Arts and Sciences, Irving K. Barber School of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Reinholz, Lindsey Kayla
Doctor of Philosophy - PhD
University of British Columbia
A congruent number n is a positive integer that is equal to the area of a right triangle with rational side lengths. Positive integers for which such a representation does not exist are called non-congruent numbers. Equivalently, n is non-congruent if and only if the arithmetic rank of the cubic curve En:y²=x³-n²x, known as a congruent number elliptic curve, is zero. Determining whether or not a given positive integer is congruent in a finite number of steps is a problem of significant interest in the field of pure mathematics. Although a complete solution to this classical problem has yet to be discovered, progress has been made in describing particular families of congruent and non-congruent numbers. The classification of numbers into such families is often done by imposing conditions on the prime divisors of the numbers and on the Legendre symbols relating the primes. This thesis focuses on the generation of both odd and even non-congruent numbers. We present a new family of even non-congruent numbers that are a product of arbitrarily many distinct primes; these non-congruent numbers have at least one prime factor in each odd congruence class modulo eight. Our main contribution is the development of a general approach for constructing families of non-congruent numbers. We show that existing families of non-congruent numbers can be extended by working over the finite field with two elements and using a formula by Monsky for computing the 2-Selmer rank of congruent number elliptic curves. The new non-congruent numbers are produced by multiplying known non-congruent numbers, corresponding to congruent number elliptic curves with 2-Selmer rank of zero, by arbitrarily many suitable primes. This novel technique allows an infinite collection of non-congruent numbers to be generated, including both odd and even non-congruent numbers with arbitrarily many distinct prime divisors in each odd congruence class modulo eight. Our results are illustrated by numerous numerical examples.
The Generation of Families ofNon-congruent Numbers withArbitrarily Many Prime FactorsbyLindsey Kayla ReinholzB.Sc., The University of British Columbia, 2011M.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)October 2019c© Lindsey Kayla Reinholz, 2019The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:The Generation of Families of Non-congruent Numbers withArbitrarily Many Prime Factorssubmitted by Lindsey Kayla Reinholz in partial fulfilment of the re-quirements of the degree of Doctor of PhilosophyDr. Qiduan Yang, Department of Computer Science, Mathematics, Physics, and StatisticsCo-supervisorDr. Sylvie Desjardins, Department of Computer Science, Mathematics, Physics, and StatisticsSupervisory Committee MemberDr. Shawn Wang, Department of Computer Science, Mathematics, Physics, and StatisticsSupervisory Committee MemberDr. Karen Perry, Department of ChemistryUniversity ExaminerDr. Yang Zhang, Department of Mathematics, University of ManitobaExternal ExamineriiAbstractA congruent number n is a positive integer that is equal to the area ofa right triangle with rational side lengths. Positive integers for which sucha representation does not exist are called non-congruent numbers. Equiv-alently, n is non-congruent if and only if the arithmetic rank of the cubiccurveEn : y2 = x3 − n2x,known as a congruent number elliptic curve, is zero. Determining whether ornot a given positive integer is congruent in a finite number of steps is a prob-lem of significant interest in the field of pure mathematics. Although a com-plete solution to this classical problem has yet to be discovered, progress hasbeen made in describing particular families of congruent and non-congruentnumbers. The classification of numbers into such families is often done byimposing conditions on the prime divisors of the numbers and on the Leg-endre symbols relating the primes.This thesis focuses on the generation of both odd and even non-congruentnumbers. We present a new family of even non-congruent numbers that area product of arbitrarily many distinct primes; these non-congruent num-bers have at least one prime factor in each odd congruence class moduloeight. Our main contribution is the development of a general approach forconstructing families of non-congruent numbers. We show that existing fam-ilies of non-congruent numbers can be extended by working over the finitefield with two elements and using a formula by Monsky for computing the2-Selmer rank of congruent number elliptic curves. The new non-congruentnumbers are produced by multiplying known non-congruent numbers, cor-responding to congruent number elliptic curves with 2-Selmer rank of zero,by arbitrarily many suitable primes. This novel technique allows an infinitecollection of non-congruent numbers to be generated, including both oddand even non-congruent numbers with arbitrarily many distinct prime divi-sors in each odd congruence class modulo eight. Our results are illustratedby numerous numerical examples.iiiLay SummaryThis thesis studies a particular set of numbers known as non-congruentnumbers. Every positive integer can be classified as either a congruentnumber or a non-congruent number. A congruent number is a positiveinteger that is equal to the area of a right-angled triangle with side lengthsthat are rational numbers. If such a representation does not exist, thenthe integer is called a non-congruent number. The classification of positiveintegers as either congruent or non-congruent is an open problem that hasbeen studied for centuries.In this thesis, we present criteria for generating non-congruent num-bers that have arbitrarily many prime divisors. Our main contribution is anew method for constructing non-congruent numbers. We show that whenexisting non-congruent numbers with a specific property are multiplied byarbitrarily many suitable primes, infinitely many new non-congruent num-bers are produced. Our results are illustrated by a collection of numericalexamples.ivPrefaceMy achievements as a graduate student would not have been possiblewithout the support, guidance, and encouragement I received from my su-pervisor, Dr. Blair Spearman, who passed away on October 1, 2017. ThoughDr. Spearman unfortunately did not have the opportunity to read my thesisor to participate in my doctoral defence, I consider him to be my Ph.D. su-pervisor. Dr. Spearman’s recognition of my research potential is what leadme to pursue graduate studies, and it is an honour to be able to completemy doctoral degree in his memory.The research work presented in this thesis is based on the following fourpapers.Published:[41] L. Reinholz, B. K. Spearman, and Q. Yang. An extension theorem forgenerating new families of non-congruent numbers, Funct. Approx. Com-ment. Math., 58 (2018), pp. 69-77.[42] L. Reinholz, B. K. Spearman, and Q. Yang. Families of even non-congruent numbers with prime factors in each odd congruence class moduloeight, Int. J. Number Theory, 14 (2018), pp. 669-692.Submitted:[43] L. Reinholz and Q. Yang. On the extension of even families of non-congruent numbers.[44] L. Reinholz and Q. Yang. On the generation of odd non-congruentnumbers with arbitrarily many prime factors.Specifically, Chapter 4 focuses on a theorem from [42]; Chapter 5 is basedon the work in [41] and [44]; and Chapter 6 is comprised of results presentedvPrefacein [42] and [43].For each of the aforementioned multi-authored papers, I developed andproved all of the theorems, and I wrote each of the manuscripts. My collabo-rators provided guidance and constructive criticism throughout the researchand publication process.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiGlossary of Notation and Symbols . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Goal of the Thesis . . . . . . . . . . . . . . . . . . . . . . 31.2 The Structure of the Thesis . . . . . . . . . . . . . . . . . . . 4Chapter 2: Preliminary Information . . . . . . . . . . . . . . . 52.1 Algebra Preliminaries . . . . . . . . . . . . . . . . . . . . . . 52.2 Number Theory Preliminaries . . . . . . . . . . . . . . . . . . 62.3 Linear Algebra Preliminaries . . . . . . . . . . . . . . . . . . 10Chapter 3: Elliptic Curves and Congruent Numbers . . . . . 123.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 An Introduction to Elliptic Curves . . . . . . . . . . . . . . . 123.3 The Group Law on Elliptic Curves and Mordell’s Theorem . 143.4 The Torsion Subgroup . . . . . . . . . . . . . . . . . . . . . . 19viiTABLE OF CONTENTS3.5 The Arithmetic Rank and the Method of Complete 2-Descent 243.6 The 2-Selmer Rank of En and Monsky’s Formula . . . . . . . 28Chapter 4: Families of Non-congruent Numbers . . . . . . . . 314.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 A Family of Even Non-congruent Numbers With ArbitrarilyMany Prime Factors . . . . . . . . . . . . . . . . . . . . . . . 324.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 40Chapter 5: The Generation of Families of Odd Non-congruentNumbers From Known Non-congruent Numbers . 415.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Extending Known Families of Odd Non-congruent Numbers . 415.3 Numerical Examples of Odd Non-congruent Numbers . . . . 455.4 An Extension Technique for Generating New Families of OddNon-congruent Numbers . . . . . . . . . . . . . . . . . . . . . 485.5 Examples of Odd Non-congruent Numbers Generated fromKnown Non-congruent Numbers . . . . . . . . . . . . . . . . 72Chapter 6: The Generation of Families of Even Non-congruentNumbers From Known Non-congruent Numbers . 806.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.2 An Extension Technique for Generating New Families of EvenNon-congruent Numbers . . . . . . . . . . . . . . . . . . . . . 806.3 Examples of Even Non-congruent Numbers Generated fromKnown Non-congruent Numbers . . . . . . . . . . . . . . . . 100Chapter 7: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 1087.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Appendix A: Maple & Magma Code . . . . . . . . . . . . . . . . . 117viiiList of TablesTable 4.1 Non-congruent numbers generated by Theorem 4.1 . . 40Table 5.1 Non-congruent numbers generated by Theorem 5.1 . . 47Table 5.2 Congruence conditions for the prime factors of the oddnumber n . . . . . . . . . . . . . . . . . . . . . . . . . 49Table 5.3 Conditions on the prime factors in the extension tailof the odd number n . . . . . . . . . . . . . . . . . . . 50Table 5.4 Diagonal entries in the matrix An in Case I.1.A ofTheorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . 53Table 5.5 Non-congruent numbers n = αsd+1sd+2 · · · st gener-ated by Cases I.1.A and I.1.B of Theorem 5.5 . . . . . 74Table 5.6 Non-congruent numbers n = αsd+1sd+2 · · · st gener-ated by Cases I.1.C and I.1.D of Theorem 5.5 . . . . . 75Table 5.7 Non-congruent numbers n = αsd+1sd+2 · · · st gener-ated by Case II.2 of Theorem 5.5 . . . . . . . . . . . . 77Table 5.8 Non-congruent numbers n = αsd+1sd+2 · · · st gener-ated by Case III.2 of Theorem 5.5 . . . . . . . . . . . . 78Table 5.9 Non-congruent numbers n = αsd+1sd+2 · · · st gener-ated by applying Case III.3 of Theorem 5.5 to numbersin Tables 5.5, 5.6, and 5.7 . . . . . . . . . . . . . . . . 79Table 6.1 Congruence conditions for the odd prime factors of theeven number n . . . . . . . . . . . . . . . . . . . . . . 81Table 6.2 Conditions on the primes in the extension tail of theeven number n . . . . . . . . . . . . . . . . . . . . . . 83Table 6.3 Lagrange’s non-congruent numbers of the form β =2pq or 2pqr, where p, q, and r are distinct odd primes 102Table 6.4 Non-congruent numbers n = βsd+1sd+2 · · · st gener-ated by Case I.1 of Theorem 6.1 . . . . . . . . . . . . . 103Table 6.5 Non-congruent numbers n = βsd+1sd+2 · · · st gener-ated by Case I.2 of Theorem 6.1 . . . . . . . . . . . . . 104ixLIST OF TABLESTable 6.6 Non-congruent numbers n = βsd+1sd+2 · · · st gener-ated by Case II.3 of Theorem 6.1 . . . . . . . . . . . . 105Table 6.7 Non-congruent numbers n = βsd+1sd+2 · · · st gener-ated by Case III.3.A of Theorem 6.1 . . . . . . . . . . 106Table 6.8 Non-congruent numbers n = βsd+1sd+2 · · · st gener-ated by Case IV.4 of Theorem 6.1 . . . . . . . . . . . . 107xList of FiguresFigure 1.1 The congruent number six is equal to the area of aright triangle with rational side lengths. . . . . . . . . 1Figure 3.1 The elliptic curve y2 = x(x− 1)(x+ 1) has three realroots. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 3.2 The elliptic curve y2 = (x + 1)(x2 − 3x + 3) with asingle real root. . . . . . . . . . . . . . . . . . . . . . 14Figure 3.3 The singular cubic curve y2 = (x + 1)3 with a tripleroot at x = −1. . . . . . . . . . . . . . . . . . . . . . 14Figure 3.4 The singular cubic curve y2 = x2(x+ 1) has a doubleroot at x = 0. . . . . . . . . . . . . . . . . . . . . . . 14Figure 3.5 The chord-tangent composition law applied to dis-tinct points P and Q, shown on the left, and a singlepoint P , shown on the right, on the elliptic curvey2 = (x+ 1)(x2 − 3x− 3). . . . . . . . . . . . . . . . 15Figure 3.6 The group law operator + applied to the points Pand Q on the elliptic curve y2 = (x+ 1)(x2 − 3x− 3). 16xiGlossary of Notation andSymbolsZ Set of integers, {. . . ,−2,−1, 0, 1, 2, . . .} p. 3N Set of natural numbers, {0, 1, 2, . . .} p. 18N+ Set of natural numbers excluding zero,{1, 2, 3 . . .}p. 2R Set of real numbers p. 9Q∗ Multiplicative group of nonzero rationalnumbersp. 26Q∗2 Group of squares of elements of Q∗ p. 26Q∗/Q∗2 Quotient group of square-free, nonzerorational numbersp. 26Qp Field of p-adic numbers p. 9Zn Cyclic group of order n,{0, 1, 2, . . . , n− 1}p. 20Zpνii The cyclic group with prime-power order p. 5Zn1 ⊕ Zn2 ⊕ · · · ⊕ Znw Direct sum of cyclic groups p. 5Zn1 × Zn2 × · · · × Znw Direct product of cyclic groups p. 20Fq Finite field with q elements p. 20|S| Number of elements in the set S p. 3〈S, ∗〉 Binary algebraic structure with non-empty set S and binary operation ∗p. 5S ∼= S′ Isomorphic structures p. 18ker(φ) Kernel of the homomorphism φ p. 6G/N Quotient group p. 26Im(b) Image of the homomorphism b p. 6(a, b) Greatest common divisor of a and b p. 6a|b a divides b p. 7a - b a does not divide b p. 7a ≡ b (mod m) a is congruent to b modulo m p. 7xiiGlossary of Notation and Symbols(ap)Legendre symbol p. 8vp(x) p-adic valuation of x p. 9In Identity matrix of order n p. 100 m× n zero matrix p. 100n Zero matrix of order n p. 10AT Transpose of the matrix A p. 10A−1 Inverse of the matrix A p. 11rank(A) Rank of the matrix A p. 10det(A) Determinant of the matrix A p. 10Cx Column x in a matrix p. 10Rx Row x in a matrix p. 10Cx −→ Cx + Cy Replacement of column Cx by the sumof columns Cx and Cyp. 10Cx ←→ Cy Interchange of columns Cx and Cy p. 10Rx −→ Rx +Ry Replacement of row Rx by the sum ofrows Rx and Ryp. 34Rx ←→ Ry Interchange of rows Rx and Ry p. 36En Congruent number elliptic curvey2 = x3 − n2xp. 2P2 Projective plane p. 12[X : Y : Z] Homogeneous coordinates or projectivecoordinates for a point in the projectiveplanep. 12O Point at infinity on an elliptic curve p. 12P +Q Addition of the points P and Q on anelliptic curvep. 15E(K) Group of K-rational points on the ellip-tic curve E over the field Kp. 15T Torsion subgroup of the elliptic curve E p. 18Tn Torsion subgroup of the elliptic curve En p. 22r(n) Arithmetic rank or Mordell-Weil rank ofthe elliptic curve Enp. 27En/Q Elliptic curve En over the field Q p. 28Sel2(En/Q) 2-Selmer group of En/Q p. 28X(En/Q)[2] Shafarevich-Tate group of En/Q p. 28s(n) 2-Selmer rank of the elliptic curve En p. 29S\T Set S excluding the elements in the setTp. 26∞ Infinite prime p. 26xiiiGlossary of Notation and SymbolsMQ Set of all places of the field Q,{∞, 2, 3, . . .}p. 26∅ Empty set p. 42xivAcknowledgementsFirst and foremost, I would like to express my deepest gratitude to mylate supervisor, Dr. Blair Spearman. I had the pleasure of working under hissupervision for eight years and am incredibly thankful for the guidance, en-couragement, patience, and unfailing support that he showed me throughoutmy educational journey. His recognition of and confidence in my potentialas a researcher are ultimately what lead me to pursue graduate studies. Iam immensely grateful for all of the time and effort that he so generouslycontributed to helping me achieve my academic and career goals.I would like to sincerely thank my co-supervisor Dr. Qiduan Yang. Hisconstructive criticism and broad knowledge have proved invaluable to myresearch work, and I am incredibly thankful for the constant support andencouragement he has given me throughout my graduate studies. I am alsoimmensely grateful that after tragic circumstances arose, he so kindly agreedto oversee the completion of my doctoral degree.In addition, I would like to express my heartfelt thanks to my committeemembers, Dr. Sylvie Desjardins and Dr. Shawn Wang. During my time asa graduate student, they have been dedicated mentors who have providedme with guidance, advice, and support. I truly appreciate the many contri-butions they each have made to help me successfully complete my educationand achieve my career goal.I would like to extend my gratitude to the members of the Departmentof Computer Science, Mathematics, Physics, and Statistics at UBC Okana-gan. In particular, I wish to thank my colleagues Dr. Chad Davis, Dr. PaulLee, Stephen Brown, and Jeewon Yoo for their advice, support, and friend-ship. Special thanks also goes to Hiroko Nakahara for her encouragement,and for the valuable experience and skills I gained while working under hersupervision.I wish to acknowledge both UBC Okanagan and the Natural Sciencesand Engineering Research Council of Canada for their financial supportthroughout my doctoral studies.Finally, I would like to sincerely thank my family and friends for theirencouragement, love, and understanding throughout every step of my ed-xvAcknowledgementsucation. This journey has been immensely challenging at times, and I amforever grateful to my parents, brothers, and Jonathon for the sacrifices theyhave made to ensure my success, and for the unconditional love and supportthey have shown me; this accomplishment would not have been possiblewithout them.xviDedicationIn memory of Dr. Blair Spearman. I never had the opportunity to fullythank you for everything you so selflessly did for me over the eight years Iknew you. Your recognition of my research potential and encouragement topursue graduate studies changed my educational and career paths. You werean incredible mentor and a dedicated teacher, who positively influenced somany lives. You are truly missed.xviiChapter 1IntroductionThe classification of positive integers as either congruent or non-congruentnumbers is an unsolved problem in the fields of algebra and number theorywith an extensive history. There exist several equivalent definitions of con-gruent numbers, but perhaps the most well-known is the one that relatescongruent numbers to right-angled triangles.Definition 1.1. A positive integer n is a congruent number if it is equal tothe area of a right triangle with rational sides. Otherwise n is said to be anon-congruent number.This means that n is a congruent number if there exist rational numbersa, b, and c such thata2 + b2 = c2 and n =ab2. (1.1)The positive integer six is an example of a congruent number.Figure 1.1: The congruent number six is equal to the area of a right trianglewith rational side lengths.There are infinitely many positive integers that cannot be written as thearea of a right triangle with rational side lengths, including the numbers1Chapter 1. Introductionone, two, three, and four; these numbers are non-congruent. A list of allcongruent numbers less than 10,000 can be found in [35].By inspecting the pair of equations in (1.1), it is clear that scaling theside lengths of a triangle changes its area by a factor of a square. Therefore,if n is a congruent number, so too is nm2, where m ∈ N+. As a result,it is a common practice to only consider square-free positive integers whenstudying congruent numbers.The search for a general solution to the problem that involves deter-mining whether a given number is congruent has fascinated mathematiciansfor centuries. Although the study of congruent numbers has its originsin ancient Greece, the first systematic discussion of the congruent num-ber problem appears in a pair of Arab manuscripts from the tenth century[5, 8, 25, 29, 56, 60]. Since that time, notable contributions to the field ofcongruent numbers have been made by many well-known mathematicians,including Fibonacci, Euler, and Fermat [5, 8]. In his book Liber Quadra-torum (1225), Fibonacci verified that both five and seven are congruentnumbers [5, 8, 32, 56]. He also conjectured that perfect squares cannot becongruent numbers, or equivalently that the integer one is non-congruent.Fibonacci’s conjecture remained unproven for four centuries, and ultimatelyled to Fermat’s significant discovery of the method of infinite descent. Fora proof that one is a non-congruent number using the method of infinitedescent, see [5].In the twentieth century, congruent numbers were shown to be related toa special type of cubic algebraic curve known as an elliptic curve [25]. Therelationship between elliptic curves and congruent numbers is summarizedby the following lemma.Lemma 1.2. A positive integer n is a congruent number if and only if therank of the elliptic curveEn : y2 = x3 − n2x = x(x− n)(x+ n)is positive. Otherwise, n is a non-congruent number. In other words, n is anon-congruent number if and only if the rank of En is zero.A proof of this lemma can be found in Section 9 of Chapter I in [25].Also, note that the properties of and theory governing elliptic curves andtheir rank will be discussed in detail in Chapter 3.In his groundbreaking paper published in 1983, Tunnell used the theoryof elliptic curves to state and prove an elegant theorem that provides asimple, but complete, characterization of congruent numbers [5, 25, 29, 46,56, 57].21.1. The Goal of the ThesisTheorem 1.3 (Tunnell’s Theorem). Let n be a square-free congruentnumber and defineAn = |{(x, y, z) ∈ Z3|n = 2x2 + y2 + 32z2}|,Bn = |{(x, y, z) ∈ Z3|n = 2x2 + y2 + 8z2}|,Cn = |{(x, y, z) ∈ Z3|n = 8x2 + 2y2 + 64z2}|,Dn = |{(x, y, z) ∈ Z3|n = 8x2 + 2y2 + 16z2}|.Then {Bn = 2An if n is odd,Dn = 2Cn if n is even.If the Birch and Swinnerton-Dyer conjecture holds for elliptic curves of theform y2 = x3 − n2x then, conversely, these equalities imply that n is acongruent number.Proof. See Tunnell’s paper [57]. Koblitz’s book [25] is also an excellentresource that provides a comprehensive discussion of Tunnell’s theorem andthe extensive collection of theory required to complete its challenging proof.Unfortunately, Tunnell’s theorem does not entirely resolve the congruentnumber problem, as one direction of it relies upon the Birch and Swinnerton-Dyer conjecture, which is currently unproven. This famous conjecture is oneof the seven Millennium Prize Problems posed by the Clay MathematicsInstitute. Because the results in this thesis do not depend