families of congruent and non-congruent numbers
TRANSCRIPT
Families of Congruent andNon-congruent Numbers
by
Lindsey Kayla Reinholz
B.Sc., The University of British Columbia, 2011
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in
The College of Graduate Studies
(Mathematics)
THE UNIVERSITY OF BRITISH COLUMBIA
(Okanagan)
August 2013
c© Lindsey Kayla Reinholz, 2013
Abstract
A positive integer n is a congruent number if it is equal to the area of a
right triangle with rational sides. Equivalently, the Mordell-Weil rank of
the elliptic curve
y2 = x(x2 − n2)
is positive. Otherwise n is a non-congruent number. Although congru-
ent numbers have been studied for centuries, their complete classification
is one of the central unresolved problems in the field of pure mathemat-
ics. However, by using algorithms such as the method of 2-descent, various
mathematicians have proven that numbers with prime factors of a specified
form that satisfy a certain pattern of Legendre symbols are either always
congruent or always non-congruent. In this thesis, we build upon these
results and not only prove the existence of new families of congruent and
non-congruent numbers, but also present a new method for generating fam-
ilies of non-congruent numbers. We begin by providing a technique for
constructing congruent numbers with three prime factors of the form 8k+3,
and then give a family of such numbers for which the rank of their asso-
ciated elliptic curves equals two, the maximal rank for congruent number
curves of this type. Following this, we offer an extension to work done by
Iskra and present our new method for generating families of non-congruent
numbers with arbitrarily many prime factors. This method employs Mon-
sky’s formula for the 2-Selmer rank. Unlike the method of 2-descent which
involves a series of lengthy and complex calculations, Monsky’s formula of-
fers an elegant approach for determining whether a given positive integer
is non-congruent. This theorem uses linear algebra, and through a series
of steps, allows one to compute the 2-Selmer rank of a congruent number
ii
Abstract
elliptic curve, which provides an upper bound for the curve’s Mordell-Weil
rank. By applying this method, we construct infinitely many distinct new
families of non-congruent numbers with arbitrarily many prime factors of
the form 8k + 3. In addition, by utilizing the aforementioned method once
again, we expand upon results by Lagrange to generate infinitely many new
families of non-congruent numbers that are a product of a single prime of
the form 8k + 1 and at least one prime of the form 8k + 3.
iii
Preface
The main results presented in my thesis are from collaborative research done
with Dr. Blair Spearman and Dr. Qiduan Yang. The contents of Chapter 3
were published in the journal Integers under the title “On congruent numbers
with three prime factors” [RSY11]. My colleagues and I contributed equally
to this article. Specifically, my research contributions to this publication
included conducting searches with the software program Magma to find
congruent numbers less than 10,000 with three prime factors of the form
8k + 3, applying Maple to solve torsors and find corresponding points on
congruent number elliptic curves, and utilizing Monsky’s formula for the
2-Selmer rank to verify that the maximal rank for our family of congruent
number elliptic curves is two. In addition, I was also an active participant
in the writing process. I was responsible for aiding in the organization of
the article’s content, for formatting the reference section, and for editing the
numerous drafts of the paper.
Chapter 5 is based on my paper “Families of non-congruent numbers
with arbitrarily many prime factors,” which was recently published in the
Journal of Number Theory [RSY13]. I played an important role in all as-
pects of the research process, from the formation of the initial hypothesis to
the submission of the completed article to The Journal of Number Theory.
When I began working on this project, I used the computer software program
Maple to carry out many numerical calculations. From these computations,
I noticed a pattern that enabled me to formulate a hypothesis. Further
numerical testing of my hypothesis allowed me to develop a method for con-
structing families of non-congruent numbers with arbitrarily many prime
factors. In addition to the research aspect of the project, I also contributed
to the writing of the paper and performed the necessary organizational and
iv
Preface
editing tasks. During the entire research process, I received assistance and
guidance from my collaborators, Dr. Spearman and Dr. Yang. As supervi-
sors of my research work, they introduced me to the topic of non-congruent
numbers and to Monsky’s formula for 2-Selmer rank. This, in turn, enabled
me to develop the hypothesis around which my paper is centred. Dr. Yang’s
linear algebra knowledge was indispensable to the proof of my hypothesis,
and Dr. Spearman’s extensive publication background was an asset to the
composition of our article.
