Families of Congruent and Non-congruent Numbers by 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 in The College of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) August 2013 c Lindsey Kayla Reinholz, 2013 Abstract A positive integer n is a congruent number if it is equal to the area of a right triangle with rational sides. Equivalently, the Mordell-Weil rank of the elliptic curve y2 = x(x2 − n2) is positive. Otherwise n is a non-congruent number. Although congru- ent numbers have been studied for centuries, their complete classification is one of the central unresolved problems in the field of pure mathemat- ics. However, by using algorithms such as the method of 2-descent, various mathematicians have proven that numbers with prime factors of a specified form that satisfy a certain pattern of Legendre symbols are either always congruent or always non-congruent. In this thesis, we build upon these results and not only prove the existence of new families of congruent and non-congruent numbers, but also present a new method for generating fam- ilies of non-congruent numbers. We begin by providing a technique for constructing congruent numbers with three prime factors of the form 8k +3, and then give a family of such numbers for which the rank of their asso- ciated elliptic curves equals two, the maximal rank for congruent number curves of this type. Following this, we offer an extension to work done by Iskra and present our new method for generating families of non-congruent numbers with arbitrarily many prime factors. This method employs Mon- sky's formula for the 2-Selmer rank. Unlike the method of 2-descent which involves a series of lengthy and complex calculations, Monsky's formula of- fers an elegant approach for determining whether a given positive integer is non-congruent. This theorem uses linear algebra, and through a series of steps, allows one to compute the 2-Selmer rank of a congruent number ii Abstract elliptic curve, which provides an upper bound for the curve's Mordell-Weil rank. By applying this method, we construct infinitely many distinct new families of non-congruent numbers with arbitrarily many prime factors of the form 8k + 3. In addition, by utilizing the aforementioned method once again, we expand upon results by Lagrange to generate infinitely many new families of non-congruent numbers that are a product of a single prime of the form 8k + 1 and at least one prime of the form 8k + 3. iii Preface The main results presented in my thesis are from collaborative research done with Dr. Blair Spearman and Dr. Qiduan Yang. The contents of 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 con- structing families of non-congruent numbers with arbitrarily many prime factors. In addition to the research aspect of the project, I also contributed to the writing of the paper and performed the necessary organizational and iv Preface editing tasks. During the entire research process, I received assistance and guidance from my collaborators, Dr. Spearman and Dr. Yang. As supervi- sors of my research work, they introduced me to the topic of non-congruent numbers and to Monsky's formula for 2-Selmer rank. This, in turn, enabled me to develop the hypothesis around which my paper is centred. Dr. Yang's linear algebra knowledge was indispensable to the proof of my hypothesis, and Dr. Spearman's extensive publication background was an asset to the composition of our article. It should also be noted that the results appearing in 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, f1; 2; 3;:::g 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