upon the Birchand Swinnerton-Dyer conjecture, we do not provide additional informationin this dissertation. A thorough discussion of the conjecture is given in anarticle by Andrew Wiles on the website of the Clay Mathematics Institute[60].A comprehensive overview of the history of the congruent number prob-lem and the progress that has been made towards its solution can be foundin [5, 8, 56].1.1 The Goal of the ThesisBecause it is difficult to find a complete solution to the congruent num-ber problem, mathematicians focus on describing and generating particu-lar families of congruent and non-congruent numbers. Such families canbe obtained by imposing certain conditions on their prime factors and the31.2. The Structure of the Thesisvalues of the Legendre symbols relating the primes; see [2, 6, 11–13, 15–17, 24, 27, 28, 36, 37, 39, 40, 49, 58].The goal of this thesis is to construct new families of odd non-congruentnumbers and even non-congruent numbers with arbitrarily many distinctprime factors. Of specific interest is the generation of families of non-congruent numbers with prime factors belonging to each odd congruenceclass modulo eight, and the development of methods that allow known non-congruent numbers to be extended to produce new families of non-congruentnumbers.1.2 The Structure of the ThesisThis section provides a brief overview of the layout and main contribu-tions of the research presented within this dissertation.Chapter 2 contains a collection of notation and basic theory from thefields of algebra, number theory, and linear algebra. Chapter 3 covers knownresults on elliptic curves with an emphasis on their connection to the con-gruent number problem. We also discuss Monsky’s formula for the 2-Selmerrank of En.Our main contributions are presented in Chapters 4, 5, and 6. Chapter4 focuses on generating a new family of even non-congruent numbers witharbitrarily many distinct prime divisors; these non-congruent numbers haveprime factors in each odd congruence class modulo eight. Chapters 5 and 6introduce our new general extension technique for constructing families ofnon-congruent numbers from other known families of non-congruent num-bers. Chapter 5 focuses on the generation of odd non-congruent numbers,whereas Chapter 6 is dedicated to the construction of even non-congruentnumbers. In each of these chapters, a detailed proof of the technique forgenerating non-congruent numbers is presented along with a collection ofnumerical examples that illustrate how the method can be applied.Finally, we summarize our research contributions and discuss future re-search avenues in Chapter 7.4Chapter 2Preliminary InformationIn this chapter, we present a collection of notation, terminology, andtheory from algebra, number theory, and linear algebra.2.1 Algebra PreliminariesWe begin by recalling some definitions and theorems from the field ofalgebra. The theory in this section closely follows that in [14] and can befound in most introductory algebra texts.Definition 2.1. Let G be a group and let a ∈ G. The element a generatesthe group G and is referred to as a generator of G if G = {an|n ∈ Z } = 〈a〉.If this is the case, then G is said to be cyclic.Definition 2.2. An abelian group 〈G,+〉 is finitely generated if it containsa finite set of elements {g1, g2, . . . , gn} such that every element g ∈ G canbe written asg = a1g1 + a2g2 + · · ·+ angn,where a1, a2, . . . , an ∈ Z.We now state an important result that completely describes the structureof all finitely generated abelian groups [14, Theorem 11.12].Theorem 2.3 (Fundamental Theorem of Finitely Generated AbelianGroups). Every finitely generated abelian group is isomorphic to a directsum of cyclic groupsZpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνωω ⊕ Z⊕ Z⊕ · · · ⊕ Z︸ ︷︷ ︸r copies,where the pi are primes that are not necessarily distinct, νi ∈ N+ for alli ∈ [1, ω], and r ∈ N.The following theorem describes the relationship between the order of afinitely generated abelian group and its subgroups.52.2. Number Theory PreliminariesTheorem 2.4. If m divides the order of a finite abelian group G, then Ghas a subgroup of order m.Definition 2.5. A field is an integral domain D in which every nonzeroelement of D has a multiplicative inverse in D.Definition 2.6. The characteristic of a field K is the least positive integern such that n · a = 0 for all a ∈ K. If no such positive integer exists, thenK has characteristic zero.Definition 2.7. A sequenceG0δ1−→ G1 δ2−→ G2 δ3−→ · · · δn−→ Gnof groups Gk and homomorphisms δk is an exact sequence ifIm(δk) = ker(δk+1).An exact sequence of the form0 −→ A f−→ B g−→ C −→ 0is called a short exact sequence if the map f is injective, the map g issurjective, and Im(f) = ker(g).2.2 Number Theory PreliminariesThis section provides a collection of definitions and theorems from thefield of number theory. Wherever not explicitly specified, the notation fol-lows that in [46].Definition 2.8. The parity of an integer defines the value as either even orodd.Definition 2.9. The greatest common divisor of two integers a and b, whichare not both equal to zero, is the largest positive integer that divides botha and b. The greatest common divisor of a and b is denoted by (a, b), and(0, 0) is defined to be zero.Definition 2.10. Consider the set of integers A = {a1, a2, . . . , an} andlet ai, aj ∈ A with i 6= j. The integers ai and aj are relatively prime if(ai, aj) = 1. The integers a1, a2, . . . , an are said to be pairwise relativelyprime, if each pair of integers from the set A is relatively prime.62.2. Number Theory PreliminariesNote that for two integers a and b, the notation a|b denotes that a dividesb, whereas a - b indicates that a does not divide b.Definition 2.11. Let m be a positive integer. If a and b are integers, thena is congruent to b modulo m if m|(a − b). To denote that a is congruentto b modulo m, we write a ≡ b (mod m). If m - (a − b), then a and b areincongruent modulo m, which is written as a 6≡ b (mod m). The integer mis called the modulus of the congruence.Definition 2.12. Let a ∈ Z and m ∈ N+. The congruence class of a modulom is the set of all integers that are congruent to a modulo m.For example, there are four congruence classes modulo four. The firstcongruence class contains all integers congruent to 0 (mod 4), the second iscomprised of integers congruent to 1 (mod 4), the third contains the integerscongruent to 2 (mod 4), and the fourth consists of integers congruent to 3(mod 4). The integers belonging to a particular one of these congruenceclasses have the form 4k + i, where k ∈ Z and i is equal to either 0, 1, 2, or3.The following theorem, known as the Chinese Remainder Theorem, pro-vides a method for solving systems of simultaneous congruences with onlyone unknown, but different moduli [46, Theorem 4.12].Theorem 2.13 (Chinese Remainder Theorem). If m1,m2, . . . ,ms arepairwise relatively prime positive integers, then the system of congruencesx ≡ a1 (modm1),x ≡ a2 (modm2),...x ≡ as (modms),has a unique solution modulo M = m1m2 · · ·ms.The proof of Theorem 2.13, along with numerical examples illustratingits application, can be found in [46].Fermat’s little theorem, which we state next, describes an importantcongruence property for the pth powers of integers modulo p, where p is aprime [46, Theorem 6.3].Theorem 2.14 (Fermat’s Little Theorem). If p is a prime and b is apositive integer with p - b, then bp−1 ≡ 1 (mod p).Next, we provide the definition of quadratic residues and quadratic non-residues to motivate our discussion of Legendre symbols.72.2. Number Theory PreliminariesDefinition 2.15. If m is a positive integer, we say that the integer b is aquadratic residue of m if (b,m) = 1 and the congruence x2 ≡ b (mod m) hasa solution. If this congruence does not have a solution, then we say that bis a quadratic nonresidue of m.The concept of a quadratic residue can be used to define a multiplicativefunction known as a Legendre symbol.Definition 2.16. Let p be an odd prime and a be an integer not divisibleby p. The Legendre symbol(ap)is defined by(ap)={1 if a is a quadratic residue of p,−1 if a is a quadratic nonresidue of p.Legendre symbols will be used extensively throughout this thesis. Wesummarize some of their useful properties in Theorems 2.17, 2.18, and 2.19[46, Theorems 11.4, 11.5, 11.6 & 11.7].Theorem 2.17. Let p be an odd prime and a and b be integers not divisibleby p. Then1) if a ≡ b (mod p), then(ap)=(bp),2)(ap)(bp)=(abp),3)(a2p)= 1.Theorem 2.18 (The Law of Quadratic Reciprocity). If p and q aredistinct odd primes, then(qp)=(pq)(−1) p−12 · q−12 .Theorem 2.19 (The First and Second Supplements to the Law ofQuadratic Reciprocity). If p is an odd prime, then1)(−1p)={1 if p ≡ 1 (mod 4),−1 if p ≡ 3 (mod 4),2)(2p)={1 if p ≡ ±1 (mod 8),−1 if p ≡ ±3 (mod 8).82.2. Number Theory PreliminariesWe now state a pair of important and well-known theorems; the firstpertains to primes in arithmetic progressions [46, Theorem 3.3], [50, ChapterVI], and the second discusses how primes are the multiplicative buildingblocks of all integers [46, Theorem 3.15].Theorem 2.20 (Dirichlet’s Theorem on Primes in Arithmetic Pro-gressions). Suppose that a and b are relatively prime positive integers. Thenthere are infinitely many primes in the arithmetic progression an+ b, wheren ∈ N+.Theorem 2.21 (The Fundamental Theorem of Arithmetic). Everypositive integer greater than one can be written as a product of primes, andthis representation is unique up to the order of the prime factors.We also recall an elementary definition from p-adic number theory [18].Definition 2.22. Let p be a prime number and x be a nonzero rationalnumber. Ifx = pα · abwhere p - ab, then the p-adic valuation, vp(x), of x isvp(x) = α.It is a well-known fact that if a polynomial with rational coefficients hasroots in Q, then it also has roots in R and in Qp for every prime p ≥ 2.Therefore, a polynomial has no rational roots when there is a prime p ≤ ∞for which it does not have any p-adic roots. This leads us to the followingdefinition [18].Definition 2.23. A polynomial equation with rational coefficients is saidto have local solutions if it has roots in R and in Qp for every prime p ≥ 2. Ifthe polynomial equation has roots over Q, these are called global solutions.Thus, a polynomial equation with global solutions also has local solutionsfor every prime. We now state an important principle by Hasse, known asthe local-global principle or the Hasse principle [18, Section 3.5].Principle 2.24 (Hasse Principle). The existence or non-existence ofglobal solutions of a Diophantine equation can be detected by studying thelocal solutions of the equation.92.3. Linear Algebra PreliminariesIn other words, an equation has a solution over Q if and only if it hassolutions over R and Qp for all primes p ≥ 2. The Hasse-Minkowski theoremstates that the Hasse principle holds for quadratic forms in m variables withcoefficients in the field K [18, 50]. Unfortunately, the Hasse principle doesnot always hold true, as many equations are locally solvable everywhere butfail to have global solutions. For example, Selmer proved that the equation3x3 + 4y3 + 5y3 = 0 has a nontrivial solution in the real numbers and inall p-adic fields, but it does not have a nontrivial rational solution [48].This shows that the Hasse-Minkowski theorem cannot be extended to cubicforms.2.3 Linear Algebra PreliminariesIn this section, we discuss various notation, terminology, and results fromlinear algebra.We denote the identity matrix of order n by In, the zero matrix of ordern by 0n, and any non-square zero matrix by 0.Let A = [aij ] be an m × n matrix. The transpose of A is denoted byAT . We introduce the following notation to describe elementary row andcolumn operations within a matrix A.Notation: For a given matrix A, let Cx denote column x and Rx denoterow x, where x ∈ N+. ThenCx −→ Cx + Cyis used to represent the replacement of column Cx by the sum of columnsCx and Cy. Finally, we denote the interchange of columns Cx and Cy byCx ←→ Cy.Analogous notation is used for row replacements and interchanges.Definition 2.25. The rank of an m × n matrix A, denoted by rank(A),is the maximal number of linearly independent columns (or rows) of A.Equivalently, the rank of A is defined to be the dimension of its columnspace (or row space).The following theorem summarizes some important properties of deter-minants [26, Theorems 2.2, 2.3 & 3.26], [33, p. 462-465].Theorem 2.26. Let A be a square matrix of order n. Then the determinant,det(A), satisfies the following properties.102.3. Linear Algebra Preliminaries1) The value of the determinant remains unchanged if a scalar multipleof one row (or column) is added to another row (or column).2) The determinant of a triangular matrix is the product of its diagonalentries.3) Taking the transpose of a matrix does not alter its determinant, sodet(AT ) = det(A).4) det(A) 6= 0 if and only if rank(A) = n.For a matrix subdivided into four blocks, the following identities canoften help to simplify the calculation of the determinant. The proofs ofthese results can be found in [33, p. 467 & 475].Proposition 2.27. If A and D are square matrices over an arbitrary field,thendet[A B0 D]= det(A) det(D) = det[A 0C D].Proposition 2.28. If A and D are square matrices, thendet[A BC D]={det (A) det(D−CA−1B) if A−1 exists,det (D) det(A−BD−1C) if D−1 exists.11Chapter 3Elliptic Curves andCongruent Numbers3.1 OverviewThis chapter provides a brief introduction to elliptic curves and theirproperties. There are no new results in this chapter; its purpose is simplyto present the background theory necessary to appreciate and understandthe results discussed in subsequent chapters. We begin by formally definingelliptic curves and their corresponding group law. Several key theorems, in-cluding Mordell’s theorem, the Nagell-Lutz theorem, and Mazur’s theorem,are also discussed. The torsion subgroup is introduced, and then calculatedfor congruent number elliptic curves. The final two sections in the chapterare devoted to studying the rank of elliptic curves, with an emphasis ontechniques for computing this quantity. The chapter concludes with a dis-cussion of an important result by Monsky that allows the 2-Selmer rank ofcongruent number elliptic curves to be calculated.3.2 An Introduction to Elliptic CurvesAn elliptic curve over a field K is a non-singular, projective, cubic al-gebraic curve with a specified base point defined over K. In the projectivespace P2, an elliptic curve has the general form, known as the projective longWeierstrass normal form,Y 2Z + a1XY Z + a3Y Z2 = X3 + a2X2Z + a4XZ2 + a6Z3, (3.1)where a1, a2, a3, a4, a6 ∈ K [23, 47, 51, 52]. The base point on this curve inhomogeneous coordinates is O = [0 : 1 : 0]. It is referred to as the point atinfinity and is the only point with Z = 0 [23, 25, 47, 51, 52]. An introductionto projective algebraic geometry, including the following definition, can befound in Appendix A of [52].123.2. An Introduction to Elliptic CurvesDefinition 3.1. The projective plane, P2, is the set of equivalence classesof triples [a : b : c] with a, b, c not all equal to zero satisfying the equivalencerelation[a : b : c] ∼ [a′ : b′ : c′] if a = ta′, b = tb′, c = tc′ for some t 6= 0.An equivalence class of triples [a, b, c] is called a point in P2, and the numbersa, b, c are called homogeneous coordinates for the point [a, b, c] in P2.By substituting x = X/Z and y = Y/Z into Equation (3.1), the Weier-strass curve can be written in non-homogeneous coordinates asy2 + a1xy + a3y = x3 + a2x2 + a4x+ a6. (3.2)This is referred to as the long Weierstrass normal form. Furthermore, if thecharacteristic of the field K is not equal to two, a change of variables canbe made [51] in Equation (3.2):y 7→ y − 12(a1x+ a3).The simplified equation that results from this substitution isE : y2 = x3 + ax2 + bx+ c, (3.3)wherea =14a21 + a2, b =12a1a3 + a4, and c =14a23 + a6.This is known as the reduced Weierstrass equation.Recall that for such a curve to be considered an elliptic curve, it must benon-singular. That is, it has distinct roots or equivalently its discriminant,given byD = −4a3c+ a2b2 + 18abc− 4b3 − 27c2, (3.4)is not equal to zero [52]. This implies that elliptic curves over K ⊂ R musteither have three real roots (as shown in Figure 3.1), or a single real root anda pair of complex conjugate roots (as shown in Figure 3.2). Cubic curveswith repeated roots are not elliptic curves; examples of singular cubic curvesare shown in Figures 3.3 and 3.4.133.3. The Group Law on Elliptic Curves and Mordell’s TheoremFigure 3.1: The elliptic curvey2 = x(x−1)(x+1) has three realroots.Figure 3.2: The elliptic curvey2 = (x + 1)(x2 − 3x + 3) with asingle real root.Figure 3.3: The singular cubiccurve y2 = (x + 1)3 with a tripleroot at x = −1.Figure 3.4: The singular cubiccurve y2 = x2(x + 1) has a dou-ble root at x = 0.3.3 The Group Law on Elliptic Curves andMordell’s TheoremA fundamental property of elliptic curves is that their points form anabelian group under a specific binary operation. To investigate this property,we begin by considering the binary algebraic structure that consists of theset of rational points on the elliptic curve E over the field K, along withthe binary operation ∗. The line connecting two rational points P and Qon E intersects the curve at a third point, R, defined to be P ∗ Q. Thisprocess, known as the chord-tangent composition law [23], can be applied143.3. The Group Law on Elliptic Curves and Mordell’s Theoremregardless of whether P and Q are distinct points (Figure 3.5). In the casewhere P = Q, the line drawn is tanget to the curve at that point, so P isa point of multiplicity two. Since P and Q are rational points on the curveE, the line joining them is a rational line, and hence R = P ∗ Q is also arational point [23, 29, 52]. Thus, given a small number of rational points,the chord-tangent law can be used to generate many other rational pointson E.Figure 3.5: The chord-tangent composition law applied to distinct points Pand Q, shown on the left, and a single point P , shown on the right, on theelliptic curve y2 = (x+ 1)(x2 − 3x− 3).Unfortunately, the chord-tangent composition law lacks an identity ele-ment, so it cannot be considered a group law. To resolve this issue, we countthe point at infinity, O, as a rational point and define the set of K-rationalpoints on the elliptic curve E to beE(K) = {(x, y) ∈ E : x, y ∈ K} ∪ {O}.By coupling the set E(K) with the chord-tangent law, we can define a newbinary operation + that makes E(K) into an abelian group; the point atinfinity serves as the identity element for this group. To add two pointsP,Q ∈ E(K), we begin by using the chord-tangent law to find the pointP ∗Q. If P ∗Q 6= O and a line is drawn through the points P ∗Q = (xpq, ypq)and O, the resulting third point of intersection between the curve E and theline is O ∗ (P ∗Q); this point is defined to be P +Q [23, 29, 47, 51, 52]. Asshown in Figure 3.6, the line through P ∗Q and O is a vertical line, so thepoint P + Q is simply the reflection of the point P ∗ Q about the x-axis.Thus, P + Q = (xpq,−ypq). Note that if P ∗Q = O, then P + Q is definedto be O.153.3. The Group Law on Elliptic Curves and Mordell’s TheoremFigure 3.6: The group law operator + applied to the points P and Q on theelliptic curve y2 = (x+ 1)(x2 − 3x− 3).This important result is summarized by the following theorem [23, 47].Theorem 3.2. For the elliptic curve E over the field K, the set E(K) ofrational points is an abelian group under the binary operation + definedabove. The point at infinity O is the identity element in the group.This means that the following properties hold for the group law operator +:1) Closure: If P,Q ∈ E(K), then P +Q ∈ E(K).2) Associativity: (P +Q) +R = P + (Q+R) for all P,Q,R ∈ E(K).3) Identity Element: P + O = O + P = P for all P ∈ E(K).4) Inverse: For every P ∈ E(K), there exists a point −P ∈ E(K) suchthat P + (−P ) = O.5) Commutativity: P +Q = Q+ P for all P,Q ∈ E(K).Each of these properties is simple to verify except for associativity; addi-tional details regarding this can be found in [47, 51, 52].The precise coordinates of the point P +Q can be determined as follows.Let P = (x1, y1) and Q = (x2, y2) be points on the elliptic curve E given byEquation (3.3)y2 = x3 + a2x2 + a4x+ a6.163.3. The Group Law on Elliptic Curves and Mordell’s TheoremBy the chord-tangent law, we know that the points P andQ are connected bya line of the form y = λx+ν. Substituting this into x3+a2x2+a4x+a6−y2 =0 and simplifying yieldsx3 + (a2 − λ2)x2 + (a4 − 2λν)x+ (a6 − ν2) = 0. (3.5)The third point of intersection between the line y = λx + ν and the curveE is P ∗ Q = (x3, y3), so the cubic polynomial in Equation (3.5) has rootsx1, x2, and x3. Therefore, we can writex3 + (a2 − λ2)x2 + (a4 − 2λν)x+ (a6 − ν2)= (x− x1)(x− x2)(x− x3)= x3 − (x1 + x2 + x3)x2 + (x2x3 + x1x3 + x1x2)x− (x1x2x3).Equating the coefficients of the x2 terms on each side of the above equation,and solving for x3 yieldsx3 = λ2 − a2 − x1 − x2, (3.6)which is the x-coordinate of the point P ∗Q.If P = (x1, y1) and Q = (x2, y2) are points with x1 6= x2, thenλ =y2 − y1x2 − x1 , and ν = y1 − λx1 = y2 − λx2. (3.7)The corresponding y-coordinate for P ∗Q = (x3, y3) can be found by usingthe equationy3 = λx3 + νwith the values of x3, λ, and ν given in Equations (3.6) and (3.7). Since P+Qis the reflection of P ∗Q about the x-axis, it follows that P +Q = (x3,−y3).If P = Q, the line y = λx + ν lies tangent to the curve E at the pointP . When y1 6= 0, the slope of this tangent line isλ =3x21 + 2a2x1 + a42y1,which is found by implicitly differentiating the elliptic curve equation Ewith respect to x and then evaluating the resulting equation at P = (x1, y1).Substituting this value for λ into Equation (3.6) and simplifying yieldsx(2P ) =x41 − 2a4x21 − 8a6x1 + a24 − 4a2a64y21. (3.8)173.3. The Group Law on Elliptic Curves and Mordell’s TheoremThis equation is known as the duplication formula for the x-coordinate ofP and is used to compute the x-coordinate of the point 2P = P + P. They-coordinate of 2P can be found by following an analogous process to theone outlined above in the case where P = (x1, y1) and Q = (x2, y2) arepoints with x1 6= x2.This procedure for determining the precise coordinates of the point P+Qis known as the group law algorithm and is described in detail for generalWeierstrass curves in [51].A well-known theorem proved by Louis Mordell in 1922 provides furtherinsight into the structure of the group of rational points on a rational ellipticcurve [23, 25, 29, 51, 52, 59].Theorem 3.3 (Mordell’s Theorem). Let E be an elliptic curve over thefield of rational numbers. The group of rational points, E(Q), is a finitelygenerated abelian group.Proof. See Chapter 6 of [23], Chapter VIII.4 of [51], or Chapter III of [52].Because the group of rational points E(Q) forms a finitely generatedabelian group, the fundamental theorem of finitely generated abelian groups(Theorem 2.3) can be applied to write E(Q) as a direct sum of cyclic groupsE(Q) ∼= Zpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνωω ⊕ Z⊕ Z⊕ · · · ⊕ Z︸ ︷︷ ︸r copies,where Z is an infinite cyclic group and Zpνii is a finite cyclic group