It should also be noted that the results appearing in Chapter 6 are
intended for publication. I was responsible for developing and proving the
main theorem presented in this chapter. The supporting corollary, and its
proof were important additions suggested by Dr. Spearman.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Congruent and Non-congruent Numbers . . . . . . . . . . . . 1
1.2 Algebra and Number Theory Preliminaries . . . . . . . . . . 7
1.2.1 Abstract Algebra . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Number Theory . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2: Congruent Numbers and Elliptic Curves . . . . . 15
2.1 Introduction to Elliptic Curves . . . . . . . . . . . . . . . . . 15
2.2 The Group Law and Mordell’s Theorem . . . . . . . . . . . . 16
2.3 The Torsion Subgroup . . . . . . . . . . . . . . . . . . . . . . 20
vi
Table of Contents
2.4 The Method of 2-Descent . . . . . . . . . . . . . . . . . . . . 22
2.5 The Relationship Between Elliptic Curves and Congruent Num-
bers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 The Method of Complete 2-Descent . . . . . . . . . . . . . . 26
2.7 Monsky’s Formula for the 2-Selmer Rank . . . . . . . . . . . 30
Chapter 3: A Family of Congruent Numbers with Three Prime
Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 41
Chapter 4: Iskra’s Family of Non-congruent Numbers . . . . 42
4.1 The Proof of Iskra’s Theorem Using the Method of Complete
2-Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 The Proof of Iskra’s Theorem Using Monsky’s Formula. . . . . 55
Chapter 5: Families of Non-congruent Numbers with Arbi-
trarily Many Prime Factors of the Form 8k + 3 . . 60
5.1 Preliminary Results Involving the Generation of Non-congruent
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 68
Chapter 6: Families of Non-congruent Numbers with One
Prime Factor of the Form 8k + 1 and Arbitrar-
ily Many Prime Factors of the Form 8k + 3 . . . . 76
6.1 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 77
6.2 A Supporting Corollary . . . . . . . . . . . . . . . . . . . . . 87
Chapter 7: Conclusion and Future Work . . . . . . . . . . . . 89
7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
vii
Table of Contents
Appendices
Chapter A: Magma Code . . . . . . . . . . . . . . . . . . . . . . 96
A.1 Elliptic Curve Calculations . . . . . . . . . . . . . . . . . . . 96
Chapter B: Maple Code . . . . . . . . . . . . . . . . . . . . . . . 99
B.1 Parametrization and 2-Selmer Rank Computations . . . . . . 99
viii
List of Tables
Table 1.1 Congruent Numbers . . . . . . . . . . . . . . . . . . . 4
Table 1.2 Non-congruent Numbers . . . . . . . . . . . . . . . . . 5
Table 3.1 Values of s(n) for n = p3q3r3 . . . . . . . . . . . . . . 40
ix
List of Figures
Figure 1.1 A rational right triangle with an area of 5. . . . . . . . 1
Figure 2.1 Elliptic curve with three real roots, y2 = (x − 1)(x −2)(x+ 1). . . . . . . . . . . . . . . . . . . . . . . . . . 16
Figure 2.2 Elliptic curve with one real root, y2 = (x+2)(x2−2x+3). 16
Figure 2.3 Cubic curve with a double root, y2 = x2(x+ 2). . . . . 16
Figure 2.4 Cubic curve with a triple root, y2 = (x+ 1)3. . . . . . 16
Figure 2.5 The chord and tangent method applied to distinct
points P and Q on the curve y2 = (x+ 2)(x2 − 2x+ 3). 17
Figure 2.6 The chord and tangent method applied to the point P
on the curve y2 = (x+ 2)(x2 − 2x+ 3). . . . . . . . . . 17
Figure 2.7 The group law applied to points P and Q on the curve
y2 = (x+ 2)(x2 − 2x+ 3). . . . . . . . . . . . . . . . . 19
