,M. E. Pohst, H. C. Williams, and H. G. Zimmer, editors, Computational Number Theory, pages 227–238. Walter de Gruyter, 1991. → pages 4,5, 25, 26, 27, 28.

[Sil09] J. H. Silverman. The Arithmetic of Elliptic Curves. Springer- Verlag, New York, second edition, 2009. → pages 17, 18, 19, 20, 21, 24, 26, 29.

[SS58] A. Schinzel and W. Sierpi´nski.Sur certain hypoth`esesconcernant les nombres premiers. Acta Arith, 4:185–208, 1958. → pages 40, 73.

[ST92] J. H. Silverman and J. Tate. Rational Points on Elliptic Curves. Springer, New York, 1992. → pages 15, 17, 18, 19, 20, 21, 22, 23, 24.

[Ste75] N. M. Stephens. Congruence properties of congruent numbers. Bull. London Math. Soc., 7:182–184, 1975. → page4.

[SZ03] S. Schmitt and H. G. Zimmer. Elliptic Curves. Walter de Gruyter, Berlin, 2003. → pages 17, 18, 20, 21.

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 Chapter3 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 A.1. Elliptic Curve Calculations

Similarly, the number n = 4731 = 3 · 19 · 83 is congruent, as its corresponding elliptic curve also has a rank of two. Input: E:=EllipticCurve([-4731^2,0]); Rank(E); AnalyticRank(E);

Output: Warning: rank computed (1) is only a lower bound (It may still be correct, though) 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: E:=EllipticCurve([-6963^2,0]); Rank(E); AnalyticRank(E);

Output: Warning: rank computed (1) is only a lower bound (It may still be correct, though) 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: E:=EllipticCurve([-7611^2,0]); 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: E:=EllipticCurve([-9339^2,0]); Rank(E); AnalyticRank(E);

Output: Warning: rank computed (1) is only a lower bound (It may still be correct, though) 1 2 42.069

We use the following Magma code to provide veriﬁcation that the two 2 2 2 points, (x1, y1) and (x2, y2) on the curve y = x(x − 7611 ) in the proof of Lemma 3.4 are linearly independent. Input: E:=EllipticCurve([-7611^2,0]); 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 p3 = − q3 . In the following, q3 p3 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: p3 = +1, p3 = +1, and q3 = +1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 6 to ﬁnd that s(n) = 0.

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

Case 2: p3 = +1, p3 = +1, and q3 = −1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 6 to deduce that s(n) = 0.

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

Case 3: p3 = +1, p3 = −1, and q3 = +1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 4 to ﬁnd that s(n) = 2.

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

Case 4: p3 = +1, p3 = −1, and q3 = −1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 6 to deduce that s(n) = 0.

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

Case 5: p3 = −1, p3 = +1, and q3 = +1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 6 to ﬁnd that s(n) = 0.

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

Case 6: p3 = −1, p3 = +1, and q3 = −1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 4 to deduce that s(n) = 2.

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

Case 7: p3 = −1, p3 = −1, and q3 = +1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 6 to ﬁnd that s(n) = 0.

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

Case 8: p3 = −1, p3 = −1, and q3 = −1 q3 r3 r3

We apply Equation (2.7) with t = 3 and rankF2 (Mo) = 6 to deduce that s(n) = 0.

107