withprime-power order for all i ∈ [1, ω]. This can be stated more concisely asE(Q) ∼= T ⊕ Zr,where T ∼= Zpν11 ⊕Zpν22 ⊕· · ·⊕Zpνωω denotes the torsion subgroup and r ∈ N iscalled the arithmetic rank, or Mordell-Weil rank, of the elliptic curve E. Thetorsion subgroup is relatively easy to compute and will be discussed in detailin Section 3.4. However, the rank is not nearly as well understood, becausethere does not exist an effective method for calculating it in all cases. Thedetermination of the rank is crucial when it comes to studying congruentand non-congruent numbers; more information regarding this mysteriousquantity will be presented in Section 3.5.183.4. The Torsion Subgroup3.4 The Torsion SubgroupWe begin by stating the definition of the torsion subgroup [23, 29, 51, 52].Definition 3.4. The torsion subgroup, T , of E(Q) is the group consistingof all rational points of finite order on the elliptic curve E.Therefore, to find the rational points that belong to the torsion subgroup,we must recall the definition of the order of a point P ∈ E(K) [52].Definition 3.5. Let E be an elliptic curve and let P = (x, y) be a point inE(K). The point P has finite order if there exists m ∈ N+ such thatmP = P + P + · · ·+ P︸ ︷︷ ︸m copies= O,but m′P 6= O for all integers m′ ∈ [1,m). The integer m is called the orderof P . If no such integer m exists, then P has infinite order.Clearly, the point at infinity has order one, and since this is a point offinite order on every elliptic curve, the torsion subgroup is guaranteed tocontain at least one element. Points P = (x, y) of order two are also simpleto find, as they statisfy 2P = O with P 6= O. An equivalent way of writing2P = P + P = O is P = −P , which means that (x, y) = (x,−y). This onlyholds if y = −y, so y = 0. Thus, points of order two have the form (x, 0)[52]. Other rational points of finite order on an elliptic curve can be foundby using a well-known theorem proven independently by Nagell and Lutz[51, 52, 59].Theorem 3.6 (Nagell-Lutz Theorem). Lety2 = f(x) = x3 + ax2 + bx+ cbe a non-singular cubic curve with a, b, c ∈ Z, and letD = −4a3c+ a2b2 + 18abc− 4b3 − 27c3be the discriminant of f(x). Let P = (x, y) be a rational point of finiteorder. Then x and y are integers and either y = 0 or else y2 divides D.Proof. See Chapter VIII.7 of [51] or Chapter II of [52].193.4. The Torsion SubgroupIt should be emphasized that the Nagell-Lutz theorem is not an if-and-only-if statement. Therefore, points on the curve with integer coordinatessatisfying y2 |D are unfortunately not guaranteed to have finite order. How-ever, if P = (x, y) is a rational point of finite order, then 2P must also havefinite order. By Theorem 3.6, rational points of finite order have integercoordinates. The duplication formula for the x-coordinate of P , stated inEquation (3.8), can be applied to compute the x-coordinate of 2P . If x(2P )is found not to be an integer, 2P is not a point of finite order, and hence Palso is not a point of finite order.An alternate technique for determining the points of finite order on anon-singular cubic curve involves defining a reduction modulo p map that isan isomorphism. This method is summarized by the following theorem [52].Theorem 3.7 (Reduction Modulo p Theorem). Let C be a non-singularcubic curvey2 = x3 + ax2 + bx+ cwith a, b, c ∈ Z, and let D be the discriminantD = −4a3c+ a2b2 + 18abc− 4b3 − 27c2.Let Φ ⊆ C(Q) be the subgroup consisting of all points of finite order. Forany prime p, let P 7→ P˜ be the reduction modulo p mapΦ −→ C˜(Fp), P 7→ P˜ ={(x˜, y˜) if P = (x, y),O˜ if P = O.If p does not divide 2D, then the reduction modulo p map is an isomorphismof Φ onto a subgroup of C˜(Fp).Proof. See Section 4.3 of [52].Both the Nagell-Lutz theorem and the reduction modulo p theorem pro-vide methods for finding the points of finite order on an elliptic curve. How-ever, to completely characterize the torsion subgroup of the curve, we utilizea powerful result by Mazur [23, 29–31, 51, 52, 59].Theorem 3.8 (Mazur’s Theorem). For an elliptic curve E defined overQ, the torsion subgroup, T , of the group of rational points, E(Q), is iso-morphic to one of the following fifteen groups: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 order2N , Z2 × Z2N , with 1 ≤ N ≤ 4.203.4. The Torsion SubgroupProof. See [30] and [31].Of specific interest to this thesis is the torsion subgroup of congruentnumber elliptic curves En : y2 = x(x− n)(x+ n). By inspection, it is clearthat these curves have exactly three points of order two, (0, 0), (n, 0), and(−n, 0). Combining these three points with the point at infinity allows usto deduce that congruent number elliptic curves have at least four rationalpoints of finite order. We would like to show that these four points are theonly torsion points on En. We follow a method similar to that presented in[22, 25].We begin by proving a lemma that enables us to calculate the numberof points on the curve En over the finite field Fp for a prime p with p ≡ 3(mod 4).Lemma 3.9. If p is a prime with p - n and p ≡ 3 (mod 4), then the curveEn over Fp has exactly p+ 1 points.Proof. The points O, (0, 0), and (±n, 0) always lie on the curve En. Wewill verify that these four points are distinct over Fp. In homogeneouscoordinates, the points O, (0, 0), and (±n, 0) are written as [0 : 1 : 0],[0 : 0 : 1], and [±n : 0 : 1], respectively. According to Definition 3.1, twosets of coordinates represent the same point in projective space if and onlyif the coordinates differ by a nonzero constant. This means that the pointat infinity cannot equal (0, 0) or (±n, 0) over Fp. Furthermore, equality ofany two of (0, 0) or (±n, 0) leads to the congruences±n ≡ 0 (mod p) or n ≡ −n (mod p).By assumption p ≡ 3 (mod 4), so p is odd. Therefore, in order for theabove congruences to hold, we require that p|n, which contradicts our initialassumption that p - n. Thus, O, (0, 0), and (±n, 0) are distinct points overFp.We treat the cases p = 3 and p > 3 separately. When p = 3,y2 = x3 − n2x = x(x2 − n2) ≡{0 (mod p) if p|x,x(1− n2) (mod p) if p - x.By assumption p - n, so Fermat’s little theorem (Theorem 2.14) implies that(1− n2) ≡ 0 (mod p)for p = 3. Therefore, irrespective of whether p divides x, we havey2 = x(x− n)(x+ n) ≡ 0 (mod p).213.4. The Torsion SubgroupThis shows that O, (0, 0), and (±n, 0) are the only possible points on Enwhen p = 3, so there are exactly p+ 1 = 4 points on En over F3.Let us now consider the case where p > 3 and examine the points forwhich x 6= 0,±n. Since we are working modulo p, there are (p − 3) suchvalues for x. By assumption p is odd, so (p− 3) is even. As a result, we cangroup the (p − 3) values for x into pairs {±x}. Clearly there is a point onEn whenever (x3 − n2x) is a square in Fp. Therefore, we treat two cases,one where (x3−n2x) is a square and one where it is not a square. We beginby assuming that (x3 − n2x) is a square in Fp, and apply Theorem 2.17 todeduce that ((x3 − n2x)p)= 1. (3.9)Since p ≡ 3 (mod 4), it follows from Theorem 2.17, Theorem 2.19, andEquation (3.9) that(((−x)3 − n2(−x))p)=(−1p)((x3 − n2x)p)= −1.This enables us to conclude that ((−x)3−n2(−x)) is a quadratic nonresidueof p, and hence it is not a square in Fp.An analogous argument can be used to show that when (x3 − n2x) isnot a square in Fp, then ((−x)3 − n2(−x)) is a square. Thus, every pair{±x} leads to exactly one point on En over Fp. This means that only halfof the (p − 3) distinct x values contained by the pairs {±x} need to beconsidered. Each of these (p−3)/2 values for x corresponds to two y values,±y. Neglecting the points O, (0, 0), and (±n, 0), there are (p− 3) points onEn over Fp. Thus, altogether there are (p+ 1) points on En over Fp.The following theorem completely characterizes the torsion subgroup ofcongruent number elliptic curves.Theorem 3.10. For the congruent number elliptic curve En, |Tn| = 4 andTn ∼= Z2 × Z2.Proof. We know that the four torsion points O, (0, 0), and (±n, 0) lie on thecurve En. By way of contradiction, assume there is another torsion pointon En. This torsion point cannot be a point of order two, so it must haveorder greater than two. Since Tn contains three points of order two andat least one with order greater than two, Mazur’s Theorem (Theorem 3.8)implies that Tn is isomorphic to Z2 × Z2N with N ∈ {2, 3, 4}. Notice thatZ2 × Z4 is a group of order eight, so Theorem 2.4 can be applied to deduce223.4. The Torsion Subgroupthat it has subgroups of order 1, 2, 4, and 8. Similarly, Theorem 2.4 impliesthat Z2 × Z6 has subgroups of order 1, 2, 3, 4, 6, and 12, and Z2 × Z8has subgroups of order 1, 2, 4, 8, and 16. Therefore, regardless of which ofthe three groups the torsion subgroup Tn is isomorphic to, Tn contains asubgroup H = {P1, P2, . . . , Pm} of order m, where m is equal to either threeor eight.Consider the reduction modulo p map P 7→ P˜ . By Theorem 3.7, weknow that if p does not divide 2D, where D is the discriminant of En, thenthe reduction modulo p map is an isomorphism of Tn onto a subgroup ofE˜n(Fp). It follows that the reduction map P 7→ P˜ is injective on H forall primes p > 2D. Thus, m divides the order of the group E˜n(Fp) for allsuch p. Notice that for the curve En, the discriminant, given by Equation(3.4), reduces to D = 4n6. Since p > 2D = 8n6, clearly p - n. Therefore, ifp ≡ 3 (mod 4), we can apply Lemma 3.9 to deduce that |E˜n(Fp)| = p + 1.For primes p with p > 2D, we know that m divides the order of E˜n(Fp), som|(p+ 1), or equivalently p ≡ −1 (mod m). This implies that for all but afinite number of primes p with p ≡ 3 (mod 4), we havep ≡ −1 (mod m).By Dirichlet’s Theorem on Primes in Arithmetic Progressions (Theorem2.20), we know that for positive integers a and b with (a, b) = 1, there areinfinitely many primes p with p ≡ b (mod a).When m = 8, all but a finite number of primes p with p ≡ 3 (mod 4)satisfy the congruence p ≡ −1 (mod 8) ≡ 7 (mod 8). This means that ifm = 8, there are only finitely many primes with p ≡ 3 (mod 8), whichcontradicts Dirichlet’s Theorem (Theorem 2.20).Now suppose m = 3. Then for all but finitely many primes p with p ≡ 3(mod 4), we have p ≡ −1 (mod 3). Solving this system of congruencesby using the Chinese Remainder Theorem (Theorem 2.13) yields p ≡ 11(mod 12). Therefore, when m = 3, there can only be finitely many primesp ≡ 7 (mod 12), which contradicts Dirichlet’s Theorem (Theorem 2.20).Thus, the only rational points of finite order on the congruent numberelliptic curve En are O, (0, 0), and (±n, 0), so |Tn| = 4 and by Mazur’sTheorem (Theorem 3.8), Tn ∼= Z2 × Z2.233.5. The Arithmetic Rank and the Method of Complete 2-Descent3.5 The Arithmetic Rank and the Method ofComplete 2-DescentIn Section 3.3, the concept of rank was introduced when we appliedMordell’s Theorem in conjunction with the fundamental theorem of finitelygenerated abelian groups to writeE(Q) ∼= T ⊕ Zr.The arithmetic rank can formally be defined as follows.Definition 3.11. Let E be an elliptic curve and let E(Q) be its group ofrational points. The arithmetic rank of E, denoted by r, is the number ofgenerators with infinite order Z in E(Q). Equivalently, the arithmetic rankis the number of independent rational points of infinite order on E. Thearithmetic rank is also known as the Mordell-Weil rank.Calculating the arithmetic rank of elliptic curves is, in most cases, adifficult and computationally challenging problem. Currently, there do notexist methods for computing the rank of all elliptic curves. It has beenconjectured that the rank can be arbitrarily large [51], but finding ellipticcurves with even moderately high rank is a difficult task. Bhargava andShankar proved that the average rank of all elliptic curves over Q is atmost 1.17 and that a positive proportion of the curves have rank zero [3].Therefore, because moderate and high-rank elliptic curves are scarce, findingcurves with rank greater than one is a problem of interest.In 2006, Elkies discovered an elliptic curve with rank equal to at least28; the precise value of the rank is still unknown. The elliptic curve withthe largest known rank isy2 + xy + y = x3 − x2 + 31368015812338065133318565292206590792820353345x+302038802698566087335643188429543498624522041683874493555186062568159847.This curve was found by Elkies in 2009, and its rank is equal to 19. Rogerscurrently holds the record for the largest-rank congruent number ellipticcurve with r(n) = 7 for n = 797507543735 [45]. Additional informationregarding high-rank elliptic curves can be found in [9, 10].Recall from Lemma 1.2 that the arithmetic rank r(n) of congruent num-ber elliptic curves, which have the form y2 = x3 − n2x, is related to thecongruent number problem; if r(n) > 0, then n is a congruent number. Al-ter, Curtz, and Kubota [1] conjectured that integers of the form n ≡ 5, 6, 7243.5. The Arithmetic Rank and the Method of Complete 2-Descent(mod 8) are congruent. By using both the Birch and Swinnterton-Dyer con-jecture and the Shafarevich-Tate conjecture, Lagrange [10, 27, 49] was ableto expand upon Alter, Curtz, and Kubota’s work and state the followingconjecture for the parity of r(n):r(n) ≡{0 (mod 2) if n ≡ 1, 2, 3 (mod 8),1 (mod 2) if n ≡ 5, 6, 7 (mod 8).This would imply that integers n satisfying n ≡ 5, 6, 7 (mod 8) are congruentnumbers.Monsky provided evidence supporting the above conjecture when he usedthe link between congruent numbers and elliptic curves to prove that certainnumbers n with at most two distinct odd prime divisors are congruent [34].Monsky’s families of congruent numbers n ≡ 5, 6, 7 (mod 8) have the form1) p5, p7, 2p7, and 2p3,2) p3p7, p3p5, 2p3p5, and 2p5p7,3) p1p5 with(p1p5)= −1,4) p1p7 and 2p1p7 with(p1p7)= −1,5) 2p1p3 with(p1p3)= −1,where pi denotes primes that are congruent to i (mod 8).Tian expanded upon Monsky’s results and described the first familiesof congruent numbers with arbitrarily many prime divisors [7, 54, 55]. Hiscongruent numbers have one prime factor congruent to 3, 5, or 7 modulo8 and arbitrarily many prime factors congruent to 1 modulo 8. Tian alsoproved the following important theorem [7, 54, 55].Theorem 3.12. For any given integer k ≥ 0, there are infinitely manysquare-free congruent numbers with exactly k+ 1 odd prime divisors in eachresidue class of 5, 6, and 7 modulo 8.Methods known as descents are often applied to determine the rank ofan elliptic curve. These techniques have an extensive history dating backto the method of infinite decent. Fermat developed this technique and usedit to prove that no right triangle with integer side lengths can have its253.5. The Arithmetic Rank and the Method of Complete 2-Descentarea equal to an integer squared [5]. A complete 2-descent is a methodthat we can apply to calculate an upper bound for the rank of an ellipticcurve. This algorithm attempts to find the generators for the quotient groupEn(Q)/2En(Q) by determining whether specific pairs of equations are solv-able. The complete 2-descent algorithm for elliptic curves with a Weierstrassequation of the form y2 = (x − e1)(x − e2)(x − e3) over a field K is statedand proved in Section X.1 of [51]. The version of this theorem for congruentnumber elliptic curves is stated in [10, Theorem 1] and [49, Theorem 3.1]and is described by the following theorem.Theorem 3.13 (Complete 2-Descent for En). Let p1, p2, . . . , pt for t ∈N+ be distinct odd primes andn = 2p1p2 · · · ptbe a positive integer with  ∈ {0, 1}. Let En be the elliptic curve over Qdefined by the equationEn : y2 = x3 − n2x = x(x− n)(x+ n),andQ(S, 2) := {c ∈ Q∗/Q∗2| vp(c) ≡ 0 (mod 2) ∀ p ∈MQ\S},where vp(c) is the p-adic valuation of c and S = {∞, 2, p1, . . . , pt} is afinite subset of MQ, the set of all places of Q. Then there is an injectivehomomorphismb : En(Q)/2En(Q) −→ Q(S, 2)×Q(S, 2) (3.10)defined for P = (x, y) byb(P ) =(1, 1) if P = O,(−1,−n) if P = (0, 0),(n, 2) if P = (n, 0),(x, x− n) if P = (x, y) 6= O, (0, 0), (n, 0).If (b1, b2) ∈ Q(S, 2)×Q(S, 2)\{(1, 1), (−1,−n), (n, 2)}, then (b1, b2) ∈ Im(b)if and only if the system of equations{b1z21 − b2z22 = n,b1z21 − b1b2z23 = −n}(3.11)has a solution (z1, z2, z3) ∈ Q∗ ×Q∗ ×Q. In this case,(b1, b2) = b(P )263.5. The Arithmetic Rank and the Method of Complete 2-DescentforP = (b1z21 , b1b2z1z2z3) = (b2z22 + n, b1b2z1z2z3).Recall from Theorem 3.10 that the torsion subgroup of En is isomorphicto Z2 ⊕ Z2, and henceEn(Q) ∼= Z2 ⊕ Z2 ⊕ Zr(n),where r(n) is the arithmetic rank of En. Furthermore,En(Q)/2En(Q) ∼= (Z2)r(n)+2,and because the homomorphism b defined in Theorem 3.13 is injective,Im(b) ∼= (Z2)r(n)+2.This means that there are 2r(n)+2 pairs (b1, b2) for which the system ofequations in (3.11) has a solution. Clearly the four torsion points O, (0, 0),(n, 0), and (−n, 0) always lie on En. However, since there is no knownmethod for determining whether equations of the form in (3.11) are solvable,it often can be difficult to conclusively ascertain which of the remaining pairs(b1, b2) are in Im(b).By defining an upper bound B for the number of pairs (b1, b2) corre-sponding to non-torsion points for which the system in (3.11) may be solv-able, the following inequality can be established.2r(n)+2 ≤ B + 4 ⇐⇒ r(n) ≤ log2(B + 44). (3.12)If the precise number of elements in Im(b) can be determined, then theinequality in (3.12) becomes an equality. In addition, if the only pairs (b1, b2)for which the system in (3.11) is solvable correspond to torsion points on En,then B is zero and consequently the rank of En is also zero. By Lemma 1.2,we know that such rank-zero elliptic curves En correspond to non-congruentnumbers n.An example that illustrates how the method of complete 2-descent canbe used to conclusively determine the rank of En and generate a family ofnon-congruent numbers can be found in [24]. In this paper, Iskra provesthat there exists an infinite set of primes of the form 8k + 3 satisfying aspecific pattern of Legendre symbols, such that any product of primes inthis set is a non-congruent number; a detailed proof of the result using themethod of complete 2-descent can also be found in Section 4.1 of [38].273.6. The 2-Selmer Rank of En and Monsky’s Formula3.6 The 2-Selmer Rank of En and Monsky’sFormulaWe now consider the space formed by the intersection of the two conicsdefined in Equation (3.11). For a given pair (b1, b2), these two equationsdefine a new curveC(b1, b2) :={b1z21 − b2z22 = n,b1z21 − b1b2z23 = −n.This is called a homogeneous space for the elliptic curve En over the fieldQ, denoted by En/Q; more information regarding homogeneous spaces canbe found in Chapter X.3 of [51].Recall from Theorem 3.13 that if there exists a solution (z1, z2, z3) ∈Q∗ × Q∗ × Q satisfying C(b1, b2) for a given pair (b1, b2), then there is acorresponding point in En(Q)/2En(Q). But determining whether or not agiven homogeneous space C(b1, b2) has a solution is a challenging task. It issometimes possible to find rational points on homogeneous spaces by inspec-tion or by conducting searches using computers. Furthermore, for certainvalues of (b1, b2), curves can be eliminated from consideration because theydo not have any points over R or over Qp for some prime p; a collection ofuseful unsolvability conditions is stated in [49]. However, some homogeneousspaces have local solutions for every prime p, but yet do not have a globalsolution. In this situation, the Hasse principle (Principle 2.24) fails, whichmakes it difficult to fully determine the elements in En(Q)/2En(Q).We define two important sets of homogeneous spaces called the 2-Selmergroup and the Shafarevich-Tate group. The 2-Selmer group, Sel2(En/Q), isthe set consisting of the homogeneous spaces C(b1, b2) that have solutionseverywhere locally [29]. Because Q ⊆ Qp for all primes p ≥ 2, the homoge-neous spaces in Im(b), where b is the injective homomorphism described inEquation (3.10), belong to the 2-Selmer group. Therefore, the set of homo-geneous spaces in Sel2(En/Q) with solutions over Q forms a subgroup that isisomorphic to En(Q)/2En(Q). The Shafarevich-Tate group,X(En/Q)[2], isthe quotient group of Sel2(En/Q) by En(Q)/2En(Q). Thus, the Shafarevich-Tate group is comprised of elements that are equal to C(1, 1) or homogeneousspaces in Sel2(En/Q) that do not have solutions over the rational numbers[29]. If the Shafarevich-Tate group is trivial, then every homogeneous spacethat is locally solvable is also globally solvable, and hence corresponds to arational point on En.The relationship between the groups En(Q)/2En(Q), Sel2(En/Q), and283.6. The 2-Selmer Rank of En and Monsky’s FormulaX(En/Q)[2] is represented by the short exact sequence0 −→ En(Q)/2En(Q) ψ−→ Sel2(En/Q) φ−→X(En/Q)[2] −→ 0,where ψ is a monomorphism, φ is an epimorphism, and ker(φ) = Im(ψ)[23, 29, 51]. A thorough discussion of the theory governing the Selmer andShafarevich-Tate groups can be found in Chapter X.4 of [51].For the congruent number elliptic curve En,|Sel2(En/Q)| = 22+s(n),where s(n) is defined to be the 2-Selmer rank of En [10, 19, 20]. Thefollowing theorem describes the parity of the 2-Selmer rank of congruentnumber elliptic curves En [20, appendix].Theorem 3.14. If n is a positive square-free integer, thens(n) ≡{0 (mod 2) if n ≡ 1, 2, 3 (mod 8),1 (mod 2) if n ≡ 5, 6, 7 (mod 8).Proof. See the appendix by Monsky in [20].The Selmer conjecture claims that s(n) and r(n) have the same parity[19]. In addition, the arithmetic rank and 2-Selmer rank of the elliptic curveEn are related by the well-known inequalityr(n) ≤ s(n). (3.13)Notice that if the 2-Selmer rank of En is equal to zero, then the aboveinequality guarantees that the Mordell-Weil rank of the curve is also zero.Thus, n is a non-congruent number.Monsky, in the appendix of Heath-Brown’s paper [20], offered a newperspective on the 2-descent algorithm when he transformed it into a com-putation involving linear algebra. In his proof of Theorem 3.14, Monskybegins with a pair of equations derived by Heath-Brown in [19], and devel-ops an elegant formula for computing s(n). Monsky’s formula calculates the2-Selmer rank of En by relating it to the rank of a matrix with entries de-fined over F2. We will refer to this matrix as the Monsky matrix. Monsky’sformula for computing the 2-Selmer rank of En is essential to the proofs ofour main results in Chapters 4, 5, and 6 and is summarized by the followingtheorem.293.6. The 2-Selmer Rank of En and Monsky’s FormulaTheorem 3.15 (Monsky’s Formula for the 2-Selmer Rank of En).Let n be a square-free positive integer with odd prime factors p1, p2, . . . , pm.The diagonal m×m matrices Dk = [di] for k ∈ {−2,−1, 2}, and the m×mmatrix A = [aij ] are defined bydi =0, if(kpi)= 1,1, if(kpi)= −1,andaij =0, if(pjpi)= 1, j 6= i,1, if(pjpi)= −1, j 6= i,aii =∑j:j 6=iaij . (3.14)Then the 2-Selmer rank of En iss(n) ={2m− rankF2(Mo), if n = p1p2 · · · pm,2m− rankF2(Me), if n = 2p1p2 · · · pm,(3.15)where Mo and Me are the 2m× 2m matrices given byMo =[A + D2 D2D2 A + D−2], (3.16)andMe =[D2 A + D2AT + D2 D−1]. (3.17)Proof. See the appendix in [20].According to Monsky’s formula, given by Equation (3.15), for an odd (oreven) integer n, En has s(n) = 0 if and only if the matrix Mo in Equation(3.16) (or Me in Equation (3.17)) has full rank, or equivalently nonzerodeterminant. Therefore, the inequality in Equation (3.13) that relates