x
List of Symbols
Z Set of integers
N+ Set of natural numbers excluding zero, {1, 2, 3, . . .}Q Set of rational numbers
Q∗ Multiplicative group of non-zero rational numbers
Q∗2 Subgroup of squares in the multiplicative group of
non-zero rational numbers
Q∗ = Q∗/Q∗2 Quotient group of square-free, non-zero rational numbers
F2 Finite field with two elements
Zn Ordered n-tuples of integers
Zn Cyclic group of order n
Zpνii Cyclic group with prime-power order
Zn1 ⊕ Zn2 ⊕ · · · ⊕ Zns Direct sum of cyclic groups
Z[x] Polynomial ring of integers in the variable x
Q(z) Ring of rational functions in the variable z
In or I Identity matrix of order n
0n or 0 Zero matrix of order n
AT Transpose of the matrix A
A−1 Inverse of the matrix A
rank(A) Rank of the matrix A
det(A) Determinant of the matrix A
gcd(a, b) Greatest common divisor of a and b
a|b a divides b
a - b a does not divide b
a ≡ b (mod m) a is congruent to b modulo m
a 6≡ b (mod m) a is incongruent to b modulo m
xi
List of Symbols
(p
q
)Legendre symbol
vp(n) p-adic valuation of n
∞ Infinite prime
MQ Set of all places of the field Q, {∞, 2, 3, . . .}O Point at infinity on an elliptic curve
x(2P ) x-coordinate of the point 2P
En Congruent number elliptic curve y2 = x(x2 − n2)E(Q) Group of rational points on the elliptic curve E
T Torsion part of E(Q)
F Free part of E(Q)
Γ (or Γ) Group of rational points on the elliptic curve E (or E)
α (or α) Homomorphism mapping Γ to Q∗ (or Γ to Q∗)r(n) Mordell-Weil rank of the elliptic curve En
s(n) 2-Selmer rank of the elliptic curve En
image(b) Image of the injective homomorphism b
M\S The set M excluding the elements in the set S∑Sum∏Product
| · | Cardinality
⊆ Subset
min{· · · } Minimum value of the elements in the set {· · · }
xii
Acknowledgements
First, and foremost, I would like to thank my supervisor, Dr. Blair Spear-
man, who has played an integral role in my educational journey. Over the
past four years, he has provided me with unfailing support, guidance, and
encouragement. Were it not for his recognition of and confidence in my po-
tential as a researcher, I never would have considered conducting research
or decided to pursue graduate studies. I am incredibly grateful for all of the
time and effort he has invested in my education, and feel truly fortunate to
have had the opportunity to work under his supervision.
I would like to thank my committee members, Dr. Qiduan Yang, Dr.
Sylvie Desjardins, and Dr. Shawn Wang. I greatly appreciate the contri-
butions that each of them has made to my education and the support and
guidance that they have given me throughout my academic studies. I am
thankful for the abundance of advice they have provided me with and the
numerous letters of reference they have written for me over the years.
I also wish to thank all of the donors who have contributed to my edu-
cation by providing me with financial support. In particular, I would like to
thank the Natural Sciences and Engineering Research Council of Canada.
Many of the ideas presented in this thesis were developed during two sum-
mers of undergraduate research funded by NSERC’s Undergraduate Student
Research Award Program. I credit this experience for sparking my interest
in research and for inspiring me to pursue graduate studies. In addition,
I would like to thank UBC Okanagan for their continued financial support
over my six years of studies. I am especially grateful that they provided me
with the opportunity to work as a teaching assistant.