thearithmetic rank to the 2-Selmer rank of En implies n is a non-congruentnumber ifdet(Mo) 6= 0 when n is odd,ordet(Me) 6= 0 when n is even.This idea will form the basis of our method for generating families of non-congruent numbers presented in Chapters 4, 5, and 6.30Chapter 4Families of Non-congruentNumbers4.1 OverviewIn this chapter, our focus is on generating a new family of even non-congruent numbers with arbitrarily many distinct prime divisors. The firstreference to families of non-congruent numbers dates back to a paper byGenocchi published in 1855 [15]. These original families of non-congruentnumbers contain a maximum of two distinct odd prime factors. In 1915Bastien [2] presented two additional families of non-congruent numbers com-prised of one or two prime factors. Over a century after the appearance ofGenocchi’s results, new families of non-congruent numbers having five orfewer prime divisors were described by Lagrange [27] and Serf [49]. Thesenew families were more complex in that they required conditions to be im-posed on the prime factors of the numbers and the associated values of theLegendre symbols relating the primes. Since then, families of non-congruentnumbers that are a product of arbitrarily many distinct prime factors havebeen described by Cheng and Guo [6], Feng [11], Feng and Xiong [12], Fengand Xue [13], Goto [16], Iskra [24], Li and Tian [28], Ouyang and Zhang[36, 37], Reinholz et al. [39, 40], and Wang [58].The purpose of this chapter is to describe a new family of even non-congruent numbers whose factorization contains arbitrarily many distinctprimes. The numbers that we generate have prime divisors in each oddcongruence class modulo eight. This characteristic distinguishes our familyof non-congruent numbers from other known families of even non-congruentnumbers that have prime factors belonging to a maximum of three oddcongruence classes modulo eight.This chapter is based on a result that appears in [42].314.2. A Family of Even Non-congruent Numbers4.2 A Family of Even Non-congruent NumbersWith Arbitrarily Many Prime FactorsOur main theorem describes a new family of even non-congruent numberscontaining prime factors in each odd congruence class modulo eight.Theorem 4.1. Let p, q, r, s1, s2, . . . , st be distinct prime numbers satisfyingthe following congruence conditions:p ≡ 1 (mod 8),q ≡ 5 (mod 8),r ≡ 3 (mod 8),sγ ≡ 7 (mod 8) ∀ 1 ≤ γ ≤ t,where t is an odd positive integer. In addition, assume that the prime factorssatisfy the following Legendre symbol conditions:(qp)=(rp)=(rq)= −1,(sip)= 1 for all i ∈ [1, t] with i ≡ 0 (mod 2),and (sjp)=(skq)=(skr)=(sksl)= −1for all j ∈ [1, t] with j ≡ 1 (mod 2) and 1 ≤ l < k ≤ t. Then by setting w =s1s2 · · · st, we have n = 2pqrw is a non-congruent number with s(n) = 0.Our method of proof for this theorem requires showing that the deter-minant of the Monsky matrix Me, stated in Equation (3.17), is nonzero.Proof. We begin by considering the case where t = 1, so n = pqrs1. TheMonsky matrix Me for n has the formMe =0 0 0 0 1 1 1 10 1 0 0 1 0 1 10 0 1 0 1 1 0 10 0 0 0 1 1 0 01 1 1 1 0 0 0 01 0 1 1 0 0 0 01 1 0 0 0 0 1 01 1 1 0 0 0 0 1.324.2. A Family of Even Non-congruent NumbersA software program, such as Maple, can be used to verify that this matrix hasfull rank, and hence by Equation (3.15), s(n) = 0 and n is a non-congruentnumber.We now let t be an odd integer with t ≥ 3 and construct the (2t +6) × (2t + 6) Monsky matrix Me for n = 2pqrw. Since some of the entriesin Me vary depending on the number of prime factors in w, we need toconsider two cases, t ≡ 1 (mod 4) or t ≡ 3 (mod 4). These cases are handledsimultaneously and the ensuing differences are carefully noted throughoutthe proof. All of our calculations are carried out over F2.The square Monsky matrix of order (2t+ 6) corresponding to n is givenby Equation (3.17),Me =[D2 A + D2AT + D2 D−1],whereD2 =0 0 00 1 00 0 100 0t, D−1 =0 0 00 0 00 0 100 It,andA =β 1 1 1 0 1 0 1 · · · 0 11 1 1 1 1 1 1 1 · · · 1 11 1 1 1 1 1 1 1 · · · 1 11 1 0 0 1 1 1 1 · · · 1 10 1 0 0 0 1 1 1 · · · 1 11 1 0 0 0 0 1 1 · · · 1 1............. . .. . .. . ................. . .. . .. . ................. . .. . .. . ....0 1 0 0 · · · · · · · · · · · · 0 0 11 1 0 0 · · · · · · · · · · · · 0 0 0,withβ ={1 if t ≡ 1 (mod 4),0 if t ≡ 3 (mod 4).334.2. A Family of Even Non-congruent NumbersNotice that the final t entries in the first row and the first column of Aexhibit an alternating pattern of ones and zeros.We perform numerous row and column operations to reduce Me, so thatthe value of its determinant can be easily computed. Note that because weare working over F2, these elementary row and column operations do notalter the determinant of Me. We list the column and row operations to beapplied to Me by using the notation introduced at the beginning of Section2.3.Ct+5 −→ Ct+4 + Ct+5C2t+6 −→ C2t+6 + Ct+5Even rowsRt+1 −→ Rt+1 +Rt+3Rt−1 −→ Rt−1 +Rt+3...R6 −→ R6 +Rt+3R4 −→ R4 +Rt+3R3 −→ R3 +Rt+3R2 −→ R2 +Rt+3Odd rowsRt −→ Rt +Rt+2Rt−2 −→ Rt−2 +Rt+2...R7 −→ R7 +Rt+2R5 −→ R5 +Rt+2R2 −→ R2 +Rt+2.In addition, if {t ≡ 1 (mod 4), then R1 −→ R1 +Rt+3,t ≡ 3 (mod 4), then R1 −→ R1 +Rt+2.When applied in chronological order, this collection of row and column op-erations transforms the (t + 3) × (t + 3) block in the upper right corner of344.2. A Family of Even Non-congruent NumbersMe intoB =0 0 1 1 0 1 0 1 · · · 0 1 0 β0 0 1 1 1 1 1 1 · · · 1 1 1 00 0 0 1 1 1 1 1 · · · 1 1 1 10 0 0 0 1 1 1 1 · · · 1 1 1 10 0 0 0 0 1 1 1 · · · 1 1 1 00 0 0 0 0 0 1 1 · · · 1 1 1 1.......... . .. . .. . .................... . .. . .. . .................... . .. . .. . .................... . . 0 1 1 00 0 0 · · · · · · · · · · · · · · · · · · 0 0 1 10 1 0 · · · · · · · · · · · · · · · · · · 0 0 0 01 0 0 · · · · · · · · · · · · · · · · · · 0 0 0 0,while leaving the remaining elements in Me unchanged. Furthermore, byapplying the row operationsR3 −→ R3 +R4R4 −→ R4 +R5...Rt−1 −→ Rt−1 +RtRt −→ Rt +Rt+1R2 −→ R2 + (Rt+1 +Rt−1 +Rt−3 + · · ·+R6 +R4 +R1)to Me in the order listed above, B becomesB′ =0 0 1 1 0 1 0 1 · · · 0 1 0 β0 0 0 0 0 0 0 0 · · · 0 0 0 10 0 0 1 0 0 0 0 · · · 0 0 0 00 0 0 0 1 0 0 0 · · · 0 0 0 10 0 0 0 0 1 0 0 · · · 0 0 0 10 0 0 0 0 0 1 0 · · · 0 0 0 1.......... . .. . .. . ................. . .. . .. . ................. . .. . .. . ................. . . 0 1 0 10 0 0 · · · · · · · · · · · · · · · · · · 0 0 1 10 1 0 · · · · · · · · · · · · · · · · · · 0 0 0 01 0 0 · · · · · · · · · · · · · · · · · · 0 0 0 0.354.2. A Family of Even Non-congruent NumbersNote that these operations do not alter the other three blocks within Me.Next, we perform numerous row interchanges in sequential order:Rt+2 ←→ Rt+1Rt+1 ←→ Rt...R6 ←→ R5R5 ←→ R4R4 ←→ R3R3 ←→ R1R2 ←→ R1R1 ←→ Rt+3.These row interchanges transform Me intoM∗e =0 0 0 0 0 · · · 00 0 0 0 0 · · · 00 0 0 0 0 · · · 00 0 1 0 0 · · · 00 0 0 0 0 · · · 0.... . ....0 1 0 0 · · · · · · 0B′′AT + D2 D−1,whereB′′ =1 0 0 0 0 0 0 0 · · · 0 0 0 00 1 0 0 0 0 0 0 · · · 0 0 0 00 0 1 1 0 1 0 1 · · · 0 1 0 β0 0 0 1 0 0 0 0 · · · 0 0 0 00 0 0 0 1 0 0 0 · · · 0 0 0 10 0 0 0 0 1 0 0 · · · 0 0 0 1.... . .. . .. . ........... . .. . .. . ........... . .. . .. . ........... . .. . .. . ........... . . 1 0 1... 0 1 10 · · · · · · · · · · · · · · · · · · · · · · · · · · · 0 0 1.364.2. A Family of Even Non-congruent NumbersOur goal is to transform the AT +D2 block of M∗e into an upper triangu-lar matrix. To accomplish this, we apply the following column replacementsin M∗e:Ct+2 −→ Ct+2 + Ct+1Ct+1 −→ Ct+1 + Ct...C3 −→ C3 + C2C2 −→ C2 + C1.These operations affect both the upper left and lower left blocks within M∗e,changing this matrix intoM∗∗e =0 0 0 0 0 · · · · · · · · · · · · 00 0 0 0 0...0 0 0 0 0...0 0 1 1 0...0 0 0 0 0................ . .... B′′............. . ....0 0 0 0 0 0 00 0 0 0 · · · · · · · · · 0 0 00 1 1 0 · · · · · · · · · 0 0 0β (β + 1) 0 0 1 1 · · · · · · 1 11 1 1 0 0 0 · · · · · · 0 11 0 1 0 0 0 · · · · · · 0 01 0 0 1 0 0......0 1 0 0 1 0......1 0 0 0 0 1...... D−1............. . ................... 1 0 00 1 0 0 · · · · · · · · · 0 1 01 0 0 0 · · · · · · · · · 0 0 0.374.2. A Family of Even Non-congruent NumbersWe then perform the following set of operations on M∗∗e .C2 −→ C2 + (Ct+2 + Ct + Ct−2 + · · ·+ C7 + C5)C1 ←→ Ct+3C1 −→ C1 + C2Rt+4 −→ Rt+4 +R1Rt+5 −→ Rt+5 +R2.Altogether, these five operations reduce the lower left block in M∗∗e to anupper triangular matrix while transforming the lower right block into theidentity matrix of order (t+ 3). This yieldsM∗∗∗e =[G B′′H It+3],whereG =0 0 0 0 0 · · · · · · · · · 00 0 0 0 0...0 0 0 0 0...0 0 1 1 0...0 0 0 0 0................ . ....0 0 0 0 0 0 00 0 0 0 · · · · · · 0 0 01 1 1 0 · · · · · · 0 0 0,andH =1 0 0 0 1 1 · · · · · · · · · 1 β0 1 1 0 0 0 · · · · · · · · · 0 10 0 1 0 0 0 0 10 0 0 1 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1. . . 0 1....... . .. . .. . .............. . . 1 0 0 0...... 0 1 0 10 0 · · · · · · · · · · · · · · · 0 0 1 00 0 · · · · · · · · · · · · · · · 0 0 0 1.384.2. A Family of Even Non-congruent NumbersFinally, we apply Proposition 2.28 to M∗∗∗e to obtaindet(M∗∗∗e ) = det(It+3) det(G−B′′I−1t+3H).Therefore,det(M∗∗∗e ) = det1 0 0 0 1 1 1 1 · · · 1 1 1 β0 1 1 0 0 0 0 0 · · · 0 0 0 10 0 1 1 0 1 0 1 · · · 0 1 0 00 0 1 0 0 0 0 0 · · · 0 0 0 10 0 0 0 1 0 0 0 · · · 0 0 0 10 0 0 0 0 1 0 0 · · · 0 0 0 00 0 0 0 0 0 1 0 · · · 0 0 0 1................ . ....................... . ....................... . ...................... 1 0 00 0 0 0 0 · · · · · · · · · · · · · · · 0 1 11 1 1 0 0 · · · · · · · · · · · · · · · 0 0 1.Notice that the entries appearing in the middle of the third row and themiddle of the last column of the above (t + 3) × (t + 3) matrix alternatebetween ones and zeros. To easily compute this determinant, we apply thefollowing operations to the matrix above to reduce it to an upper triangularmatrix with ones along its diagonal.Rt+3 −→ Rt+3 +R2C3 ←→ C4R1 ←→ Rt+3Rt+3 −→ Rt+3 + (R1 +R5 +R6 +R7 + · · ·+Rt+1 +Rt+2).Since we are working over F2, the determinant of this upper triangular ma-trix is equal to that of M∗∗∗e . Therefore,det(Me) = det(M∗∗∗e ) = 1,so Me has full rank. Thus, by Equation (3.15), s(n) = 0 and n is a non-congruent number.394.3. Numerical Examples4.3 Numerical ExamplesThe purpose of this section is to give some examples of non-congruentnumbers generated by Theorem 4.1. The prime divisors of these numbers arerequired to satisfy certain Legendre symbols conditions, and hence Dirich-let’s theorem on primes in arithmetic progressions (Theorem 2.20) guar-antees that our theorem produces infinitely many non-congruent numbers.We list examples of non-congruent numbers described by Theorem 4.1 inthe following table.Table 4.1: Non-congruent numbers generated by Theorem 4.1n = 2pqrs1s2 · · · st t (mod 4)2·41·13·19·71·31·1319·743·3191·28151·52879·33287·160583 12·73·5·83·47·223·7927·2287·6247·7127·8647·26863·252463 12 · 17 · 5 · 3 · 23 · 263 · 503 · 1583 · 743 · 18143 · 18047 32 · 17 · 29 · 131 · 31 · 127 · 743 · 967 · 2207 · 3391 · 2879· 59671 · 118247 · 350447 · 13784393In Appendix A, we show how Maple can be used to generate non-congruent numbers, such as the ones listed in Table 4.1, satisfying the con-ditions imposed in Theorem 4.1.40Chapter 5The Generation of Familiesof Odd Non-congruentNumbers From KnownNon-congruent Numbers5.1 OverviewIn this chapter, we present a novel technique for generating families ofodd non-congruent numbers. Unlike other known results on non-congruentnumbers that impose congruence conditions on each prime factor of a non-congruent number and on the Legendre symbols relating these primes, thenew technique described in this chapter provides a general approach for find-ing families of non-congruent numbers. That is, given an odd non-congruentnumber α corresponding to a congruent number elliptic curve with 2-Selmerrank equal to zero, new non-congruent numbers can be constructed by mul-tiplying α by primes of a specified form. This process allows existing familiesof non-congruent numbers to be extended and infinitely many non-congruentnumbers to be produced.This chapter is based on results that appear in [41] and [44].5.2 Extending Known Families of OddNon-congruent NumbersThis section provides an introduction to our extension technique for gen-erating new families of non-congruent numbers. We begin by introducingsome notation.Let pi ≡ 5 (mod 8) and qj ≡ 3 (mod 8) with i, j ∈ N+ be distinct oddprimes, and let a, b ∈ N with (a+ b) > 0. We define the sets415.2. Extending Known Families of Odd Non-congruent NumbersP ={∅ if a = 0,{p1, p2, . . . , pa} if a > 0,andQ ={∅ if b = 0,{q1, q2, . . . , qb} if b > 0.We now state the following theorem from [41].Theorem 5.1. Letα =∏pi∈Ppi∏qj∈Qqj,and suppose that the elliptic curvey2 = x(x2 − α2)has s(α) = 0. Define the square-free positive integer n byn = αr1r2 · · · rv,where r1, r2, . . . , rv are primes satisfying rk ≡ 1 (mod 8) for all k ∈ [1, v]with v ∈ N+. If for each k with 1 ≤ k ≤ v the set Sk defined bySk ={(rkpi),(rkqj),(rkrh)with 1 ≤ i ≤ a, 1 ≤ j ≤ b, and 1 ≤ h < k ≤ v}contains exactly one Legendre symbol equal to −1, then s(n) = 0 and n is anon-congruent number.Proof. We work over F2 and use properties from the field of linear algebrain conjunction with Monsky’s formula for the 2-Selmer rank of En to provethis result. We use Equation (3.16) to construct the (2a + 2b) × (2a + 2b)Monsky matrix for α = p1p2 · · · paq1q2 · · · qb. This matrix has the formMα =[Aα + Dα2 Ia+bIa+b Aα + Dα−2], (5.1)whereDα2 = Ia+b and Dα−2 =[Ia 00 0b]425.2. Extending Known Families of Odd Non-congruent Numbersrepresent the diagonal matrices for α, and Aα is the square A matrix oforder (a + b) corresponding to α. We use the specified Legendre symbolconditions imposed on the prime divisors of n = αr1r2 · · · rv and Equation(3.14) to form the A matrix for n, denoted by An. As a result, the Monskymatrix for n can be written asMn =[An + Dn2 Dn2Dn2 An + Dn−2],whereDn2 =[Dα2 00 0v]and Dn−2 =[Dα−2 00 0v].Guided by the structure of the matrix An, we use elementary row andcolumn operations to reduce Mn until the value of its determinant canbe easily computed. Since we are working over F2, the operations yield amatrix with the same determinant. To describe the sequence of steps usedto reduce Mn, we let mij denote the entry in the ith row and jth columnof Mn. We consider those entries with mij = 1, where 1 ≤ i ≤ (a + b + v),(a+b) < j ≤ (a+b+v), and i < j. Note that for every j ∈ [a+b+1, a+b+v],there exists a single value of i ∈ [1, a+ b+ v] satisfying mij = 1. Therefore,beginning with j = (a+b+v), we determine the corresponding value of i forwhich mij = 1. We subtract column j from column i, and then subtract rowj from row i. Following this, we decrease the value of j by one and repeat thepreviously described column and row subtraction operations. We continuethis process for each j = (a+ b+v−1), (a+ b+v−2), . . . , (a+ b+ 1). Uponcompleting the v column subtractions and v row subtractions, we find thatthe upper left block of Mn is reduced to[Aα + Dα2 00 Iv],while the remaining three blocks in Mn are left unaltered.The structure of the original An + Dn2 and An + Dn−2 blocks in Mn issimilar, so for each j ∈ [2a+ 2b+ v + 1, 2a+ 2b+ 2v], there is precisely onevalue of i ∈ [a + b + v + 1, 2a + 2b + 2v] for which mij = 1. Therefore, werepeat the aforementioned procedure, but with the rows i and the columns jsatisfying (a+b+v+1) ≤ i ≤ (2a+2b+2v), (2a+2b+v) < j ≤ (2a+2b+2v),and i < j. We begin with j = (2a+ 2b+ 2v) and complete the necessary v435.2. Extending Known Families of Odd Non-congruent Numberscolumn subtractions and v row subtractions, thus reducing the lower rightblock of Mn to [Aα + Dα−2 00 Iv].Altogether, these operations transform Mn intoM∗n =Aα + Dα2 0 Ia+b 00 Iv 0 0Ia+b 0 Aα + Dα−2 00 0 0 Iv.We now add rows (2a+ 2b+v+ 1) through (2a+ 2b+ 2v) to rows (a+ b+ 1)through (a+ b+ v) respectively to getM∗∗n =Aα + Dα2 00 IvIa+b+vDn2Aα + Dα−2 00 Iv.Following this, we perform (a + b + v) row interchanges to M∗∗n to obtainthe matrixM∗∗∗n =Dn2Aα + Dα−2 00 IvAα + Dα2 00 IvIa+b+v.Note that since we are working over F2,det(Mn) = det(M∗n) = det(M∗∗n ) = det(M∗∗∗n ). (5.2)Applying Proposition 2.28 to M∗∗∗n allows us to deduce that its determinantis equal todet(Ia+b+v) det(Dn2 −[Aα + Dα−2 00 Iv]I−1a+b+v[Aα + Dα2 00 Iv]),445.3. Numerical Examples of Odd Non-congruent Numberswhich, when simplified, reduces todet(M∗∗∗n ) = det([Ia+b 00 0]−[(Aα + Dα−2)(Aα + Dα2) 00 Iv])= det(Ia+b −(Aα + Dα−2)(Aα + Dα2))det (Iv)= det(Ia+b −(Aα + Dα−2)(Aα + Dα2)). (5.3)To compute this determinant, we need to consider the matrix Mα de-scribed by Equation (5.1). By assumption s(α) = 0, so Equation (3.15)implies that Mα has full rank, and hencedet(Mα) 6= 0. (5.4)In addition, if we perform (a + b) row interchanges to Mα to transform itintoM∗α =[Ia+b Aα + Dα−2Aα + Dα2 Ia+b],and apply Proposition 2.28 to M∗α, thendet (Mα) = det (M∗α) = det (Ia+b) det(Ia+b −(Aα + Dα−2)I−1a+b (Aα + Dα2))= det(Ia+b −(Aα + Dα−2)(Aα + Dα2)). (5.5)Combining Equations (5.2), (5.3), (5.4), and (5.5) enables us to concludethatdet(Mn) 6= 0.Thus, s(n) = 0 and n is a non-congruent number.5.3 Numerical Examples of Odd Non-congruentNumbersIn this section, we provide examples to demonstrate how Theorem 5.1can be used to generate new non-congruent numbers from known families ofnon-congruent numbers. Because these numbers are specified by conditionson the Legendre symbol relating their prime factors, Dirichlet’s theorem onprimes in arithmetic progressions (Theorem 2.20) guarantees that infinitelymany non-congruent numbers are produced by Theorem 5.1. Also, notethat the examples we list in Table 5.1 clearly belong to new families of non-congruent numbers because their prime factorizations differ from those of455.3. Numerical Examples of Odd Non-congruent Numbersexisting families of non-congruent numbers [2, 6, 11–13, 15–17, 24, 27, 28,36, 37, 39, 40, 49, 58].We begin by extending Iskra’s family of non-congruent numbers [24].Theorem 5.2 (Iskra). Let a ∈ N+ and suppose that p1, p2, . . . , pa are dis-tinct primes satisfying pi ≡ 3 (mod 8) and(pjpi)= −1 for j < i. Thenα = p1p2 · · · pa is a non-congruent number.In Reinholz’s master’s thesis [38, Section 4.2], the congruent numberelliptic curves associated with numbers described by Iskra’s theorem areshown to have 2-Selmer rank of zero. As a result, new non-congruent num-bers can be produced by using Theorem 5.1 to append a tail of primes ofthe form 8k + 1 to Iskra’s non-congruent numbers.Next, we apply Theorem 5.1 to the following result by Reinholz et al.[39].Theorem 5.3 (Reinholz et al.). Let m be a fixed nonnegative even integerand let a be any positive integer satisfying a ≥ m. Let Nm denote the set ofpositive integers with prime factorization p1p2 · · · pa, where p1, p2, . . . , pa aredistinct primes of the form 8k + 3 such that(pjpi)={ −1 if 1 ≤ j < i and (j, i) 6= (1,m),1 if 1 ≤ j < i and (j, i) = (1,m).If α ∈ Nm, then α is non-congruent.In the proof of this theorem [39], the congruent number elliptic curvescorresponding to these non-congruent numbers are shown to have 2-Selmerrank equal to zero. Therefore, Theorem 5.1 can be directly applied to The-orem 5.3 to generate infinitely many new non-congruent numbers.Finally, we use Theorem 5.1 to extend an important result by Ouyangand Zhang [37].Theorem 5.4 (Ouyang and Zhang). Let[xh]=(1−(xh))2,and suppose that α = p1p2 · · · pk ≡ 1, 3 (mod 8) with pi ≡ ±3 (mod 8) forall i ∈ [1, k]. Define B to be the k × k matrix with (i, j)-entries[pjpi]fori 6= j and with (i, i)-entries[α/pipi], andC = diag{[−1p1],[−1p2], . . . ,[−1pk]}.465.3. Numerical Examples of Odd Non-congruent NumbersIf B2 + CB + C is invertible, then α is a non-congruent number.With a little effort, one can prove that for the integer α in Theorem 5.4,the condition that B2 + CB + C is invertible is equivalent to the Monskymatrix having full rank. Thus, the matrix B2 + CB + C is invertible if andonly if s(α) = 0. As a result, Theorem 5.1 can be used to extend Ouyangand Zhang’s work and generate new non-congruent numbers containing arbi-trarily many prime factors belonging to two or three odd congruence classesmodulo eight.Numerical examples obtained by applying Theorem 5.1 to the non-congruentnumbers in Theorems 5.2-5.4 are given in Table 5.1.Table 5.1: Non-congruent numbers generated by Theorem 5.1TheoremSatisfiedby αn = (α) · r1r2 · · · rk Extension Tail LegendreSymbols that Equal −1Theorem5.2(19 · 11 · 163 · 419) · 97 · 313 ·617 · 1697 · 1721· 6521 · 15361 · 16889(9719),(313419),(617163),(1697163),(17211697),(65211721),(15361419),(168891721)Theorem5.2(347·83·11·3·499·1123·2803)·673 · 2953 · 3617 · 7417 · 8713(67311),(29531123),(36173),(74171123),(87132953)Theorem5.3(11 · 59 · 163 · 307 · 947) · 41 ·1361 · 2017 · 4057 · 4673 · 8969(4111),(136111),(201759),(4057947),(4673163),(89694673)Theorem5.3(3 · 11 · 67 · 163 · 691 · 1483 ·3019 · 2179 · 16987)· 2137 · 4273 · 13553 · 36793(2137163),(427367),(135533),(367931483)Theorem5.4(3·11·19·43·59·5·13·29·37)·27481 · 31321 · 52561 · 78049(2748129),(3132137),(5256113),(7804929)Theorem5.4(3 · 19 · 67 · 83 · 13 · 61 · 101 ·149) · 4177 · 9649 · 9721· 17449 · 26953 · 49297(417761),(964961),(972119),(1744983),(269539721),(4929767)Note that the non-congruent numbers presented in Table 5.1 were con-structed using similar Maple code to that described in Appendix A.475.4. An Extension Technique for Generating Odd Non-congruent Numbers5.4 An Extension Technique for Generating NewFamilies of Odd Non-congruent NumbersIn this section, we present a method for generating new families of oddnon-congruent numbers by extending other known families of non-congruentnumbers. Of significance is the fact that our method produces non-congruentnumbers with arbitrarily many distinct prime divisors in each of the fourodd congruence classes modulo eight. This is a distinguishing feature ofour result, as all other existing theorems on non-congruent numbers imposerestrictions on the number of prime factors belonging to the odd congruenceclasses modulo eight.We introduce some notation that will be used throughout this section.Let pi, qj , rk, and sl with i, j, k, l ∈ N+ be distinct odd primes, and leta, b, c, d ∈ N with (a+ b+ c+ d) > 0. We define the setP ={∅ if a = 0,{p1, p2, . . . , pa} if a > 0.The sets Q,R, and S are defined analogously with |Q| = b, |R| = c, and|S| = d. In addition, we letW = P ∪Q ∪R ∪ S.Our main result [44] provides a general extension technique for construct-ing new families of odd non-congruent numbers.Theorem 5.5. Defineα =∏pi∈Ppi∏qj∈Qqj∏rk∈Rrk∏sl∈Ssl, (5.6)and suppose that the elliptic curvey2 = x(x2 − α2)has 2-Selmer rank of zero. Let t > d with t ∈ N+, and define the oddsquare-free positive integer n byn = αsd+1sd+2 · · · st,where the prime factors of n satisfy the congruence conditions described inone of the three cases in Table 5.2.485.4. An Extension Technique for Generating Odd Non-congruent NumbersTable 5.2: Congruence conditions for the prime factors of the odd numbernCondition pi(mod 8)∀ pi ∈ Pqj(mod 8)∀ qj ∈ Qrk(mod 8)∀ rk ∈ Rsγ(mod 8)∀ γ ∈ [1, t]I 1 5 7 3II 1 7 3 5III 5 3 7 1In addition, assume that the primes appended onto α satisfy one of thefollowing Legendre symbol conditions.Condition 1:For all h ∈ [d + 1, t], pi ∈ P, qj ∈ Q, rk ∈ R, and g ∈ [1, h), one of thefollowing four sets of Legendre symbol conditions hold.A)(shpi)=(shqj)=(shrk)=(shsg)= 1.B)(pish)=(qjsh)=(rksh)=(sgsh)= 1.C)(shpi)=(shqj)=(shrk)=(shsg)= −1.D)(pish)=(qjsh)=(rksh)=(sgsh)= −1.Condition 2:For all h, β ∈ [d+ 1, t] with h 6= β(sβsh)= 1,and define T ⊆W withT ={µ∣∣∣∣ (shµ)= −1 ∀ h ∈ [d+ 1, t]}and |T | ≡ 1 (mod 2). For all primes ε ∈W\T,(shε)= 1 ∀ h ∈ [d+ 1, t].495.4. An Extension Technique for Generating Odd Non-congruent NumbersCondition 3:For each h ∈ [d+ 1, t], the set Lh defined byLh ={(shpi),(shqj),(shrk),(shsg)with pi ∈ P, qj ∈ Q, rk ∈ R, g ∈ [1, h)}has exactly one Legendre symbol equal to −1.Then, in each of the cases described in Table 5.3, n is a non-congruentnumber and the 2-Selmer rank of the elliptic curve y2 = x(x2 − n2) is zero.Table 5.3: Conditions on the prime factors in the extension tail of the oddnumber nCase CongruenceConditionLegendre SymbolConditionParity of (t− d)I.1.A I 1.A No restrictionI.1.B I 1.B No restrictionI.1.C I 1.C EvenI.1.D I 1.D EvenII.2 II 2 EvenIII.2 III 2 EvenIII.3 III 3 No restrictionCase I.1To prove this theorem, we use Monsky’s formula for the 2-Selmer rankof En along with various properties of determinants, including the one de-scribed by the following lemma.Lemma 5.6. LetQ1 =In1 0 0 00 0 In3 00 In2 0 00 0 0 In4, Q2 =In1 0 0 00 0 In2 00 In3 0 00 0 0 In4,R1 =In1 0 0 00 0 In3 00 0 0 In40 In2 0 0, and R2 =In1 0 0 00 0 0 In20 In3 0 00 0 In4 0,505.4. An Extension Technique for Generating Odd Non-congruent Numberswhere n1, n2, n3, n4 ∈ N+. ThenQ1Q2 = Q2Q1 = R1R2 = R2R1 = In1+n2+n3+n4 .Define the square matrix M of order (n1 + n2 + n3 + n4) byM =M11 M12 M13 M14M21 M22 M23 M24M31 M32 M33 M34M41 M42 M43 M44,where Mij is a ni × nj matrix. ThenM = Q1MQ2 =M11 M13 M12 M14M31 M33 M32 M34M21 M23 M22 M24M41 M43 M42 M44, (5.7)MN = R1MR2 =M11 M13 M14 M12M31 M33 M34 M32M41 M43 M44 M42M21 M23 M24 M22, (5.8)M∗ = R1MQ2 =M11 M13 M12 M14M31 M33 M32 M34M41 M43 M42 M44M21 M23 M22 M24, (5.9)andM′ = Q1MR2 =M11 M13 M14 M12M31 M33 M34 M32M21 M23 M24 M22M41 M43 M44 M42. (5.10)Thus, over F2,det(M) = det(M) = det(MN) = det(M∗) = det(M′).The proofs of the separate cases in Theorem 5.5 are sufficiently different,so we provide details for each of them individually. Throughout the proofs,we work over the finite field F2 and letδ = (a+ b+ c) and ω = (2δ + t+ d). (5.11)515.4. An Extension Technique for Generating Odd Non-congruent NumbersWe also let Aα denote the (δ + d)× (δ + d) A matrix for the integer α; theentries in this matrix are defined in Equation (3.14).We begin by presenting the proof of the first four cases listed in Table5.3.Case I.1 Proof. Consider n = αsd+1sd+2 · · · st and recall the Monsky matrixdescribed in Equation (3.16). We form the diagonal D2 and D−2 matricescorresponding to n; they can be written asDn2 =0a 0 0 0 00 Ib 0 0 00 0 0c 0 00 0 0 Id 00 0 0 0 It−d= Dα2 00 It−d (5.12)andDn−2 =0a 0 0 00 Ib+c 0 00 0 0d 00 0 0 0t−d = Dα−2 00 0t−d. (5.13)We also define a pair of (t− d)× (δ + d) matricesT = 0 · · · 0 0 · · · 0 1 · · · 1 1 · · · 1... ... ... ... ... ... ... ...︸ ︷︷ ︸a columns0 · · · 0 ︸ ︷︷ ︸b columns0 · · · 0 ︸ ︷︷ ︸c columns1 · · · 1 ︸ ︷︷ ︸d columns1 · · · 1 (5.14)andU = 1 · · · 1 1 · · · 1 0 · · · 0 0 · · · 0... ... ... ... ... ... ... ...︸ ︷︷ ︸a columns1 · · · 1 ︸ ︷︷ ︸b columns1 · · · 1 ︸ ︷︷ ︸c columns0 · · · 0 ︸ ︷︷ ︸d columns0 · · · 0. (5.15)We now consider the four separate subcases that are dependent upon theLegendre symbols of the prime factors appended onto α.525.4. An Extension Technique for Generating Odd Non-congruent NumbersCase I.1.A:For the given pattern of Legendre symbols, the (δ + t) × (δ + t) A matrixdescribed in Theorem 3.15 has the formAn =Aα 0Tad+1 0 · · · · · · 01 ad+2. . ........ . .. . .. . ........ . . at−1 01 · · · · · · 1 at=[Aα 0T A∗]. (5.16)Note that the entries along the diagonal in A∗ vary according to the parityof the numbers (c+d) and (t−d). By knowing the parity of these quantities,we can determine whether there are an even or odd number of ones in each ofthe final (t−d) rows in An. This information, when combined with Equation(3.14), allows us to easily deduce the values of the diagonal elements in A∗.These values are listed in Table 5.4.Table 5.4: Diagonal entries in the matrix An in Case I.1.A of Theorem 5.5Case Parity of(c+ d)Parity of(t− d)Values of the Entriesad+1, ad+2, . . . , at−1, at1 Even Odd 0, 1, 0, 1, 0, . . . , 1, 02 Even Even 0, 1, 0, 1, . . . , 0, 13 Odd Odd 1, 0, 1, 0, 1, . . . , 0, 14 Odd Even 1, 0, 1, 0, . . . , 1, 0We consider the first two cases listed in Table 5.4, and form the (2δ +2t) × (2δ + 2t) Monsky matrix Mo for n by using Equation (3.16). It canbe written asMo =Aα + Dα2 0 Dα2 0T A∗ + It−d 0 It−dDα2 0 Aα + Dα−2 00 It−d T A∗,535.4. An Extension Technique for Generating Odd Non-congruent Numberswhere Dα2 , Dα−2, T, and A∗ are described in Equations (5.12), (5.13), (5.14),and (5.16), respectfully. By applying Equation (5.9) from Lemma 5.6, wecan transform Mo intoM∗o =Aα + Dα2 Dα2 0 0Dα2 Aα + Dα−2 0 00 T It−d A∗T 0 A∗ + It−d It−d.Note that by Lemma 5.6,det(Mo) = det(M∗o).To compute the determinant of Mo, and hence determine the rank ofthis matrix, we use Proposition 2.27. However, before doing this, we carryout (t−d−1) row replacements in M∗o to reduce the A∗ block in M∗o to the(t− d)× (t− d) zero matrix. Because the operations vary slightly for eachof cases 1 and 2, we list them below using the notation defined in Section2.3 and Equation (5.11).Case 1:R2δ+2d+2 −→ R2δ+2d+2 + (Rω+2 +Rω+1)R2δ+2d+3 −→ R2δ+2d+3 + (Rω+2 +Rω+1)R2δ+2d+4 −→ R2δ+2d+4 + (Rω+4 +Rω+3 +Rω+2 +Rω+1)R2δ+2d+5 −→ R2δ+2d+5 + (Rω+4 +Rω+3 +Rω+2 +Rω+1)...Rω−3 −→ Rω−3 + (R2δ+2t−3 +R2δ+2t−4 + · · ·+Rω+2 +Rω+1)Rω−2 −→ Rω−2 + (R2δ+2t−3 +R2δ+2t−4 + · · ·+Rω+2 +Rω+1)Rω−1 −→ Rω−1 + (R2δ+2t−1 +R2δ+2t−2 + · · ·+Rω+2 +Rω+1)Rω −→ Rω + (R2δ+2t−1 +R2δ+2t−2 + · · ·+Rω+2 +Rω+1)545.4. An Extension Technique for Generating Odd Non-congruent NumbersCase 2:R2δ+2d+2 −→ R2δ+2d+2 + (Rω+2 +Rω+1)R2δ+2d+3 −→ R2δ+2d+3 + (Rω+2 +Rω+1)R2δ+2d+4 −→ R2δ+2d+4 + (Rω+4 +Rω+3 +Rω+2 +Rω+1)R2δ+2d+5 −→ R2δ+2d+5 + (Rω+4 +Rω+3 +Rω+2 +Rω+1)...Rω−4 −→ Rω−4 + (R2δ+2t−4 +R2δ+2t−5 + · · ·+Rω+2 +Rω+1)Rω−3 −→ Rω−3 + (R2δ+2t−4 +R2δ+2t−5 + · · ·+Rω+2 +Rω+1)Rω−2 −→ Rω−2 + (R2δ+2t−2 +R2δ+2t−3 + · · ·+Rω+2 +Rω+1)Rω−1 −→ Rω−1 + (R2δ+2t−2 +R2δ+2t−3 + · · ·+Rω+2 +Rω+1)Rω −→ Rω + (R2δ+2t +R2δ+2t−1 + · · ·+Rω+2 +Rω+1)Since we are working over F2, the A∗ matrix in M∗o is now reduced to thezero block, while all other entries in M∗o remain unchanged. For both case1 and case 2, M∗o is transformed intoM∗∗o =Aα + Dα2 Dα2 0 0Dα2 Aα + Dα−2 0 00 T It−d 0t−dT 0 A∗ + It−d It−d.Since [It−d 0t−dA∗ + It−d It−d]is a lower triangular matrix with ones along its diagonal, its determinant isequal to one. Therefore, by applying Proposition 2.27 to M∗∗o , we deducethatdet(Mo) = det(M∗∗o ) = det[Aα + Dα2 Dα2Dα2 Aα + Dα−2]. (5.17)Notice that the square matrix of order (2δ + 2d) within the determinantcalculation above is precisely the Monsky matrix for α. Recall that by ouroriginal assumption, s(α) = 0. As a result, Equation (3.15) implies that theMonsky matrix corresponding to α has full rank. It follows from Equation(5.17) that Mo has full rank. Thus, when (c + d) is even in Case I.1.A,s(n) = 0 and n is a non-congruent number.555.4. An Extension Technique for Generating Odd Non-congruent NumbersWe now consider cases 3 and 4 in Table 5.4. As in cases 1 and 2, we firstconstruct the (2δ+ 2t)× (2δ+ 2t) Monsky matrix Mo for n. We then applyEquation (5.10) from Lemma 5.6 to transform Mo intoM′o =Aα + Dα2 Dα2 0 0Dα2 Aα + Dα−2 0 0T 0 It−d A∗ + It−d0 T A∗ It−d,where the matrices Dα2 , Dα−2, T, and A∗ are given by Equations (5.12),(5.13), (5.14), and (5.16), respectfully. Notice that the A∗ + It−d block incase 3 is identical to the A∗ block in case 1. Therefore, the set of (t− d− 1)row operations described in case 1 can also be applied in case 3 to transformM′o intoM′′o =Aα + Dα2 Dα2 0 0Dα2 Aα + Dα−2 0 0T 0 It−d 0t−d0 T A∗ It−d. (5.18)Similarly, the A∗ + It−d block in case 4 is identical to the A∗ block incase 2. This means that the matrix M′′o, given by Equation (5.18), can beobtained by applying the same row replacement operations used in case 2.The remainder of the proof of cases 3 and 4 follows the process described inthe proof of cases 1 and 2. We conclude that in Case I.1.A, s(n) = 0 and nis a non-congruent number.Case I.1.B:We split the proof of Case I.1.B into two subcases according to the parityof (t − d), and start by considering the even case. The square A matrix oforder (δ + t) corresponding to n has the form565.4. An Extension Technique for Generating Odd Non-congruent NumbersAn =Aα TT01 1 1 · · · · · · · · · 10 0 1...0 0 1. . ........ . .. . .. . ........ . . 0 1 1... 0 1 10 · · · · · · · · · 0 0 0, (5.19)where TT is the transpose of the matrix defined in Equation (5.14). Since weare working over F2 and TT has dimension (δ+d)×(t−d) with (t−d) even,the elements along the diagonal in the upper left block of An are unaffectedby the entries in TT . Also, notice that the block in the lower right cornerof An is a (t− d)× (t− d) matrix with ones above its diagonal; this resultsin an alternating pattern of ones and zeros along the main diagonal of theblock.We construct the (2δ+ 2t)× (2δ+ 2t) Monsky matrix Mo for n by usingEquation (3.16) and apply Equation (5.9) from Lemma 5.6 to transform itintoM∗o =Aα + Dα2 Dα2 TT 0Dα2 Aα + Dα−2 0 TT0 0 It−d1 1 · · · · · · 10 0. . ........ . .. . .. . ........ . . 1 10 · · · · · · 0 00 00 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 0 10 · · · · · · 0 1It−d,575.4. An Extension Technique for Generating Odd Non-congruent Numberswhere Dα2 and Dα−2 are described in Equations (5.12) and (5.13), respec-tively. Next, we carry out the following (t− d− 1) row operations to reduceM∗o so that its determinant can be easily calculated using Proposition 2.27.R2δ+2d+1 −→ R2δ+2d+1 + (R2δ+2t +R2δ+2t−1 + · · ·+Rω+2 +Rω+1)R2δ+2d+2 −→ R2δ+2d+2 + (R2δ+2t +R2δ+2t−1 + · · ·+Rω+4 +Rω+3)R2δ+2d+3 −→ R2δ+2d+3 + (R2δ+2t +R2δ+2t−1 + · · ·+Rω+4 +Rω+3)R2δ+2d+4 −→ R2δ+2d+4 + (R2δ+2t +R2δ+2t−1 + · · ·+Rω+6 +Rω+5)...Rω−4 −→ Rω−4 + (R2δ+2t +R2δ+2t−1 +R2δ+2t−2 +R2δ+2t−3)Rω−3 −→ Rω−3 + (R2δ+2t +R2δ+2t−1 +R2δ+2t−2 +R2δ+2t−3)Rω−2 −→ Rω−2 + (R2δ+2t +R2δ+2t−1)Rω−1 −→ Rω−1 + (R2δ+2t +R2δ+2t−1)These row operations transform the matrix M∗o intoM∗∗o =Aα + Dα2 Dα2 TT 0Dα2 Aα + Dα−2 0 TT0 0 It−d 0t−d0 00 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 0 10 · · · · · · 0 1It−d,withdet(Mo) = det(M∗∗o ).The (2t− 2d)× (2t− 2d) block in the bottom right corner of M∗∗o is a lowertriangular matrix with determinant equal to one, so Proposition 2.27 impliesthatdet(Mo) = det[Aα + Dα2 Dα2Dα2 Aα + Dα−2],which is identical to Equation (5.17). Therefore, as in Case I.1.A, we con-clude that s(n) = 0 and n is a non-congruent number when (t− d) is even.585.4. An Extension Technique for Generating Odd Non-congruent NumbersFinally, we consider the case where (t−d) is odd, and form the A matrixfor n,An =Aα + J TT00 1 1 · · · · · · 10 1 1...0 0 0. . ........ . .. . .. . ........ . . 1 10 · · · · · · · · · 0 0,whereJ =[0a+b 00 Ic+d]. (5.20)The matrix J is introduced to account for the fact that when (t− d) is odd,the (c + d) rows of ones in TT affect the diagonal entries in the upper leftblock of An. Also, notice that the elements along the main diagonal inthe lower right block of An alternate between zeros and ones, but that thepattern is different from the one in the matrix described by Equation (5.19).We form the Monsky matrix Mo corresponding to n and follow the seriesof steps outlined in the proof of the case where (t− d) is even. That is, weperform a set of row operations and apply Proposition 2.27 to deduce thatdet(Mo) = det[Aα + J + Dα2 Dα2Dα2 Aα + J + Dα−2]. (5.21)Since we are working over F2,Dα2 = Dα−2 + J and Dα−2 = Dα2 + J,where the matrices Dα2 , Dα−2, and J are given by Equations (5.12), (5.13),and (5.20), respectfully. Therefore, Equation (5.21) becomesdet(Mo) = det[Aα + Dα−2 Dα2Dα2 Aα + Dα2].This can be rewritten in the form of Equation (5.17) by applying elementaryrow and column operations to the block matrix. As before, we conclude thats(n) = 0 and n is a non-congruent number when (t− d) is odd.595.4. An Extension Technique for Generating Odd Non-congruent NumbersCase I.1.C:The square A matrix of order (δ + t) corresponding to n isAn =1 · · · · · · · · · 1Aα......1 · · · · · · · · · 1ad+1 1 · · · · · · 10 ad+2. . ....U.... . .. . .. . ........ . . at−1 10 · · · · · · 0 at,where U is given by Equation (5.15). Because we are working over F2 andthe upper right block of An has an even number ones in each of its rows,the elements along the diagonal in the upper left block of An remain thesame as those in Aα. However, the diagonal entries in the lower right blockin An vary according to the parity of (a+ b). Since (t−d) is even, the blockin the lower right corner of An has an even number of columns, and hence(ad+1, ad+2, . . . , at−1, at) ={(1, 0, . . . , 1, 0) if (a+ b) is even,(0, 1, . . . , 0, 1) if (a+ b) is odd.We construct the Monsky matrix Mo for n by using Equation (3.16).When (a + b) is even, we apply Equation (5.9) from Lemma 5.6 to Mo, so605.4. An Extension Technique for Generating Odd Non-congruent Numbersthat it becomesM∗o =1 · · · · · · · · · 1Aα + Dα2 Dα2...... 01 · · · · · · · · · 11 · · · · · · · · · 1Dα2 Aα + Dα−2 0......1 · · · · · · · · · 11 1 · · · · · · 10 0. . ....0 U It−d.... . .. . .. . ........ . . 1 10 · · · · · · 0 00 1 · · · · · · 10 1. . ....U 0.... . .. . .. . .... It−d.... . . 0 10 · · · · · · 0 1.We modify this matrix by implementing a series of row operations. Weadd all of the final (t − d) rows in M∗o to each of rows (δ + d + 1) through(2δ+2d) in M∗o. These elementary operations transform the rightmost blockof ones in M∗o into a zero block while leaving the remaining entries in M∗ounchanged. We reduce the nonzero block immediately above the identitymatrix in the lower right corner of M∗o to the zero matrix by carrying outthe (t − d − 1) row replacements listed in Case I.1.B. Finally, we add eachof rows (2δ + 2d + 1) through (2δ + t + d) to the first (δ + d) rows in M∗o.615.4. An Extension Technique for Generating Odd Non-congruent NumbersAltogether, these operations transform M∗o intoM∗∗o =Aα + Dα2 Dα2 0 0Dα2 Aα + Dα−2 0 00 U It−d 0t−dU 00 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 0 10 · · · · · · 0 1It−d.We apply Proposition 2.27 to M∗∗o to obtain Equation (5.17). As before, itfollows that s(n) = 0 and n is a non-congruent number when (a+ b) is even.Finally, we consider the case where (a + b) is odd, and use Equation(5.10) from Lemma 5.6 to transform Mo intoM′o =1 · · · · · · · · · 1Aα + Dα2 Dα2 0......1 · · · · · · · · · 11 · · · · · · · · · 1Dα2 Aα + Dα−2...... 01 · · · · · · · · · 11 1 · · · · · · 10 0. . ....U 0 It−d.... . .. . .. . ........ . . 1 10 · · · · · · 0 00 1 · · · · · · 10 1. . ....0 U.... . .. . .. . .... It−d.... . . 0 10 · · · · · · 0 1.Notice that this matrix is very similar to the matrix M∗o in the case where(a + b) is even; of specific interest is the fact that the lower right square625.4. An Extension Technique for Generating Odd Non-congruent Numbersblocks of order (2t − 2d) in each matrix are identical. Therefore, the casewhere (a+ b) is odd can be proved analogously to the case where (a+ b) iseven. Thus, we conclude that s(n) = 0 and n is a non-congruent numberirrespective of the parity of (a+ b).Case I.1.D:We construct the (δ+ t)× (δ+ t) A matrix for n from the Legendre symbolconditions imposed on the tail of primes appended onto α. This matrix hasthe formAn =Aα UT1 · · · 1..................1 · · · 1ad+1 0 · · · · · · 01 ad+2. . ........ . .. . .. . ........ . . at−1 01 · · · · · · 1 at,where UT is the transpose of the matrix in Equation (5.15). Since the (t−d)elements in each row of UT are either all zeroes or all ones, the diagonalentries in the upper left block of An are not affected by the elements inUT , and hence remain the same as those in Aα. However, the entries alongthe diagonal in the lower right block in An depend upon the parity of thequantity (δ + d). By assumption (t− d) is even, so we can deduce that(ad+1, ad+2, . . . , at−1, at) ={(0, 1, . . . , 0, 1) if (δ + d) is even,(1, 0, . . . , 1, 0) if (δ + d) is odd.We consider the case where (δ+ d) is even and form the Monsky matrixMo for n. We apply Equation (5.9) from Lemma 5.6 to rearrange the blocks635.4. An Extension Technique for Generating Odd Non-congruent Numbersin Mo and transform it intoM∗o =Aα + Dα2 D2 UT 0Dα2 Aα + Dα−2 0 UT01 · · · 1..................1 · · · 1It−d0 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 0 01 · · · · · · 1 11 · · · 1..................1 · · · 101 0 · · · · · · 01 0. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 0It−d.To apply Proposition 2.27 to this matrix, we must reduce the two UTblocks in the upper right corner of M∗o to zero blocks by carrying out rowoperations. Specifically, we add the final (t − d) rows in M∗o to each ofrows (δ + d + 1) through (δ + d + a + b). Notice that these (a + b) rowreplacements only affect the elements in the rightmost UT block of M∗o,leaving the remaining entries in M∗o unchanged. We then carry out thesame set of (t− d− 1) row replacements as in case 2 of Case 1.I.A to reducethe (2t− 2d)× (2t− 2d) block in the bottom right corner of M∗o to a lowertriangular matrix. Finally, we add all of rows (2δ+2d+1) through (2δ+t+d)to each of the first (a+b) rows in M∗o. This set of row operations transformsM∗o into645.4. An Extension Technique for Generating Odd Non-congruent NumbersM∗∗o =Aα + Dα2 D2 0 0Dα2 Aα + Dα−2 0 001 · · · 1......1 · · · 1It−d 0t−d1 · · · 1..................1 · · · 101 0 · · · · · · 01 0. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 0It−d.Sincedet(Mo) = det(M∗∗o ),Proposition 2.27 yieldsdet(Mo) = det[Aα + Dα2 Dα2Dα2 Aα + Dα−2].This is identical to Equation (5.17). It follows that s(n) = 0 and n isnon-congruent when (δ + d) is even.In the case where (δ + d) is odd, we construct the Monsky matrix Mo655.4. An Extension Technique for Generating Odd Non-congruent Numbersfor n and use Equation (5.10) from Lemma 5.6 to transform Mo intoM′o =Aα + Dα2 D2 0 UTDα2 Aα + Dα−2 UT 01 · · · 1..................1 · · · 10 It−d0 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 0 01 · · · · · · 1 101 · · · 1..................1 · · · 11 0 · · · · · · 01 0. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 0It−d.Because the matrices M∗o for (δ + d) even and M′o for (δ + d) odd aresimilar, we can follow the same process to prove the case where (δ + d) isodd. We conclude that, regardless of the parity of (δ + d), s(n) = 0 and nis a non-congruent number.Next, we prove that Case II.2.Case II.2 Proof. We construct the (δ + t) × (δ + t) D2 and D−2 matricesfor n = αsd+1sd+2 · · · st,Dn2 =[Dα2 00 It−d]and Dn−2 =[Dα−2 00 It−d],where Dα2 and Dα−2 are the (δ+d)× (δ+d) diagonal matrices correspondingto α. We also define the (δ+d)× (t−d) matrix B with the elements in eachof its rows being either all equal to one or all equal to zero. In addition, werequire that B contains an odd number of rows with entries all equal to one.665.4. An Extension Technique for Generating Odd Non-congruent NumbersThe Monsky matrix Mo for n can be written asMo =[An + Dn2 Dn2Dn2 An + Dn−2]=Aα + Dα2 B Dα2 0BT 0t−d 0 It−dDα2 0 Aα + Dα−2 B0 It−d BT 0t−d.Notice that since (t−d) is even, the sum of the elements in an arbitrary rowof B is congruent to zero modulo two. Because we are working over F2, theelements along the main diagonal of the upper left blocks of An + Dn2 andAn + Dn−2 in Mo are not affected by the entries in B. Furthermore, seeingthat there are an odd number of rows in B with elements all equal to one,there are an odd number of entries in each row of BT that are equal to one.This allows us to determine that the diagonal elements in the lower rightblocks of An + Dn2 and An + Dn−2 are all zero.We apply Equation (5.9) from Lemma 5.6 to rearrange the blocks in Moand transform it intoM∗o =Aα + Dα2 Dα2 B 0Dα2 Aα + Dα−2 0 B0 BT It−d 0t−dBT 0 0t−d It−d. (5.22)Our goal is to reduce the two B blocks in M∗o to zero matrices by carryingout a set of row operations. We begin by adding rows (2δ+ 2d+ 1) through(2δ + d + t) in M∗o to each of the first (δ + d) rows in M∗o that correspondto rows in B containing all ones. Notice that when we apply these rowoperations, the entries in BT do not affect those in the Dα2 block. This isbecause each column in BT contains elements that are either all ones or allzeros, and we are adding all of the (t− d) rows of BT , where (t− d) is even,to a given row in Dα2 . An analogous procedure can be applied to transform675.4. An Extension Technique for Generating Odd Non-congruent Numbersthe remaining B block in M∗o into the zero matrix; this reduces M∗o toM∗∗o =Aα + Dα2 Dα2 0 0Dα2 Aα + Dα−2 0 00 BT It−d 0t−dBT 0 0t−d It−d. (5.23)Since we are working over F2, the elementary row and column operationsthat we applied to transform Mo into M∗∗o do not affect the value of thedeterminant, sodet(Mo) = det(M∗∗o ).Furthermore, by applying Proposition 2.27 to M∗∗o , we can deduce thatdet(Mo) = det[Aα + Dα2 Dα2Dα2 Aα + Dα−2]. (5.24)The block matrix in the determinant calculation immediately above is theMonsky matrix for α. Our original assumption that s(α) = 0 implies thatthe Monsky matrix corresponding to α has full rank. Thus, by Equation(5.24), we conclude that Mo also has full rank, so s(n) = 0 and n is anon-congruent number.We now provide a proof of Case III.2.Case III.2 Proof. We define Dα2 and Dα−2 to be the (δ+d)×(δ+d) diagonalD2 and D−2 matrices for α. The diagonal matrices corresponding to n =αsd+1sd+2 · · · st have the formDn2 =[Dα2 00 0t−d]and Dn−2 =[Dα−2 00 0t−d].We write the (2δ + 2t)× (2δ + 2t) Monsky matrix for n asMo =Aα + Dα2 B Dα2 0BT It−d 0 0t−dDα2 0 Aα + Dα−2 B0 0t−d BT It−d,685.4. An Extension Technique for Generating Odd Non-congruent Numberswhere the block B is identical to that described in Case II.2.As in Case II.2, we transform the two B blocks in Mo into zero blocksby carrying out row operations. Specifically, we add rows (δ+d+1) through(δ + t) in Mo to each of the first (δ + d) rows in Mo that correspond torows in B containing all ones. This reduces the B block in the upper leftquadrant of Mo to a zero block, while leaving the remaining entries in Mounchanged. In addition, we add each of rows (2δ+d+t+1) through (2δ+2t)to any of rows (δ + t + 1) through (2δ + t + d) in Mo that correspond torows in B containing all ones. The matrix that results from applying thisset of row operations to Mo isM∗o =Aα + Dα2 0 Dα2 0BT It−d 0 0t−dDα2 0 Aα + Dα−2 00 0t−d BT It−d.By interchanging rows and columns in M∗o, it can be transformed into thematrix given by Equation (5.23) in Case II.2. Therefore, the remainder ofthe proof follows as in Case II.2, and we conclude that in Case III.2, s(n) = 0and n is a non-congruent number.Finally, we complete the proof of Theorem 5.5 by verifying that CaseIII.3 generates non-congruent numbers. Note that Case III.3 of Theorem5.5 provides a generalization of Theorem 5.1.Case III.3 Proof. LetMα =[Aα + Dα2 Dα2Dα2 Aα + Dα−2](5.25)be the Monsky matrix corresponding to α. We