Last, but certainly not least, I would like to thank my family and friends
for their endless reassurance, love, and encouragement throughout my edu-
xiii
Acknowledgements
cational journey. I appreciate the interest they have shown in my work, the
hours they have spent listening to my mathematical tirades, and the sanity-
preserving distractions they have provided. Most of all, I am thankful for
their unconditional support, which has made my accomplishments possible.
xiv
To my parents, who have instilled in me a strong work ethic
and have provided me with love, support, and encouragement along every
step of my educational journey.
xv
Chapter 1
Introduction
1.1 Congruent and Non-congruent Numbers
A positive integer n is a congruent number if it is equal to the area of a
right triangle with rational sides. In other words, there must exist rational
numbers a, b, and c such that
a2 + b2 = c2 and1
2ab = n.
Otherwise n is said to be a non-congruent number. For example, 5 is a
congruent number as it is equal to the area of a right triangle with side
lengths 203 ,
32 , and 41
6 [Cha98].
Figure 1.1: A rational right triangle with an area of 5.
In contrast, the integer 1 is non-congruent because no combination of ratio-
nal side lengths can be found to generate a right triangle with an area of 1
[Cha98, Joh09].
1
1.1. Congruent and Non-congruent Numbers
For centuries scholars have studied congruent numbers to find a solution to
a question known as the congruent number problem [Cha98, Hem06, Joh09]:
For a given positive integer n, is it possible to determine whether or
not n is a congruent number in a finite number of steps?
The first reference to this problem appears in an Arab manuscript written
in the tenth century [Alt80, Cha98]. Since then, many famous mathemati-
cians, including Fibonacci, Fermat, and Euler have studied congruent num-
bers [Alt80, Cha98, Joh09]. Fibonacci made a notable contribution to the
field by proving that both 5 and 7 are congruent numbers. He also conjec-
tured without proof that numbers that are perfect squares are not congruent
[Cha98]. This fact remained unproven until four centuries later when Fermat
developed the method of infinite descent. By applying this technique, Fer-
mat was able to prove that 1 is not a congruent number, which is equivalent
to showing that squares are not congruent [Cha98, Joh09]. In the twentieth
century, a link between congruent numbers and elliptic curves was estab-
lished [Kob93]. This significant discovery lead Tunnell to state and prove a
theorem that provides a simple criterion for determining whether or not a
given positive integer is a congruent number [Cha98, Kob93].
Theorem 1.1 (Tunnell’s Theorem). Let n be a square-free congruent
number and define
An = #{(x, y, z) ∈ Z3|n = 2x2 + y2 + 32z2},Bn = #{(x, y, z) ∈ Z3|n = 2x2 + y2 + 8z2},Cn = #{(x, y, z) ∈ Z3|n = 8x2 + 2y2 + 64z2},Dn = #{(x, y, z) ∈ Z3|n = 8x2 + 2y2 + 16z2}.
Then {2An = Bn if n is odd,
2Cn = Dn if n is even.
If the Birch and Swinnerton-Dyer conjecture holds for elliptic curves of the
form y2 = x3 − n2x then, conversely, these equalities imply that n is a
congruent number.
2
1.1. Congruent and Non-congruent Numbers
Proof. See Section 4 of Chapter IV in [Kob93].
Note that one direction of Tunnell’s theorem relies upon the Birch and
Swinnerton-Dyer conjecture, which has never been proven. This well-known
conjecture is widely believed to be true and is one of Clay Mathematics
Institute’s Millennium Prize Problems [Hem06, Joh09]. However, since the
results presented in this thesis do not require the use of the Birch and
Swinnerton-Dyer conjecture, additional details regarding it will be excluded
from the discussion.
Before Tunnell presented his ground-breaking theorem, classifying num-
bers as either congruent or non-congruent had been a difficult task. It took
until 1915 for all of the square-free congruent numbers less than 100 to be
discovered [Bas15, Alt80]. Following this, various mathematicians including
Gerardin, Alter, Curtz, Kubota, Godwin, and Hunter worked to assem-
ble a list containing all congruent numbers less than 1000 [Ger15, ACK72,
AC74, God78, Alt80]. However, it was not until 1983, when Tunnell proved
his theorem, that this list was officially completed [Joh09]. By 1993, this
list had been expanded to include all congruent numbers less than 10,000
[NW93]. Recently, computer software utilized in conjunction with Tunnell’s
theorem has enabled mathematicians to broaden their search and identify
all square-free congruent numbers less than one trillion [Joh09].