use induction to prove thats(n) = 0 and n is non-congruent. We begin by considering the case whereα is multiplied by a single prime, sd+1, satisfying the pattern of Legendresymbols described in the theorem statement, son1 = αsd+1.695.4. An Extension Technique for Generating Odd Non-congruent NumbersThe (2δ + 2d+ 2)× (2δ + 2d+ 2) Monsky matrix for n1 can be written asMn1 =[An1 + Dn12 Dn12Dn12 An1 + Dn1−2]=F v Dα2 0vT 1 0 0Dα2 0 C v0 0 vT 1,where v is a column vector with only a single element equal to one and allof its remaining elements equal to zero. Due to the form of v, we introducethe matrices F and C, whereF = Aα + Dα2 + vvTandC = Aα + Dα−2 + vvT .Notice that F is nearly identical to the Aα+Dα2 block in Mα, with the onlydifference between them being a single element along their main diagonals.This difference is a consequence of the column vector v having a singleelement equal to one. Likewise, the only difference between the C blockin Mn1 and the Aα + Dα−2 block in Mα is a single element along theirdiagonals.We carry out row and column operations on Mn1 to reduce it, so that itsdeterminant can be easily computed. Let mγ denote the entry in row  andcolumn γ of Mn1 , and consider the element mγ = 1 with γ = (δ + d + 1)and  ∈ [1, δ + d]. We subtract column (δ + d+ 1) from column , and thensubtract row (δ + d + 1) from row . Similarly, there is a single elementmγ = 1 in Mn1 with γ = (2δ + 2d+ 2) and  ∈ [δ + d+ 2, 2δ + 2d+ 1]. Bysubtracting column (2δ + 2d+ 2) from column , and then subtracting row(2δ + 2d+ 2) from row , we transform Mn1 intoM∗n1 =Aα + Dα2 0 Dα2 00 1 0 0Dα2 0 Aα + Dα−2 00 0 0 1.705.4. An Extension Technique for Generating Odd Non-congruent NumbersApplying row and column interchanges to M∗n1 allows us to write it asM∗∗n1 =Aα + Dα2 Dα2 0 0Dα2 Aα + Dα−2 0 00 0 1 00 0 0 1.Since we are working over F2, and M∗∗n1 was obtained from Mn1 via a seriesof elementary row and column operations, we havedet(Mn1) = det(M∗∗n1). (5.26)Furthermore, Proposition 2.27 implies thatdet(M∗∗n1) = det[Aα + Dα2 Dα2Dα2 Aα + Dα−2].We combine this equation with Equations (5.25) and (5.26) to deduce thatdet(Mn1) = det(Mα).By assumption, s(α) = 0, so we know that the Monsky matrix correspondingto α has full rank. Thus, we conclude that Mn1 also has full rank, and hences(n1) = 0 and n1 = αsd+1 is a non-congruent number.We now assume that the theorem statement is true for the integern∗ = αsd+1sd+2 · · · st−2st−1 formed by appending a tail of (t−d− 1) primessatisfying the specified Legendre symbol conditions onto α. By assumption,we know thats(n∗) = 0.Consider the integern = (αsd+1sd+2 · · · st−2st−1)stthat is generated by appending a single prime, st, satisfying the requiredLegendre symbol conditions onto n∗. Proving that n is non-congruent byknowing that n∗ has s(n∗) = 0 is analogous to showing that n1 is non-congruent when α satisfies s(α) = 0. This is because in each of the cases, asingle prime of the specified form is appended onto a number whose corre-sponding congruent number elliptic curve has 2-Selmer rank equal to zero.Thus, an identical argument to the one that was used to verify that n1 is anon-congruent number can be applied here to deduce that s(n) = 0 and nis non-congruent.715.5. Examples of Odd Non-congruent NumbersThe families of non-congruent numbers generated by Cases I.1.A andI.1.B of Theorem 5.5 exhibit an interesting property that is summarized bythe following corollary.Corollary 5.7. LetW ∪ {sd+1, sd+2, . . . , st}be a collection of distinct primes satisfying the hypotheses of either CaseI.1.A or Case I.1.B in Theorem 5.5. Any product of integers from the setH = {α, sd+1, sd+2, . . . , st},where α is given by Equation (5.6), is a non-congruent number.Proof. Let λ be a product of integers belonging to the setH. If λ is a productof two or more of the primes sd+1, sd+2, . . . , st, but does not contain α, thenλ is non-congruent by Theorem 5.2. If λ is a product of α and at least oneof the remaining primes in H, then λ is guaranteed to be non-congruent byeither Case I.1.A or Case I.1.B of Theorem 5.5.5.5 Examples of Odd Non-congruent NumbersGenerated from Known Non-congruentNumbersIn this section, we provide a collection of non-conguent numbers sat-isfying the criterion described in Theorem 5.5. Any odd non-congruentnumber corresponding to an elliptic curve with 2-Selmer rank equal to zerocan be extended to produce other non-congruent numbers by using Theo-rem 5.5. Since various Legendre symbol conditions are imposed upon theprimes that are appended onto the existing non-congruent numbers, Dirich-let’s theorem on primes in arithmetic progressions (Theorem 2.20) ensuresthat each case in Theorem 5.5 generates infinitely many non-congruent num-bers. Depending on the prime divisors of the non-congruent number chosenfor α, the numbers constructed by applying Theorem 5.5 to α may over-lap with those described by other known families of non-congruent num-bers. However, the numerical examples that we present in Tables 5.5, 5.6,5.7, 5.8, and 5.9 clearly belong to new families of non-congruent numbers,because their prime factorizations differ from those in [2, 6, 11–13, 15–17, 24, 27, 28, 36, 37, 39, 40, 49, 58].Appendix A offers insight into how Maple can be used to construct thenon-congruent numbers listed in the tables in this section. In particular,725.5. Examples of Odd Non-congruent Numbersthe appendix provides the Maple code used to produce one of the numericalexamples in Table 5.5.We begin by using Case I.1 in Theorem 5.5 to generate new families ofnon-congruent numbers from existing families of non-congruent numbers.First, we consider Lagrange’s paper [27], which describes many differentfamilies of non-congruent numbers containing a maximum of three odd primefactors. We apply Case I.1 of our extension theorem to two of Lagrange’sfamilies of non-congruent numbers.The first family has the form qr, where q ≡ 5 (mod 8) and r ≡ 7 (mod 8),and the condition (qr)= −1is satisfied. The numbers in the second family are a product of three primefactors, pqr, with p ≡ 1 (mod 8), q ≡ 5 (mod 8), and r ≡ 7 (mod 8) satisfy-ing(pq)=(pr)= −1,(qr)=(qp)= −1, or(rp)=(rq)= −1.In Section 9 of [17], Goto states that the numbers belonging to either oneof Lagrange’s two families described above correspond to congruent numberelliptic curves with 2-Selmer rank of zero. Therefore, new families of non-congruent numbers can be generated using Theorem 5.5 to extend theseexisting families of non-congruent numbers. Some numerical examples aregiven in Tables 5.5 and 5.6.Furthermore, since the numbers described by Theorem 5.1 were shownto have congruent number elliptic curves with 2-Selmer rank equal to zero,Theorem 5.5 can be applied to these non-congruent numbers to produceinfinitely many non-congruent numbers. Tables 5.5 and 5.6 include severalnumerical examples that extend some of the non-congruent numbers givenin Table 5.1.It is worthwhile to mention that it is possible to apply Theorem 5.5to odd non-congruent numbers that do not belong to existing families ofnon-congruent numbers. For a given odd integer α, one can compute s(α)by using Monsky’s formula. If s(α) = 0, then infinitely many new non-congruent numbers can be generated by applying Theorem 5.5 to α. Tables5.5 and 5.6 list a few numerical examples that extend a non-congruent num-ber α that does not belong to a known family of non-congruent numbersbut satisfies s(α) = 0.735.5. Examples of Odd Non-congruent NumbersTable 5.5: Non-congruent numbers n = αsd+1sd+2 · · · st generated by CasesI.1.A and I.1.B of Theorem 5.5Theorem Extension Tail sd+1sd+2 · · · stSatisfied by α α Case I.1.A Case I.1.BLagrange with( qr)= −15 · 7 11 · 179 · 499 · 2179 ·2531 · 2699 · 21211 ·38459 ·43019 ·14869119·59·811·1459·1931·2371 · 2579 · 13331 ·14699 ·65579 ·164771Lagrange with(pq)=(pr)= −117 · 5 · 7 179 · 1019 · 2531 ·5779 · 10259 · 41771 ·64891 · 74699 ·220579 · 25489919 · 59 · 1291 · 3251 ·5011 · 23099 · 33851 ·41411 · 45779 · 77419Lagrange with(rp)=(rq)= −141 · 5 · 7 379 · 1171 · 1451 ·3259 · 14891 · 17011 ·54251 · 67979 ·280219 · 28049959 · 131 · 419 · 1931 ·5659 · 12011 · 12659 ·110459 · 189251 ·442139Lagrange with( qr)=(qp)= −141 · 29 · 31 59 · 107 · 1619 · 3203 ·9371 · 20771 · 33923 ·48523 · 210187 ·308051 · 92622783 · 139 · 1499 · 6067 ·7043 · 9931 · 13171 ·19843 ·32939 ·285451Theorem 5.1 19·11·163·419·97 · 313 · 6176011 · 11867 · 69931 ·83339 · 133387 ·236339811 · 8059 · 45979 ·64451 · 131779 ·454379 · 562091Theorem 5.1 347 · 83 · 11 · 3 ·499 · 1123 ·2803 ·673 ·2953140827 · 172507 ·191227 · 6700995051 · 43787 · 46691 ·147179 · 1174091Theorem 5.1 11 · 59 · 163 ·307 · 947 · 41 ·1361 · 20174651 · 15139 · 68611 ·119827 · 186019 ·3567316947 · 12547 · 43403 ·149027 · 119027 ·696827 · 783779Theorem 5.1 3 · 11 · 67 · 163 ·691·1483·3019·2179 · 16987 ·2137 · 427323203 · 121531 ·938491 · 152985127299 · 137363 ·1557443 · 1734827Theorem 5.1 3 · 11 · 19 · 43 ·59 · 5 · 13 · 29 ·37 · 27481124459 · 376819 ·467899 · 58957973259 · 159059 ·86291 · 394811 ·930179 · 954971Theorem 5.1 3 · 19 · 67 · 83 ·13 · 61 · 101 ·149 ·4177 ·9649138451 · 172507 ·290443 · 73185114771 · 66491 ·657947 · 680003None 13 · 29 · 37 · 23 ·31 · 714003 · 5867 · 41947 ·60779 · 135131 ·196387 · 296299 ·3298912731 · 3467 · 1483 ·16363 · 32083 ·76883 · 269851 ·274811 · 659611745.5. Examples of Odd Non-congruent NumbersTable 5.6: Non-congruent numbers n = αsd+1sd+2 · · · st generated by CasesI.1.C and I.1.D of Theorem 5.5Theorem Extension Tail sd+1sd+2 · · · stSatisfied by α α Case I.1.C Case I.1.DLagrange with( qr)= −15 · 7 3 · 83 · 467 · 2243 ·3923 · 3947 · 22067 ·35363 · 59723 ·111443 · 185363 ·58184343 · 67 · 107 · 2027 ·8563 · 8803 · 12923 ·16363·29803·78467Lagrange with(pq)=(pr)= −117 · 5 · 7 3 · 227 · 1907 ·3947 · 7643 · 22307 ·24443 · 44483 ·106907 · 151787107 · 163 · 547 ·1187 · 8803 · 14387 ·5987 · 48563 ·62563 · 142907 ·215723 · 899467Lagrange with( qr)=(qp)= −141 · 29 · 31 3 · 11 · 827 · 1667 ·6899 · 28283 ·47819 · 80603 ·179483 · 45392319 · 1163 · 1867 ·2467 · 2531 · 9091 ·32971 · 71387 ·93187 · 203659Theorem 5.1 19·11·163·419 · 97 ·313 · 617211 · 17027 · 22739 ·82387 · 85571 ·1146591483 · 15643 ·26339 · 60899 ·174443 · 191299 ·396091 · 235723None 13 ·29 ·37 ·23 · 31 · 71827 · 2683 · 7451 ·7963 ·39827 ·48563 ·255443 · 2759232699 · 3491 · 8123 ·27763 · 21491 ·121379 · 133843 ·156619We also provide some examples of non-congruent numbers that can begenerated by Case II.2 of Theorem 5.5. The first family we extend is de-scribed by the following theorem from [40].Theorem 5.8. Let m be a fixed positive integer and let c be any integersatisfying c ≥ m. Let Tm denote the set of positive integers with prime fac-torization pr1r2 · · · rc, where p is a prime of the form 8k+1 and r1, r2, . . . , rcare distinct primes of the form 8k + 3 such that(pri)={ −1 if i = m,1 if i 6= m,755.5. Examples of Odd Non-congruent Numbersand (rjri)= −1 if j < i.If α ∈ Tm, then α is non-congruent.The congruent number elliptic curves corresponding to the non-congruentnumbers produced by this theorem are shown to have 2-Selmer rank equal tozero [40]. Therefore, Theorem 5.5 can be applied to these numbers to con-struct infinitely many new non-congruent numbers, including those listed inTable 5.7.The non-congruent numbers generated by Case I.1 of Theorem 5.5 alsocorrespond to congruent number elliptic curves with 2-Selmer rank equal tozero, so any of the numerical examples in Tables 5.5 and 5.6 can be extendedby using Case II.2 of Theorem 5.5. We list several numerical examples inTable 5.7.765.5. Examples of Odd Non-congruent NumbersTable 5.7: Non-congruent numbers n = αsd+1sd+2 · · · st generated by CaseII.2 of Theorem 5.5TheoremSatisfiedby αα Case II.2 ExtensionTail sd+1sd+2 · · · stPrime(s)µ ∈ TTheorem5.8 withm = 541 · 43 · 59 ·107 · 251 ·547 · 13073541 · 15061 · 31469 ·52301 · 595717 · 70395741Theorem5.8 withm = 541 · 43 · 59 ·107 · 251 ·547 · 13075381 · 5717 · 31357 ·125101 · 214189 · 217981 ·414157 · 2844701107Theorem5.8 withm = 541 · 43 · 59 ·107 · 251 ·547 · 13079157 · 12517 · 15773 ·122069 · 277741 · 444557 ·544877 · 283606941, 547, 1307Theorem5.8 withm = 541 · 43 · 59 ·107 · 251 ·547 · 13078293 · 9437 · 13109 · 84589 ·119173 · 251501 · 687461 ·236617341, 43, 59,107, 251, 547,1307Theorem5.5 CaseI.1.A17 · 5 · 7 ·179 · 1019 ·2531 · 5779541 · 1229 · 2069 · 67349 ·405749 · 671269 · 786941 ·83842917Theorem5.5 CaseI.1.A17 · 5 · 7 ·179 · 1019 ·2531 · 57795557 · 17477 · 58733 ·114197 · 128813 · 136237 ·337973 · 5291575, 7, 179,2531, 5779Theorem5.5 CaseI.1.A17 · 5 · 7 ·179 · 1019 ·2531 · 57793533 · 18133 · 51133 ·64333 · 77797 · 37227717, 5, 7, 179,1019, 2531,5779None 13 · 29 · 37 ·23 · 31 · 71877·4877·8933·9397·15173·125197 · 414629 · 49513313None 13 · 29 · 37 ·23 · 31 · 71173 · 4813 · 6269 · 58237 ·60733 · 94709 · 140773 ·35305371None 13 · 29 · 37 ·23 · 31 · 714013 · 6917 · 12373 · 14869 ·23981 · 157141 · 414413 ·42970129, 37, 23None 13 · 29 · 37 ·23 · 31 · 71709 · 13037 · 14029 · 21221 ·68141 · 79669 · 80221 ·347813 · 1186621 · 277361313, 37, 23, 31,71775.5. Examples of Odd Non-congruent NumbersFinally, we provide some examples of non-congruent numbers generatedby Cases III.2 and III.3 of Theorem 5.5. The non-congruent numbers α,described by Theorems 5.2, 5.3, and 5.4, have s(α) = 0. This means thatwe can apply Cases III.2 and III.3 of Theorem 5.5 to these numbers. Inaddition, since the non-congruent numbers listed in Tables 5.5, 5.6, and 5.7correspond to elliptic curves with 2-Selmer rank equal to zero, it is possibleto append a tail of primes of the form described in either Case III.2 orCase III.3 of Theorem 5.5 onto any of the numbers in Tables 5.5, 5.6, and5.7 to produce new non-congruent numbers. Examples of non-congruentnumbers that result from applying Case III.2 of our extension theorem tonon-congruent numbers given by Theorems 5.2, 5.3, and 5.4, and to a couplenumbers from Tables 5.5 and 5.7 are listed in Table 5.8.Table 5.8: Non-congruent numbers n = αsd+1sd+2 · · · st generated by CaseIII.2 of Theorem 5.5TheoremSatisfiedby αα Case III.2 ExtensionTail sd+1sd+2 · · · stPrime(s)µ ∈ TTheorem5.219 · 11 · 163 · 419 1361 ·1889 ·2833 ·14401 ·44497 · 79537 · 98689 ·217169 · 250433 · 96904119, 11, 419Theorem5.311 · 59 · 163 ·307 · 947809 · 1009 · 5881 · 8681 ·58153 · 124673 · 361961 ·43540111, 59, 163,307, 947Theorem5.43 · 19 · 67 · 83 ·13 · 61 · 101 · 14916481 · 46049 · 70937 ·78233 · 521777 · 13876493, 19, 67, 83,61, 101, 149None 13 · 29 · 37 · 23 ·31 · 718017 · 24337 · 31121 ·49481 · 81689 · 214033 ·532801 · 126539371Theorem5.5 CaseI.1.A41 · 29 · 31 · 59 ·107 · 16191801 · 2393 · 6841 · 9001 ·23057 · 75041 · 289841 ·122529741, 29, 31,59, 1619Theorem5.5 CaseII.217 · 5 · 7 · 179 ·1019·2531·5779·3533 · 1813356401 · 88321 · 189961 ·191969 ·551321 ·166964917, 2531,18133785.5. Examples of Odd Non-congruent NumbersBecause Case III.3 of Theorem 5.5 is a generalization of Theorem 5.1,the non-congruent numbers, n, listed in Table 5.1 that arise from applyingTheorem 5.1 to the non-congruent numbers, α, produced by Theorems 5.2,5.3, and 5.4 can also be generated by Case III.3 of Theorem 5.5. Therefore,the examples that we list in Table 5.9 are constructed by applying Case III.3of Theorem 5.5 to a few of the numbers in Tables 5.5, 5.6, and 5.7. Notethat the numbers in Table 5.9 cannot be produced by Theorem 5.1.Table 5.9: Non-congruent numbers n = αsd+1sd+2 · · · st generated by ap-plying Case III.3 of Theorem 5.5 to numbers in Tables 5.5, 5.6, and 5.7α Case III.3 ExtensionTail sd+1sd+2 · · · stExtension Tail LegendreSymbols that Equal −113 · 29 · 37 ·23 · 31 · 71593 · 521 · 857 · 6977 ·38561 · 36433 · 75193 ·56377 · 54833 · 331553(59313),(52123),(85737),(697729),(38561521),(3643313),(7519338561),(56377521),(5483371),(33155375193)17 · 5 · 7 · 179 ·1019 · 2531 ·57795881 · 7561 · 7841 ·7481 · 12601 · 59921 ·28729 · 47681 · 324361(58812531),(75611019),(78411019),(74817),(12601179),(599215881),(287297841),(4768128729),(3243615881)5 · 7 · 43 · 67 ·107 · 20274649 · 5009 · 2801 ·18289 · 19001 · 20441 ·56809 · 62969 · 221729(464943),(500967),(28015009),(182897),(190017),(2044118289),(568095009),(629692027),(22172919001)17 · 5 · 7 · 179 ·1019 · 2531 ·5779·541·12292129 · 7481 · 5849 ·44641 · 59921 · 39089 ·70289 · 710873(2129179),(74817),(58497481),(446411229),(599211229),(3908917),(7028917),(7108735)Notice that the different cases of Theorem 5.5 can be combined to gener-ate new non-congruent numbers. This is illustrated by the numerical exam-ples listed in the final row of Table 5.8 and Table 5.9 that apply Case I.1.A,Case II.2, and either Case III.2 or Case III.3 of Theorem 5.5 to producenon-congruent numbers.79Chapter 6The Generation of Familiesof Even Non-congruentNumbers From KnownNon-congruent Numbers6.1 OverviewIn this chapter, we present a criterion for generating new families ofeven non-congruent numbers. Our approach is similar to that describedin Chapter 5 for producing families of odd non-congruent numbers. Weconstruct infinitely many new even non-congruent numbers by appendingprimes of a certain form onto existing even non-congruent numbers, whosecorresponding congruent number elliptic curves have 2-Selmer rank equal tozero. This allows us to generate even non-congruent numbers with arbitrar-ily many prime factors in each of the four odd congruence classes moduloeight. This characteristic distinguishes our result from other theorems oneven non-congruent numbers that place restrictions on the prime divisorsof the non-congruent numbers, only allowing an unlimited number of primefactors in at most two odd congruence classes modulo eight.This chapter is based on results that appear in [42] and [43].6.2 An Extension Technique for Generating NewFamilies of Even Non-congruent NumbersWe begin by stating our main theorem. We let pi, qj , rk, and sl withi, j, k, l ∈ N+ be distinct odd primes, and let a, b, c, d ∈ N with(a+ b+ c+ d) > 0. We define the setP ={∅ if a = 0,{p1, p2, . . . , pa} if a > 0.806.2. Extension Technique for Generating Even Non-congruent NumbersThe sets Q,R, and S with |Q| = b, |R| = c, and |S| = d are defined analo-gously. In addition, we letW = P ∪Q ∪R ∪ S.Theorem 6.1. Defineβ = 2∏pi∈Ppi∏qj∈Qqj∏rk∈Rrk∏sl∈Ssl,and suppose that the elliptic curvey2 = x(x2 − β2)has 2-Selmer rank of zero. Let t > d with t ∈ N+, and define the evensquare-free positive integer n byn = βsd+1sd+2 · · · st,where the odd prime factors of n satisfy the congruence conditions describedin one of the four cases in Table 6.1.Table 6.1: Congruence conditions for the odd prime factors of the evennumber nCondition pi(mod 8)∀ pi ∈ Pqj(mod 8)∀ qj ∈ Qrk(mod 8)∀ rk ∈ Rsγ(mod 8)∀ γ ∈ [1, t]I 5 3 7 1II 1 5 7 3III 1 7 3 5IV 1 5 3 7In addition, assume that the primes appended onto β satisfy one of thefollowing Legendre symbol conditions.Condition 1:For all h, σ ∈ [d+ 1, t] with h 6= σ(sσsh)= 1,816.2. Extension Technique for Generating Even Non-congruent Numbersand define T ⊆W withT ={µ∣∣∣∣ (shµ)= −1 ∀ h ∈ [d+ 1, t]}and |T | ≡ 1 (mod 2). For all primes ε ∈W\T ,(shε)= 1 ∀ h ∈ [d+ 1, t].Condition 2:For each h ∈ [d+ 1, t], the set Lh defined byLh ={(shpi),(shqj),(shrk),(shsg)with pi ∈ P, qj ∈ Q, rk ∈ R, g ∈ [1, h)}has exactly one Legendre symbol equal to −1.Condition 3:For all h ∈ [d + 1, t], pi ∈ P, qj ∈ Q, rk ∈ R, and g ∈ [1, h), one of thefollowing four sets of Legendre symbol conditions hold.A)(shpi)=(shqj)=(shrk)=(shsg)= 1.B)(shpi)=(shqj)=(shrk)=(shsg)= −1.C)(pish)=(qjsh)=(rksh)=(sgsh)= 1.D)(pish)=(qjsh)=(rksh)=(sgsh)= −1.Condition 4:Let h ∈ [d+ 1, t], ε ∈ N+ and define H ⊆W , where|H| ={Odd if (t− d) ≡ 0 (mod 4),1 if (t− d) ≡ 2 (mod 4),and µ ∈ H if (µsh)={1 for h = d+ (2ε− 1),−1 for h = d+ 2ε.826.2. Extension Technique for Generating Even Non-congruent NumbersSet (sσsh)=(λsh)= 1for all h ∈ [d+ 1, t], σ ∈ [d+ 1, h), and all primes λ ∈W\H.Then, in each of the cases described in Table 6.2, n is a non-congruentnumber with s(n) = 0.Table 6.2: Conditions on the primes in the extension tail of the even numbernCase CongruenceConditionLegendre SymbolConditionParity of (t− d)I.1 I 1 EvenI.2 I 2 No restrictionII.3 II 3.A, 3.B, 3.C, 3.D EvenIII.3.A III 3.A No restrictionIV.4 IV 4 EvenOur technique for proving this theorem utilizes theory from the field oflinear algebra along with Monsky’s formula for the 2-Selmer rank of En. Wework over F2 and make use of Lemma 5.6 to simplify the calculations in ourproof. As in Section 5.4, we setδ = (a+ b+ c) and ω = (2δ + t+ d).In addition, we let Aβ denote the square A matrix of order (δ + d) for β,and Dβ2 and Dβ−1 represent the diagonal matrices corresponding to β; theentries in these matrices are defined in Theorem 3.15. We now work throughthe proofs of the cases described in Theorem 6.1 separately, beginning withCase I.1.Case I.1 Proof. We use Equation (3.17) to form the (2δ + 2d) × (2δ + 2d)Monsky matrix,Mβe = Dβ2 Aβ + Dβ2AβT + Dβ2 Dβ−1,for the even number β. By assumption s(β) = 0, so Equation (3.15) impliesthat Mβe has full rank.836.2. Extension Technique for Generating Even Non-congruent NumbersWe now consider n = βsd+1sd+2 · · · st and construct its corresponding AmatrixAn = Aβ BBT It−d,where (t− d) is even. Note that B is a (δ + d)× (t− d) matrix containingrows with entries all equal to zero, except for an odd number of rows withelements all equal to one. Because (t − d) is even, the entries along thediagonal in the upper left block of An are not affected by the elements inB, so they remain the same as in Aβ. The matrix BT has an odd numberof columns with entries all equal to one, and so there is an identity block inthe lower right corner of An. Thus, the Monsky matrix corresponding to ncan be written asMe =Dβ2 0 Aβ + Dβ2 B0 0t−d BT It−dAβT + Dβ2 B Dβ−1 0BT It−d 0 0t−d.By applying Equation (5.10) from Lemma 5.6, Me can be transformed intoM′e =Dβ2 Aβ + Dβ2 B 0AβT + Dβ2 Dβ−1 0 B0 BT It−d 0t−dBT 0 0t−d It−d,withdet(Me) = det(M′e).Notice that the matrix M′e is similar to M∗o given by Equation (5.22) inthe proof of Case II.2 of Theorem 5.5, with the only difference being theelements in the (2δ + 2d) × (2δ + 2d) blocks located in their upper leftcorners. Therefore, we can follow the process outlined in the proof of CaseII.2 of Theorem 5.5 to reduce M′e to a form where Proposition 2.27 can beapplied. This allows us to deduce thatdet(M′e) = det Dβ2 Aβ + Dβ2AβT + Dβ2 Dβ−1 = det(Mβe).846.2. Extension Technique for Generating Even Non-congruent NumbersSince Mβe has full rank, its determinant is nonzero, so the above equationimplies that M′e also has full rank. Thus, by Equation (3.15), we concludethat s(n) = 0 and n is non-congruent.We now verify that Case I.2 also yields non-congruent numbers. Notethat this result is a generalization of Theorem 1.1 Case 1 in [42].Case I.2 Proof. We use induction and follow a process analogous to thatused in the proof of Case III.3 of Theorem 5.5. We begin by considering theintegern′ = βsd+1formed by appending the prime sd+1, satisfying the specified pattern ofLegendre symbols, onto the non-congruent number β. The (2δ + 2d+ 2)×(2δ + 2d+ 2) Monsky matrix for n′ isMn′ =Dβ2 0 K v0 0 vT 1KT v Dβ−1 0vT 1 0 0,where v is a column vector containing (δ+d−1) elements equal to zero anda single element equal to one, andK = Aβ + Dβ2 + vvT .The matrix Mn′ can be transformed intoM∗n′ =Dβ2 Aβ + Dβ2 0 0AβT + Dβ2 Dβ−1 0 00 0 1 00 0 0 1by carrying out similar row and column operations to those used in the proofof Case III.3 of Theorem 5.5 and applying Equation (5.9) from Lemma 5.6.Therefore, by Proposition 2.27, we havedet(Mn′) = det(M∗n′) = det(Mβe).856.2. Extension Technique for Generating Even Non-congruent NumbersBy assumption s(β) = 0, so as in the proof for Case I.1 of Theorem 6.1,we can conclude that s(n′) = 0 and n′ is a non-congruent number. Theremainder of the proof can be completed by following the process outlinedin the proof of Case III.3 in Theorem 5.5.We now prove that Case II.3 generates non-congruent numbers. Notethat Case II.3.A generalizes Theorem 1.2 in [42]. This is because Case II.3.Aallows β to have arbitrarily many prime factors belonging to each odd con-gruence class modulo eight, whereas Theorem 1.2 in [42] imposes restrictionson the number of primes in certain odd congruence classes modulo eight.Case II.3 Proof. The diagonal matrices for n areDn2 =[Dβ2 00 It−d]and Dn−1 =[Dβ−1 00 It−d]. (6.1)We consider the four subcases described in the theorem statement sep-arately, beginning with Case II.3.A.Case II.3.A:By using Equation (3.14), we can write the A matrix corresponding to n asAn =Aβ 0Tad+1 0 · · · · · · 01 ad+2. . ........ . .. . .. . ........ . . at−1 01 · · · · · · 1 at=[Aβ 0T A∗], (6.2)where T is given by Equation (5.14). The diagonal entries in A∗ are depen-dent upon the parity of the quantity (c + d) and, since (t − d) is even, itfollows that(ad+1, ad+2, . . . , at−1, at) ={(0, 1, . . . , 0, 1) if (c+ d) is even,(1, 0, . . . , 1, 0) if (c+ d) is odd.866.2. Extension Technique for Generating Even Non-congruent NumbersThe Monsky matrix for n can be written asMe =Dβ2 0 Aβ + Dβ2 00 It−d T A∗ + It−dAβT + Dβ2 TT Dβ−1 00 A∗T + It−d 0 It−dby combining Equations (3.17), (6.1), and (6.2).In the case where (c+ d) is even, we apply Equation (5.8) from Lemma5.6 to Me to obtainMNe =Dβ2 Aβ + Dβ2 0 0AβT + Dβ2 Dβ−1 0 TT0 0 It−d A∗T + It−d0 T A∗ + It−d It−d,withdet(Me) = det(MNe ).We modify this matrix by carrying out the set of (t−d−1) row replacementslisted in the proof of Case I.1.B of Theorem 5.5. These operations transformthe A∗T + It−d block into the (t − d) zero matrix, while leaving the otherelements in MNe unchanged. We then reduce the T block to the zero matrixby adding all of the final (t− d) columns in MNe to each of columns (δ+ d+a + b + 1) through (2δ + 2d). The entries in the TT block do not modifythe elements in the Dβ−1 block, because we are adding an even number ofcolumns to each of the aforementioned (c+ d) columns in MNe . Altogether,these operations change MNe intoMNNe =Dβ2 Aβ + Dβ2 0 0AβT + Dβ2 Dβ−1 0 TT0 0 It−d 0t−d0 0 A∗ + It−d It−d.We apply Proposition 2.27 to MNNe and, because the (2t − 2d) × (2t − 2d)block in the lower right corner of MNNe is a lower triangular matrix with876.2. Extension Technique for Generating Even Non-congruent Numbersdeterminant equal to one, we are able to conclude thatdet(Me) = det(MNNe ) = det Dβ2 Aβ + Dβ2AβT + Dβ2 Dβ−1 = det(Mβe).Thus, it follows that s(n) = 0 and n is non-congruent when (c+ d) is even.In the case where (c+ d) is odd, we transform Me intoMe =Dβ2 Aβ + Dβ2 0 0AβT + Dβ2 Dβ−1 TT 00 T It−d A∗ + It−d0 0 A∗T + It−d It−d (6.3)by applying Equation (5.7) from Lemma 5.6. We need to reduce Me so thatProperty 2.27 can be used to calculate its determinant. We carry out the(t− d− 1) row operations stated in the proof of case 2 of Theorem 5.5 CaseI.1.A to change the A∗ + It−d block in Me into the zero matrix. We alsotransform the TT block into the zero matrix by adding rows (2δ + 2d + 1)through (2δ+ d+ t) to each of rows (δ+ d+ a+ b+ 1) through (2δ+ 2d) inMe. Upon completing these row operations, Me has the formMe =Dβ2 Aβ + Dβ2 0 0AβT + Dβ2 Dβ−1 0 00 T It−d 0t−d0 0 A∗T + It−d It−d.The remainder of the proof can be completed analogously to the case where(c + d) is even, allowing us to reach the desired conclusion that s(n) = 0and n is non-congruent when (c+ d) is odd.Case II.3.B:We recall Equation (3.14), and use it, along with the specified pattern ofLegendre symbols corresponding to the primes appended onto β, to form886.2. Extension Technique for Generating Even Non-congruent Numbersthe A matrix for n. This matrix can be written asAn =1 · · · · · · · · · 1Aβ......1 · · · · · · · · · 1ad+1 1 · · · · · · 10 ad+2. . ....U.... . .. . .. . ........ . . at−1 10 · · · · · · 0 at= Aβ1 · · · 1......1 · · · 1U A∗,where U is given by Equation (5.15), and(ad+1, ad+2, . . . , at−1, at) ={(1, 0, . . . , 1, 0) if (a+ b) is even,(0, 1, . . . , 0, 1) if (a+ b) is odd.Note that there are an even number of ones in each row of the upper rightblock of An. Because we are working over F2, the elements along the diago-nal in the upper left block of An remain the same as those in Aβ. We formthe Monsky matrix for n,Me =1 · · · 1Dβ2 0 Aβ + Dβ2......1 · · · 10 It−d U A∗ + It−dAβT + Dβ2 UT Dβ−1 01 · · · 1...... A∗T + It−d 0 It−d1 · · · 1,by using An and the diagonal matrices given by Equation (6.1).896.2. Extension Technique for Generating Even Non-congruent NumbersWe consider the case where (a+ b) is even and transform Me intoMNe =1 · · · 1Dβ2 Aβ + Dβ2...... 01 · · · 1AβT + Dβ2 Dβ−1 0 UT1 · · · 1...... 0 It−d A∗T + It−d1 · · · 10 U A∗ + It−d It−dby applying Equation (5.8) from Lemma 5.6. We carry out the (t − d − 1)row replacement operations listed in the proof of case 2 of Theorem 5.5 CaseI.1.A to reduce the A∗T + It−d block in MNe to the (t − d) zero block. Wethen add all of the final (t − d) rows in MNe to each of rows (δ + d + 1)through (δ + d + a + b) to convert all of the entries in the UT block tozeros. Finally, we reduce the block consisting entirely of ones in the upperright quadrant of MNe to the zero matrix by adding all of rows (2δ+ 2d+ 1)through (2δ + d+ t) to each of the first (δ + d) rows in MNe . This collectionof row operations transforms MNe intoMNNe =Dβ2 Aβ + Dβ2 0 0AβT + Dβ2 Dβ−1 0 01 · · · 1...... 0 It−d 0t−d1 · · · 10 U A∗ + It−d It−dand, since we are working over F2,det(Me) = det(MNe ) = det(MNNe ).Applying Proposition 2.27 to MNNe allows us to determine thatdet(MNNe ) = det Dβ2 Aβ + Dβ2AβT + Dβ2 Dβ−1 = det(Mβe).906.2. Extension Technique for Generating Even Non-congruent NumbersThe above determinant cannot be equal to zero because of our originalassumption that s(β) = 0. Therefore, Equation (3.15) implies that s(n) = 0and n is non-congruent when (a+ b) is even.In the case where (a+ b) is odd, we use Equation (5.7) from Lemma 5.6to transform Me intoMe =1 · · · 1Dβ2 Aβ + Dβ2 0......1 · · · 1AβT + Dβ2 Dβ−1 UT 00 U It−d A∗ + It−d1 · · · 1...... 0 A∗T + It−d It−d1 · · · 1,withdet(Me) = det(Me).We carry out the (t− d− 1) row operations listed in the proof of Theorem5.5 Case I.1.B to convert the A∗ + It−d block in Me into the zero matrix.We then reduce all of the nonzero entries in the upper right quadrant of Meto zero by performing analogous row operations to the ones that we used totransform MNe into MNNe in the case where (a+ b) is even. Completing theremaining portion of the proof can be done by simply following the seriesof steps outlined in case with (a+ b) even. Thus, regardless of the parity ofthe quantity (a+ b), s(n) = 0 and n is non-congruent.Case II.3.C:We use the pattern of Legendre symbols for the prime factors of n andEquations (3.14), (3.17), and (6.1) to construct the Monsky matrix for n.916.2. Extension Technique for Generating Even Non-congruent NumbersThis matrix has the formMe =Dβ2 0 Aβ + Dβ2 TT0 It−d 00 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 0 10 · · · · · · 0 1AβT + Dβ2 0 Dβ−1 0T0 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 0 01 · · · · · · 1 10 It−d,where T is given by Equation (5.14). Applying Equation (5.8) from Lemma5.6 to Me transforms it intoMNe =Dβ2 Aβ + Dβ2 TT 0AβT + Dβ2 Dβ−1 0 0T 0 It−d0 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 0 01 · · · · · · 1 10 00 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 0 10 · · · · · · 0 1It−d.By inspection, it is clear that the structure of this matrix is similar to theone in Equation (6.3). Therefore, an analogous process to that described in926.2. Extension Technique for Generating Even Non-congruent Numbersthe proof of Theorem 6.1 Case II.3.A with odd (c+ d) can be implementedto deduce that s(n) = 0 and n is a non-congruent number.Case II.3.D:We use the Legendre symbol values and Equation (3.14) to constructAn =Aβ UT1 · · · 1..................1 · · · 1ad+1 0 · · · · · · 01 ad+2. . ........ . .. . .. . ........ . . at−1 01 · · · · · · 1 at=Aβ UT1 · · · 1......1 · · · 1A∗,where UT is the transpose of the matrix described in Equation (5.15). Noticethat the elements along the diagonal in A∗ depend upon the parity of thequantity (δ + d), where(ad+1, ad+2, . . . , at−1, at) ={(1, 0, . . . , 1, 0) if (δ + d) is odd,(0, 1, . . . , 0, 1) if (δ + d) is even.It follows that the Monsky matrix for n has the formMe =Dβ2 0 Aβ + Dβ2 UT1 · · · 10 It−d...... A∗ + It−d1 · · · 11 · · · 1AβT + Dβ2...... Dβ−1 01 · · · 1U A∗T + It−d 0 It−d.When (δ + d) is odd, Equation (5.7) from Lemma 5.6 can be applied torearrange the blocks in Me. The resulting matrix is similar in structure toMNe in the proof of Theorem 6.1 Case II.3.B with (a + b) even. Therefore,we can complete the proof of the case where (δ + d) is odd by following theprocedural details described in the proof of Theorem 6.1 Case II.3.B with(a+ b) even.936.2. Extension Technique for Generating Even Non-congruent NumbersSimilarly, when (δ+d) is even, we transform Me by using Equation (5.8)from Lemma 5.6. The remainder of the proof can be completed by followingthe process described in the proof of Theorem 6.1 Case II.3.B with (a + b)odd.Thus, s(n) = 0 and n is a non-congruent number irrespective of theparity of (δ + d).Next, we provide the proof of Case III.3.A.Case III.3.A Proof. We form the (2δ + 2t) × (2δ + 2t) Monsky matrix forn = βsd+1sd+2 · · · st. This matrix is described by Equation (3.17) and canbe written in the following general formMe =Dβ2 0 Aβ + Dβ2 00 It−d 0 It−dAβT + Dβ2 0 Dβ−1 00 It−d 0 0t−d.In addition, the (δ + t)× (δ + t) diagonal matrices D2 and D−1 that corre-spond to n are given byD2 = Dβ2 00 It−d =0a+b 0 00 Ic+d 00 0 It−dandD−1 = Dβ−1 00 0t−d =0a 0 0 00 Ib+c 0 00 0 0d 00 0 0 0t−d.We apply Equation (5.10) from Lemma 5.6 to Me to transform it intoM′e =Dβ2 Aβ + Dβ2 0 0AβT + Dβ2 Dβ−1 0 00 0 It−d It−d0 0 0t−d It−d,946.2. Extension Technique for Generating Even Non-congruent Numberswithdet(Me) = det(M′e).Notice that the lower right corner square block of order (2t−2d) is an uppertriangular matrix with determinant equal to one. Therefore, by applyingProposition 2.27, we can determine thatdet(Me) = det[Dβ2 Aβ + Dβ2AβT + Dβ2 Dβ−1]. (6.4)The block matrix in Equation (6.4) is recognizable as the Monsky matrix cor-responding to β. Recall that by assumption s(β) = 0. Therefore, Equation(3.15) implies that the Monsky matrix for β has full rank, so by Equation(6.4), we conclude thatdet(Me) 6= 0.Thus, s(n) = 0 and n is a non-congruent number.Finally, we complete the proof of Theorem 6.1 by verifying that CaseIV.4 produces families of non-congruent numbers.Case IV.4 Proof. First, we construct the diagonal matrices for n. Becausethe primes appended onto β are of the form 8k + 7,Dn2 =[Dβ2 00 0t−d]and Dn−1 =[Dβ−1 00 It−d].We defineu =[1 0 · · · 1 0 ] (6.5)andv =[0 1 · · · 0 1 ] (6.6)to be a pair of row vectors containing (t−d) elements that alternate between956.2. Extension Technique for Generating Even Non-congruent Numbersones and zeros. The Monsky matrix for n can be written asMe =Dβ2 0 E F0 0t−d G1 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 1 10 · · · · · · 0 1ET GT Dβ−1 0FT1 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 10 It−d, (6.7)where the structure of the blocks E, F, and G vary depending on whether(t− d) ≡ 2 (mod 4) or (t− d) ≡ 0 (mod 4).If (t − d) ≡ 2 (mod 4), G is a (t − d) × (δ + d) matrix with all of itselements equal to zero, except for a single column of the form vT , where v isgiven by Equation (6.6). The structure of G is important, because it ensuresthat the block immediately to the right of G has nonzero determinant; if Ghad simply been the zero matrix, then the upper triangular block situatedbeside G would have exhibited an alternating pattern of ones and zeros alongits diagonal. The significance of this upper triangular block with nonzerodeterminant becomes apparent later in the proof when we apply Proposition2.27.When (t − d) ≡ 2 (mod 4), the structure of the matrix F is dependentupon the prime that belongs to the set H. If pi ∈ H for any i ∈ [1, a] orqj ∈ H for any j ∈ [1, b], then F has (c+ d) rows of ones, (a+ b− 1) rows ofzeros, and a single row of the form v. Similarly, if rk ∈ H for any k ∈ [1, c]or sl ∈ H for any l ∈ [1, d], then F has (a+ b) rows of zeros, (c+ d− 1) rowsof ones, and a single row of the form u, where u is given by Equation (6.5).The rows in F that are composed either entirely of zeros or entirely of onesdo not affect the elements along the diagonal in the block E, because we areworking over F2 and (t−d) is even. The remaining row in F of the form u orv contains an odd number of ones because (t− d) ≡ 2 (mod 4). Therefore,966.2. Extension Technique for Generating Even Non-congruent Numbersthe block E is nearly identical to Aβ +Dβ2, with the only difference betweenthese two matrices being a single element along their main diagonals.We now use Equation (5.9) from Lemma 5.6 to transform Me intoM∗e =Dβ2 E 0 FET Dβ−1 GT 0FT 01 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 1It−d0 G 0t−d1 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 1 10 · · · · · · 0 1. (6.8)We convert the G block into the zero matrix by adding each of the final(t − d) columns in M∗e to the column in M∗e whose final (t − d) elementsalternate between zeros and ones. This single column replacement operationtransforms the block E into Aβ + Dβ2 and the matrix M∗e intoM∗∗e =Dβ2 Aβ + Dβ2 0 FET Dβ−1 GT 0FT K1 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 1It−d0 0 0t−d1 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 1 10 · · · · · · 0 1,976.2. Extension Technique for Generating Even Non-congruent Numberswhere K has a single column of ones and all of its remaining entries equalto zero.We reduce the GT block to the zero matrix by carrying out a single rowreplacement operation in M∗∗e . Specifically, we add each of rows (2δ+2d+1)through (2δ + d + t) to the row in M∗∗e with nonzero entries in GT . Notethat this row replacement operation also transforms ET into AβT + Dβ2, soM∗∗e becomesM∗∗∗e =Dβ2 Aβ + Dβ2 0 FAβT + Dβ2 Dβ−1 0 KTFT K1 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 1It−d0 0 0t−d1 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 1 10 · · · · · · 0 1, (6.9)withdet(Me) = det(M∗∗∗e ). (6.10)Applying Proposition 2.27 to M∗∗∗e twice allows us to deduce thatdet(M∗∗∗e ) = detDβ2 Aβ + Dβ2 0AβT + Dβ2 Dβ−1 0FT K1 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 1= det[Dβ2 Aβ + Dβ2AβT + Dβ2 Dβ−1]. (6.11)986.2. Extension Technique for Generating Even Non-congruent NumbersThe square matrix of order (2δ+2d) immediately above is recognizable as theMonsky matrix corresponding to β. By assumption s(β) = 0, so Equation(3.15) implies that the Monsky matrix for β has nonzero determinant. Thisfact when coupled with Equations (6.10) and (6.11) enables us to deducethatdet(Me) 6= 0.Therefore, when (t − d) ≡ 2 (mod 4), s(n) = 0 and n is a non-congruentnumber.If (t−d) ≡ 0 (mod 4), |H| is odd, so the G matrix in Equation (6.7) hasan odd number of columns of the form vT , where v is given by Equation(6.6), and all of the remaining entries in G are equal to zero. Since thereare an odd number of columns of the form vT in G, the block immediatelyto the right of G in Me is an upper triangular matrix with ones along itsmain diagonal.Suppose that the number of primes pi with i ∈ [1, a] and qj with j ∈ [1, b]belonging to the set H is η, and the number of primes rk with k ∈ [1, c] andsl with l ∈ [1, d] in H is κ. Then η+ κ = |H|, and F has (a+ b− η) rows ofzeros, η rows of the form v, (c+ d−κ) rows of ones, and κ rows of the formu, where u is given by Equation (6.5). Since u and v are row vectors oflength (t−d) and (t−d) ≡ 0 (mod 4), there are an even number of elementsequal to one in each of these two vectors. Therefore, the entries in the rowsof F do not affect the elements along the diagonal in the block E andE = Aβ + Dβ2.We follow an analogous process to the one described in the case where(t−d) ≡ 2 (mod 4) to verify that n is a non-congruent number when (t−d) ≡0 (mod 4). First, we apply Equation (5.9) from Lemma 5.6 to transform Meinto the matrix M∗e, given by Equation (6.8). To reduce the G block in M∗eto a zero block, we add all of the final (t− d) columns in M∗e to each of the|H| columns in M∗e whose final (t − d) entries alternate between zeros andones. This process does not alter the Aβ + Dβ2 block in M∗e because we areadding all of the final (t− d) columns of M∗e to another column in M∗e, andeach row in F has either no ones or an even number of ones. The column996.3. Examples of Even Non-congruent Numbersoperations transform M∗e intoM∗∗e =Dβ2 Aβ + Dβ2 0 FAβT + Dβ2 Dβ−1 GT 0FT K1 0 · · · · · · 01 1. . ........ . .. . .. . ........ . . 1 01 · · · · · · 1 1It−d0 0 0t−d1 1 · · · · · · 10 1. . ........ . .. . .. . ........ . . 1 10 · · · · · · 0 1,where K is a matrix with |H| columns of ones and (δ+ d− |H|) columns ofzeros.Next, we complete a series of |H| row replacement operations to convertthe GT block to the zero matrix. This involves adding all of rows (2δ+2d+1)through (2δ + d + t) in M∗∗e to each of the |H| rows in GT of the form v.These row operations do not affect the Dβ−1 or AβT + Dβ2 blocks in M∗∗e .Therefore, M∗∗e is transformed into the matrix M∗∗∗e , given by Equation(6.9), where the K block has |H| columns composed entirely of ones and allremaining columns composed entirely of zeros.The rest of the proof of the case where (t − d) ≡ 0 (mod 4) can becompleted by following the steps outlined in the proof of the case with(t − d) ≡ 2 (mod 4). Thus, when (t − d) ≡ 0 (mod 4), we conclude thats(n) = 0 and n is non-congruent.6.3 Examples of Even Non-congruent NumbersGenerated from Known Non-congruentNumbersIn this section, we provide examples of non-congruent numbers that aregenerated by Theorem 6.1. These numbers are specified by values of Leg-1006.3. Examples of Even Non-congruent Numbersendre symbols, and hence Dirichlet’s theorem on primes in arithmetic pro-gressions (Theorem 2.20) guarantees that our theorem produces infinitelymany non-congruent numbers. Of significance is the fact that any evennon-congruent number β with s(β) = 0 can be extended to produce in-finitely many non-congruent numbers by using Theorem 6.1. Dependingon the value chosen for β, the numbers generated by applying Theorem6.1 may belong to existing families of non-congruent numbers. However,the numerical examples presented in Tables 6.4, 6.5, 6.6, 6.7, and 6.8 arefrom new families of non-congruent numbers, because their prime factor-izations differ from those of known families of even non-congruent numbers[2, 6, 11, 12, 15–17, 27, 37, 49].It is worthwhile to mention that the non-congruent numbers listed in thetables in this section can be generated with the aid of the Maple code givenin Appendix A.We begin by considering some existing families of non-congruent num-bers, so that we can apply Theorem 6.1 to construct new non-congruentnumbers.Lagrange describes several families of even non-congruent numbers witha maximum of three distinct odd prime factors [27]. We summarize some ofhis results in Table 6.3.1016.3. Examples of Even Non-congruent NumbersTable 6.3: Lagrange’s non-congruent numbers of the form β = 2pq or 2pqr,where p, q, and r are distinct odd primesp (mod 8) q (mod 8) r (mod 8) Legendre SymbolConditions Imposed onthe Odd Prime Factorsof β1) 3 3 – None2) 1 5 –(pq)= −13) 7 3 –(pq)= 14) 1 3 3(pq)= − (pr )5) 1 5 5(pq)= − (pr )6) 7 3 5(pq)=(pr)7) 7 7 5(pq)=(pr)= − ( qr)8) 1 7 3(pq)=(pr)= −1or(rp)=(rq)= −1In Section 9 of [17], Goto asserts that numbers β of the form describedby Table 6.3 correspond to congruent number elliptic curves with 2-Selmerrank equal to zero. Therefore, Theorem 6.1 can be used to extend thesenumbers to produce new families of even non-congruent numbers containingarbitrarily many distinct prime factors. In addition, since the even integersn described by Theorem 4.1 were proven to have s(n) = 0, they too can beextended by Theorem 6.1. In fact, any even non-congruent number n withs(n) = 0 can be extended to produce an infinite collection of non-congruentnumbers by using Theorem 6.1.We begin by applying Case I.1 of Theorem 6.1 to a few non-congruentnumbers described by Lagrange’s results in Table 6.3 and a non-congruentnumber that is listed in Table 4.1. The resulting numerical examples gener-ated by Case I.1 of Theorem 6.1 are stated in Table 6.4.1026.3. Examples of Even Non-congruent NumbersTable 6.4: Non-congruent numbers n = βsd+1sd+2 · · · st generated by CaseI.1 of Theorem 6.1TheoremSatisfied by ββ Case I.1 ExtensionTail sd+1sd+2 · · · stPrime(s)µ ∈ TLagrange Case1 in Table 6.32 · 43 · 83 281·337·1433·1601·2593·4129·6529·17393·98737·337121 ·490001 ·201503383Lagrange Case3 in Table 6.32 · 23 · 11 89 · 97 · 881 · 1433 ·2777 · 22697 · 25793 ·37489 · 51217 · 14968923Lagrange Case6 in Table 6.32·19·5·103 761 · 769 · 1489 · 2129 ·12641 · 25409 · 38321 ·191089 ·339161 ·1185601103Lagrange Case7 in Table 6.32 ·79 ·7 ·13 41 · 353 · 1097 · 6089 ·10601 · 10993 · 24169 ·93481 · 252913 · 41208179, 7, 13Theorem 4.1 2 · 17 · 5 · 3 ·23·263·5036857 · 11393 · 16553 ·53897 · 58337 · 68993 ·583673 · 86833717, 5, 3, 263,503None 2 · 5 · 7 · 71 ·179 · 4991201·1801·12401·13841·43649 · 57649 · 105449 ·290441 ·347729 ·107884171, 179, 499Notice that the value of β listed in the final row of Table 6.4 does notbelong to a known family of non-congruent numbers. However, Monsky’sformula can be used to verify that s(2 · 5 · 7 · 71 · 179 · 499) = 0, and soTheorem 6.1 can be applied to this value.Table 6.5 lists a collection of non-congruent numbers that result fromusing Case I.2 of Theorem 6.1 to extend some of the non-congruent numbersdescribed in Tables 4.1 and 6.3. Note that the first four numerical examplesgiven in Table 6.5 also appear in Table 2 in [42]. These numbers have beenincluded to illustrate how the non-congruent numbers generated by Case 1of Theorem 1.1 in [42] are a subset of those produced by Case I.2 of Theorem6.1.1036.3. Examples of Even Non-congruent NumbersTable 6.5: Non-congruent numbers n = βsd+1sd+2 · · · st generated by CaseI.2 of Theorem 6.1TheoremSatisfiedby ββ Case I.2Extension Tailsd+1sd+2 · · · stExtension Tail LegendreSymbols that Equal −1LagrangeCase 6 inTable 6.32·79·3·13 73 · 97 · 313 · 937 ·1433 · 1249 · 10729 ·12601·19249·17137(7313),(9713),(31373),(93779),(14333),(12491433),(1072997),(1260173),(192491433),(1713719249)LagrangeCase 6 inTable 6.32 · 23 ·11 · 13113 · 233 · 257 ·1049 · 1193 · 3433 ·6337 · 5641 · 49201 ·64793 · 58217(11323),(23311),(257233),(104923),(119323),(3433113),(633713),(56416337),(49201113),(647936337),(5821764793)LagrangeCase 6 inTable 6.32 · 103 ·19 · 517 · 137 · 409 · 1721 ·4409 · 7681 · 7753 ·8209 · 13001 · 26449(175),(1375),(40919),(1721103),(440917),(768117),(77535),(8209409),(13001409),(2644917)LagrangeCase 6 inTable 6.32·7·11·13 137 · 257 · 433 · 641 ·2129 · 3697 · 8969 ·14561·23761·34057(13713),(2577),(4337),(641257),(212911),(369713),(8969433),(1456111),(237617),(34057641)LagrangeCase 1 inTable 6.32 · 3 · 11 73 · 313 · 577 · 97 ·1433 · 6689 · 2689 ·7297 · 7129(7311),(31373),(57773),(97577),(14333),(66893),(26891433),(72971433),(71297297)Theorem4.12 ·41 ·13 ·19 · 71 ·31 · 1319617 · 3769 · 4001 ·4937 · 7673 · 40897 ·45161 · 25913(61713),(376941),(400141),(493741),(767341),(408974937),(451614001),(2591371)Next, we use Case II.3 of Theorem 6.1 to extend several non-congruentnumbers β with s(β) = 0. For each of the first five examples listed inTable 6.6, β belongs to a known family of non-congruent numbers. For theremaining three examples, s(β) can easily be shown to equal zero by formingthe Monsky matrix corresponding to β and verifying that it has full rank.Also, note that the examples listed in the last two rows of Table 6.6 appearin Table 3 of [42]. This is because Case II.3.A of Theorem 6.1 generalizesTheorem 1.2 in [42].1046.3. Examples of Even Non-congruent NumbersTable 6.6: Non-congruent numbers n = βsd+1sd+2 · · · st generated by CaseII.3 of Theorem 6.1TheoremSatisfied by ββ Case II.3Extension Tailsd+1sd+2 · · · stCase(s)Satisfied by theTail of PrimesLagrange Case2 in Table 6.32 · 17 · 5 3 · 107 · 347 · 947 ·1163 · 3803 · 11243 ·12203 · 22787 · 31643 ·373187 · 562763Case II.3.BLagrange Case2 in Table 6.32 · 17 · 5 19 · 251 · 491 · 1259 ·1531 · 6971 · 3947 ·44563·115883·304907·662323 · 1205123Case II.3.A (firstsix primes) andCase II.3.B (finalsix primes)Lagrange Case8 in Table 6.32·41·47·3 331 ·1051 ·1867 ·3307 ·3907 · 4003 · 25819 ·59707 · 72763 · 110419Case II.3.DLagrange Case8 in Table 6.32·41·47·3 107 · 419 · 443 · 1091 ·3499 · 8803 · 21211 ·93139·135043·873043·1217683 · 1396987Case II.3.C (firstfour primes) andCase II.3.D (finaleight primes)Theorem 6.1Case I.12 · 79 · 7 ·13·41·353859·1291·8363·22571·58763 · 76579 · 215123 ·325043·326219·890683Case II.3.CNone 2·5·7·71·179 · 4991019 · 19259 · 40699 ·55931·130099·180539·239171 · 313331Case II.3.ANone 2 · 41 · 5 ·31 · 3379 · 3931 · 3691 ·5011 · 10651 · 32299Case II.3.ANone 2 · 73 · 29 ·47 · 283467 · 971 · 1531 · 4003 ·14243 · 3467 · 63691 ·84299Case II.3.B (firsttwo primes) andCase II.3.A (finalsix primes)In Table 6.7, we state some non-congruent numbers that can be producedwhen Case III.3.A of Theorem 6.1 is used to extend non-congruent numberseither described by Lagrange’s work, or listed in Tables 4.1, 6.5, or 6.6.1056.3. Examples of Even Non-congruent NumbersTable 6.7: Non-congruent numbers n = βsd+1sd+2 · · · st generated by CaseIII.3.A of Theorem 6.1Theorem Satisfiedby ββ Case III.3.AExtension Tailsd+1sd+2 · · · stLagrange Case 8 inTable 6.32 · 17 · 7 · 3 373 · 421 · 1381 · 1429 ·4813 · 17077 · 22453Lagrange Case 8 inTable 6.32 · 41 · 23 · 19 197 · 821 · 1373 · 5717 ·20149 · 26573 · 29741Theorem 4.1 2 ·17 ·5 ·3 ·23 ·263 ·503 3109 · 6949 · 31741 ·33469 · 101149Theorem 4.1 2 · 41 · 13 · 19 · 71 · 31 ·1319 · 743 · 319111701 · 18773 · 58733 ·459749 · 578213Theorem 6.1 CaseI.22 ·79 ·3 ·13 ·73 ·97 ·313 997 · 3301 · 12373 · 20029 ·42013Theorem 6.1 CaseI.22·23·11·13·113·233·257 653 · 1013 · 6653 · 20333 ·126949Theorem 6.1 CaseI.22·103·19·5·17·137·409 5381 · 5861 · 10429 ·48109 · 50261Theorem 6.1 CaseI.22·7·11·13·137·257·433 5581 · 6029 · 7253 · 20549 ·59557Theorem 6.1 CaseII.3.A2 ·41 ·5 ·31 ·3 ·379 ·3931 3229 · 9781 · 26701 ·27901 · 28429 · 74149Theorem 6.1 CaseII.3.C2 · 79 · 7 · 13 · 41 · 353 ·859 · 12911117·12373·30781·51949·129581 · 225941 · 678773It is also worthwhile to mention that the numbers described by Case (c)of Theorem 2 in Goto’s paper [16] are a subset of those generated by CaseIII.3.A of Theorem 6.1. Goto’s non-congruent numbers are a product of theinteger two and arbitrarily many primes of the form 8k + 5. His numbersfollow the same pattern of Legendre symbols as the tail of primes of theform 8k + 5 in Case III.3.A of Theorem 6.1.Finally, in Table 6.8, we present some non-congruent numbers constructedaccording to Case IV.4 of Theorem 6.1.1066.3. Examples of Even Non-congruent NumbersTable 6.8: Non-congruent numbers n = βsd+1sd+2 · · · st generated by CaseIV.4 of Theorem 6.1TheoremSatisfied by ββ Case IV.4 ExtensionTail sd+1sd+2 · · · stElement(s)in HLagrange Case4 in Table 6.32 · 73 · 3 · 11 359 · 167 · 1823 · 6599 ·2063 · 20327 · 20063 ·85439 · 431903 · 13895973Lagrange Case4 in Table 6.32 · 73 · 3 · 11 479 · 79 · 359 · 5623 · 6863 ·8887 · 8087 · 77743 · 38543 ·989113Lagrange Case4 in Table 6.32 · 73 · 3 · 11 503 · 31 · 1487 · 823 · 10247 ·4519·70583·65839·278879·218887 · 541439 · 26806373, 3, 11Lagrange Case5 in Table 6.32 ·17 ·13 ·29 103 · 127 · 1223 · 599 ·10039 · 5087 · 26399 · 9103 ·253751 · 10835929Lagrange Case5 in Table 6.32 ·17 ·13 ·29 919 · 239 · 1223 · 463 ·5407 · 1879 · 18199 · 3583 ·200983 · 6854313Lagrange Case5 in Table 6.32 ·17 ·13 ·29 647 · 23 · 103 · 199 · 4943 ·14303 · 38047 · 16007 ·430847 · 10462317Theorem 6.1Case III.3.A2 ·79 ·7 ·13 ·41·353·859·1291 · 11172287 · 239 · 207847 · 74687 ·392831 · 275039 · 650543 ·216503979, 7, 13,41, 353Theorem 6.1Case III.3.A2 ·79 ·7 ·13 ·41·353·859·1291 · 111714551 · 12959 · 161503 ·142543 · 986543 · 594119 ·1492063 · 37038237, 13, 41,353, 859,1291, 1117Theorem 6.1Case III.3.A2 ·79 ·7 ·13 ·41·353·859·1291 · 11172287 · 607 · 27583 · 60623 ·298847 · 4017431117Note that it is possible to combine the cases in Theorem 6.1 to produceeven non-congruent numbers with prime factors in each odd congruence classmodulo eight. The numerical examples in the final three rows of Table 6.8are generated by applying Cases I.1, II.3.C, III.3.A, and IV.4 of Theorem6.1 to a non-congruent number described by Lagrange [27].107Chapter 7ConclusionThis thesis focused on the generation of non-congruent numbers witharbitrarily many distinct prime divisors. Our results are significant becausethey provide a new technique for constructing non-congruent numbers, andproduce families of both odd and even non-congruent numbers containingprime factors belonging to each odd congruence class modulo eight. Webegin by summarizing our main research contributions and then proceed todiscuss some interesting avenues for future work.7.1 Main ResultsIn Chapter 4, we described a particular family of even non-congruentnumbers that are a product of arbitrarily many primes. These non-congruentnumbers have at least one prime factor in each odd congruence class moduloeight. This is a distinguishing feature of our result, as all known families ofeven non-congruent numbers are comprised of primes belonging to no morethan three odd congruence classes modulo eight.Chapters 5 and 6 developed and described a general approach for con-structing non-congruent numbers. Our method allows any non-congruentnumber α, for which the elliptic curve y2 = x3−α2x has 2-Selmer rank equalto zero, to be extended to produce other non-congruent numbers. Chapter5 focused on odd non-congruent numbers, and showed that infinitely manynon-congruent numbers can be generated by appending arbitrarily manyprimes of the form 8k+ 1, 8k+ 3, or 8k+ 5 onto known odd non-congruentnumbers. In Chapter 6, we considered even non-congruent numbers and pro-vided criteria for constructing non-congruent numbers with arbitrarily manyprime factors in possibly all of the four odd congruence classes modulo eight.Our results in Chapters 5 and 6 allow existing families of non-congruentnumbers to be extended and thus, considerably broaden the collection ofnon-congruent numbers that can be generated and described.1087.2. Future Work7.2 Future WorkIn this section, we identify and discuss some directions for future researchwork.1) Theorem 5.5 produces an infinite collection of non-congruent numbersby multiplying existing odd non-congruent numbers α with s(α) = 0by primes sd+1, sd+2, . . . , st of the form 8k + 1, 8k + 3, or 8k + 5. Isit possible to find primes sd+1, sd+2, . . . , st of the form 8k + 7 witht even, such that for any odd number α with s(α) = 0, the integern = αsd+1sd+2 · · · st is a non-congruent number with s(n) = 0?2) It would be interesting to determine whether Theorems 5.5 and 6.1 canbe expanded to describe a larger collection of non-congruent numbers.In Theorem 5.5, the primes sd+1, sd+2, . . . , st that are appended ontothe odd non-congruent number α are required to satisfy specific Legen-dre symbol conditions. Do there exist Legendre symbol conditions thatdiffer from those in the statement of Theorem 5.5 that, when imposedupon the primes sd+1, sd+2, . . . , st, guarantee that αsd+1sd+2 · · · st is anon-congruent number with s(α) = 0? Similarly, are there Legendresymbol conditions other than the ones stated in Theorem 6.1 that al-low the even non-congruent number β with s(β) = 0 to be extended toproduce infinitely many non-congruent numbers βsd+1sd+2 · · · st? Fur-thermore, if Theorems 5.5 and 6.1 can be expanded, is there a better,more concise way to describe all of the Legendre symbol conditions?3) There exist non-congruent numbers whose corresponding congruentnumber elliptic curves have nonzero 2-Selmer rank. For example,Bastien [2] described a family of non-congruent numbers of the formn = 2p with p ≡ 9 (mod 16) for which s(n) > 0. Congruent numberelliptic curves with rank equal to zero and non-trivial Shafarevich-Tategroup are also given by Wang [58]; the non-congruent numbers that hedescribes have arbitrarily many prime divisors. It would be desirableto find other families of non-congruent numbers with nonzero 2-Selmerrank.4) The work in this thesis strictly focused on the construction of non-congruent numbers with arbitrarily many prime factors. However,it would be of interest to find families of congruent numbers witharbitrarily many prime divisors. Families of congruent numbers withtwo or fewer odd prime factors were given by Heegner [21], Birch [4],1097.2. Future WorkStephens [53], and Monsky [34]. More recently Tian described somefamilies of congruent numbers that have an unlimited number of primedivisors [7, 54, 55]. Because the 2-Selmer rank provides an upperbound for the arithmetic rank, Monsky’s formula for computing the2-Selmer rank of En cannot be used in isolation to generate families ofcongruent numbers. This makes the search for families of congruentnumbers with arbitrarily many distinct prime factors a substantiallymore challenging task than the search for families of non-congruentnumbers of a similar form.5) Monsky’s formula, which is described in Section 3.6, computes the 2-Selmer rank of congruent number elliptic curves y2 = x3 − n2x. Anatural extension would be to examine whether or not it is possible touse Monsky’s approach to calculate the 2-Selmer rank of other ellipticcurves. In [17], Goto studies a more general version of the congruentnumber problem, known as the θ-congruent number problem. Thisinvolves determining whether a positive integer is a θ-congruent num-ber. A positive integer n is called a θ-congruent number if n√r2 − s2occurs as the area of a triangle with rational side lengths and an angle0 < θ < pi, where cos(θ) = s/r with s, r ∈ Z, |s| ≤ r, and (r, s) = 1.The θ-numbers are related to elliptic curves of the formy2 = x(x+ (r + s)n)(x− (r − s)n).Note that pi/2-congruent numbers are regular congruent numbers. Gotoclaims that he “could not apply Monsky’s method to the θ-congruentnumber problem with θ 6= pi/2”. However, he does not provide anyfurther details regarding this, and so it is unclear whether it is in factimpossible to derive a formula for computing the 2-Selmer rank ofelliptic curves for θ-congruent numbers with θ 6= pi/2.110Bibliography[1] R. Alter, T. B. Curtz, and K. K. Kubota, Remarks and results oncongruent numbers, Proc. Third Southeastern Conf. on Combinatorics,Graph Theory and Computing, (1972), pp. 27–35. → pages 24[2] L. Bastien, Nombres congruents, Interme´diaire des Math., 22 (1915),pp. 231–232. → pages 4, 31, 46, 72, 101, 109[3] M. Bhargava and A. Shankar, Ternary cubic forms having boundedinvariants, and the existence of a positive proportion of elliptic curveshaving rank 0, Ann. of Math. (2), 181 (2015), pp. 587–621. → pages 24[4] B. J. Birch, Elliptic curves and modular functions, in Symposia Math-ematica, Indam Rome 1968/1969, Vol. 4, Academic Press, 1970, pp. 27–32. → pages 109[5] V. Chandrasekar, The congruent number problem, Resonance, 3(1998), pp. 33–45. → pages 2, 3, 26[6] W. Cheng and X. Guo, Some new families of non-congruent num-bers, J. Number Theory, 196 (2019), pp. 291–305. → pages 4, 31, 46,72, 101[7] J. Coates, Congruent numbers, Acta Math. Vietnam, 39 (2014), pp. 3–10. → pages 25, 110[8] L. E. Dickson, History of the Theory of Numbers, II, Carnegie Insti-tution of Washington, Washington, 1920. → pages 2, 3[9] A. Dujella, History of elliptic curves rank records. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html. Accessed: 2019-07-13. → pages 24[10] A. Dujella, A. S. Janfada, and S. Salami, A search for high rankcongruent number elliptic curves, J. Integer Seq., 12 (2009). Article09.5.8. → pages 24, 25, 26, 29111Bibliography[11] K. Feng, Non-congruent numbers, odd graphs and the Birch-Swinnerton-Dyer conjecture, Acta Arith., 75 (1996), pp. 71–83. →pages 4, 31, 46, 72, 101[12] K. Feng and M. Xiong, On elliptic curves y2 = x3 − n2x with rankzero, J. Number Theory, 109 (2004), pp. 1–26. → pages 31, 101[13] K. Feng and Y. Xue, New series of odd non-congruent numbers, Sci.China Ser. A, 49 (2006), pp. 1642–1654. → pages 4, 31, 46, 72[14] J. B. Fraleigh, A First Course in Abstract Algebra, Addison Wesley,Boston, seventh ed., 2003. → pages 5[15] A. Genocchi, Note analitiche sopra tre scritti, Annali di ScienzeMatematiche e Fisiche, 6 (1855), pp. 273–317. → pages 4, 31, 46,72, 101[16] T. Goto, A note on the Selmer group of the elliptic curve y2 = x3+Dx,Proc. Japan Acad. Ser. A Math. Sci., 77 (2001), pp. 122–125. → pages31, 106[17] , A Study on the Selmer Groups of Elliptic Curves with a Rational2-Torsion, PhD thesis, Kyushu University, 2002. → pages 4, 46, 72, 73,101, 102, 110[18] F. Q. Gouveˆa, p-adic Numbers: An Introduction, Springer-Verlag,New York, second ed., 1997. → pages 9, 10[19] D. R. Heath-Brown, The size of Selmer groups for the congruentnumber problem, Invent. Math., 111 (1993), pp. 171–195. → pages 29[20] , The size of Selmer groups for the congruent number problem, II.With an appendix by P. Monsky, Invent. Math., 118 (1994), pp. 331–370. → pages 29, 30[21] K. Heegner, Diophantische analysis und modulfunktionen, Math Z.,56 (1952), pp. 227–253. → pages 109[22] B. Hemenway, On Recognizing Congruent Primes, Master’s thesis,Simon Fraser University, 2006. → pages 21[23] D. Husemo¨ller, Elliptic Curves, Graduate Texts in Mathematics,111, Springer-Verlag, New York, second ed., 2004. → pages 12, 14,15, 16, 18, 19, 20, 29112Bibliography[24] B. Iskra, Non-congruent numbers with arbitrarily many prime factorscongruent to 3 modulo 8, Proc. Japan Acad. Ser. A Math. Sci., 72(1996), pp. 168–169. → pages 4, 27, 31, 46, 72[25] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Grad-uate Texts in Mathematics, 97, Springer-Verlag, New York, second ed.,1993. → pages 2, 3, 12, 18, 21[26] J. H. Kwak and S. Hong, Linear Algebra, Springer, New York, sec-ond ed., 2004. → pages 10[27] J. Lagrange, Nombres congruents et courbes elliptiques, Se´minaireDelange-Pisot-Poitou, The´orie des nombres, 16e anne´e (1974/1975). →pages 4, 25, 31, 46, 72, 73, 101, 107[28] D. Li and Y. Tian, On the Birch-Swinnerton-Dyer conjecture ofelliptic curves ED : y2 = x3 − D2x, Acta Math. Sin. (Engl. Ser.), 16(2000), pp. 229–236. → pages 4, 31, 46, 72[29] A. Lonzano-Robledo, Elliptic Curves, Modular Forms, and TheirL-functions (Student Mathematical Library), American MathematicalSociety, 2011. → pages 2, 15, 18, 19, 20, 28, 29[30] B. Mazur, Modular curves and the Eisenstein ideal, Inst. HautesE´tudes Sci. Publ. Math., 47 (1977), pp. 33–186. → pages 21[31] , Rational isogenies of prime degree, Invent. Math., 44 (1978),pp. 129–162. → pages 20, 21[32] R. B. McClenon, Leonardo of Pisa and his Liber Quadratorum,Amer. Math. Monthly, 26 (1919), pp. 1–8. → pages 2[33] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM,Philadelphia, 2000. → pages 10, 11[34] P. Monsky, Mock Heegner points and congruent numbers, Math. Z.,204 (1990), pp. 45–67. → pages 25, 110[35] K. Noda and H. Wada, All congruent numbers less than 10000, Proc.Japan Acad. Ser. A Math. Sci., 69 (1993), pp. 175–178. → pages 2[36] Y. Ouyang and S. Zhang, On non-congruent numbers with 1 modulo4 prime factors, Sci. China Math., 57 (2014), pp. 649–658. → pages 4,31, 46, 72113Bibliography[37] , On second 2-descent and non-congruent numbers, Acta Arith.,170 (2015), pp. 343–360. → pages 4, 31, 46, 72, 101[38] L. Reinholz, Families of Congruent and Non-congruent Numbers,Master’s thesis, The University of British Columbia, 2013. → pages27, 46[39] L. Reinholz, B. K. Spearman, and Q. Yang, Families of non-congruent numbers with arbitrarily many prime factors, J. NumberTheory, 133 (2013), pp. 318–327. → pages 4, 31, 46, 72[40] , On the prime factors of non-congruent numbers, Colloq. Math.,138 (2015), pp. 271–282. → pages 4, 31, 46, 72, 75, 76[41] , An extension theorem for generating new families of non-congruent numbers, Funct. Approx. Comment. Math., 58 (2018),pp. 69–77. → pages v, 41, 42[42] , Families of even non-congruent numbers with prime factors ineach odd congruence class modulo eight, Int. J. Number Theory, 14(2018), pp. 669–692. → pages v, vi, 31, 80, 85, 86, 103, 104[43] L. Reinholz and Q. Yang, On the extension of even families of non-congruent numbers. Submitted for publication. → pages v, vi, 80[44] , On the generation of odd non-congruent numbers with arbitrarilymany prime factors. Submitted for publication. → pages v, 41, 48[45] N. Rogers, Elliptic curves x3 + y3 = k with high rank, PhD thesis,Harvard University, 2004. → pages 24[46] K. H. Rosen, Elementary Number Theory and its Applications, Addi-son Wesley, Boston, fifth ed., 2005. → pages 2, 6, 7, 8, 9[47] S. Schmitt and H. G. Zimmer, Elliptic Curves: A ComputationalApproach, Walter de Gruyter, Berlin, 2003. → pages 12, 15, 16[48] E. S. Selmer, The Diophantine equation ax3 + by3 + cz3 = 0, ActaMath., 85 (1951), pp. 203–362. → pages 10[49] P. Serf, Congruent numbers and elliptic curves, in ComputationalNumber Theory, A. Petho¨, M. E. Pohst, H. C. Williams, and H. G.Zimmer, eds., Walter de Gruyter, 1991, pp. 227–238. → pages 4, 25,26, 28, 31, 46, 72, 101114Bibliography[50] J. P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics,7, Springer-Verlag, New York, 1973. → pages 9, 10[51] J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Textsin Mathematics, 106, Springer, New York, second ed., 2009. → pages12, 13, 15, 16, 18, 19, 20, 24, 26, 28, 29[52] J. H. Silverman and J. T. Tate, Rational Points on Elliptic Curves,Undergraduate Texts in Mathematics, Springer, New York, second ed.,2015. → pages 12, 13, 15, 16, 18, 19, 20[53] N. M. Stephens, Congruence properties of congruent numbers, Bull.London Math. Soc., 7 (1975), pp. 182–184. → pages 110[54] Y. Tian, Congruent numbers with many prime factors, Proc. Natl.Acad. Sci. USA, 109 (2012), pp. 21256–21258. → pages 25, 110[55] , Congruent numbers and Heegner points, Camb. J. Math., 2(2014), pp. 117–161. → pages 25, 110[56] J. Top and N. Yui, Congruent number problems and their variants,Algorithmic number theory: lattices, number felds, curves and cryp-tography, 44 (2008), pp. 613–639. → pages 2, 3[57] J. B. Tunnell, A classical Diophantine problem and modular formsof weight 3/2, Invent. Math., 72 (1983), pp. 323–334. → pages 2, 3[58] Z. Wang, Congruent elliptic curves with non-trivial Shafarevich-Tategroups, Sci. China Math., 59 (2016), pp. 2145–2166. → pages 4, 31, 46,72, 109[59] L. C. Washington, Elliptic Curves: Number Theory and Cryptogra-phy, Chapman and Hall/CRC, New York, 2003. → pages 18, 19, 20[60] A. Wiles, The Birch and Swinnerton-Dyer Conjecture, 2000. http://www.claymath.org/sites/default/files/birchswin.pdf. → pages2, 3115Appendix116Appendix AMaple & Magma CodeIn Sections 4.3, 5.3, 5.5, and 6.3, we presented numerical examples gen-erated by our main theorems. In this section, we show how to use Mapleto construct those numbers as well as infinitely many other non-congruentnumbers satisfying the constraints imposed in the statements of our theo-rems in Chapters 4, 5, and 6.We illustrate the process by considering the non-congruent number n =(α)s1s2 · · · s8 = (13 · 29 · 37 · 23 · 31 · 71) · 4003 · 5867 · 41947 · 60779 · 135131 ·196387 · 296299 · 329891 listed in Table 5.5. Since α does not belong to aknown family of non-congruent numbers, we begin by verifying that α =(13 · 29 · 37 · 23 · 31 · 71) satisfies s(α) = 0. We do this by using Monsky’sformula, given by Equation (3.15). The following code shows how Maple isused to compute the rank of the Monsky matrix, Mo, for α. Note that inthe code, the diagonal matrices D−2 and D2 are denoted by Dneg2 and D2,respectively.117Appendix A. Maple & Magma CodeSince rankF2(Mo) = 12, Equation (3.15) implies that s(α) = 0, as re-quired.If α is even, an analogous set of Maple calculations can be completed todetermine the rank of the matrix Me, given by Equation (3.17).Our next step is to generate new non-congruent numbers by finding atail of primes to append onto α that satisfy the Legendre symbol conditionsspecified in the statement of Theorem 5.5 Case I.1.A. This can be doneefficiently by using the following Maple code.Notice that in this loop, the six required Legendre symbol conditions arespecified and primes of the form 8k + 3 satisfying the conditions are listedin the output. For k values from 0 to 1000, three primes are stated in the118Appendix A. Maple & Magma Codeoutput. We choose to append the first prime listed, 4003, onto α. We thenupdate the Legendre symbol conditions to include the condition imposed onthe prime 4003, and run the code again to find another prime to appendonto α · 4003. Our search yields the prime 5867, as shown by the followingcode.This process can be repeated indefinitely, allowing arbitrarily many primesof the form 8k+3 to be appended onto α, and infinitely many non-congruentnumbers to be produced. Note that if the code fails to return an output,then the bounds for k need to be adjusted.The numbers generated by our theorems can be verified to be non-congruent by using the computer algebra system Magma. This is foundonline athttp://magma.maths.usyd.edu.au/calc/.The code below shows how Magma is used to calculate the Mordell-Weilrank of the congruent number elliptic curve corresponding to the integerαs1 = (13 · 29 · 37 · 23 · 31 · 71) · 4003.Input:E:=EllipticCurve([-(13*29*37*23*31*71*4003)^2,0]);Rank(E);Output:0 trueMagma confirms that the rank of the curve y2 = x(x2 − (αs1)2) is zero,so αs1 = (13 · 29 · 37 · 23 · 31 · 71) · 4003 is a non-congruent number.119
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The generation of families of non-congruent numbers with arbitrarily many prime factors
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