Due to the reliance of Tunnell’s theorem on the unproven Birch and
Swinnerton-Dyer conjecture, many scholars have chosen to avoid using this
theorem when studying the congruent number problem. Some of the results
that have been proven without the use of the Birch and Swinnerton-Dyer
conjecture include the ones listed in Tables 1.1 and 1.2. Note that pi, qi,
and ri denote distinct primes of the form 8k + i for k ∈ Z, or equivalently
pi ≡ qi ≡ ri ≡ i (mod 8) (see Definition 1.24). In addition,(piqj
)is the
Legendre symbol (see Definition 1.28).
3
1.1. Congruent and Non-congruent Numbers
Table 1.1: Congruent Numbers
Heegner, 1952 [Hee52] −→ 2p3 and 2p7and Birch, 1968 [Bir68]
Stephens, 1975 [Ste75] −→ p5 and p7
Monsky, 1990 [Mon90] −→ p3q7, p3q5, 2p3q5, and 2p5q7
−→ p1q5 with(p1q5
)= −1
−→ p1q7 with(p1q7
)= −1
−→ 2p1q3 with(p1q3
)= −1
−→ 2p1q7 with(p1q7
)= −1
Serf, 1991 [Ser91] −→ p3q3r5, p3q3r7, 2p3q3r7, 2p3q5r5, and 2p5q5r7
−→ p7q7r7 with(p7q7
)= −
(p7r7
)=(q7r7
)−→ 2p7q7r7 with
(p7q7
)= −
(p7r7
)=(q7r7
)−→ p1q3r3s5 with(
p1q3
)=(p1r3
)=(p1s5
)= +1
or −(p1q3
)= −
(p1r3
)=(p1s5
)= +1,
(q3s5
)=(r3s5
)−→ 2p1q3r5s5 with(
p1q3
)=(p1r5
)=(p1s5
)= +1
or(p1q3
)= −
(p1r5
)= −
(p1s5
)= +1,
(q3r5
)=(q3s5
)
4
1.1. Congruent and Non-congruent Numbers
Table 1.2: Non-congruent Numbers
Genocchi, 1855 [Gen55] −→ p3, p3q3, 2p5, and 2p5q5
Lagrange, 1974 [Lag75] −→ p1q3 with(p1q3
)= −1
−→ p5q7 with(p5q7
)= −1
−→ 2p3q3
−→ 2p1q5 with(p1q5
)= −1
−→ 2p3q7 with(p3q7
)= −1
−→ p1q3r3 with(p1q3
)= −
(p1r3
)−→ p3q5r7 with
(q5r7
)= −1
−→ p3q7r7 with(p3q7
)= −
(p3r7
)=(q7r7
)−→ 2p1q3r3 with
(p1q3
)= −
(p1r3
)−→ 2p1q5r5 with
(p1q5
)= −
(p1r5
)−→ 2p3q5r7 with
(p3r7
)= −
(q5r7
)−→ 2p5q7r7 with
(p5q7
)= −
(p5r7
)=(q7r7
)Serf, 1991 [Ser91] −→ p5q5r7s7 with(
p5r7
)= −
(p5s7
)= −
(q5r7
)= +1
or −(p5r7
)=(p5s7
)= −
(q5s7
)= +1
or −(p5r7
)= −
(p5s7
)= +1,
(q5r7
)= −
(q5s7
)−→ 2p1q1r3s3 with(
p1q1
)= +1,
(p1r3
)= −
(p1s3
),(q1r3
)= −
(q1s3
)or
(p1q1
)= −1,
(p1r3
)=(p1s3
),(q1r3
)= −
(q1s3
)or
(p1q1
)= −1,
(p1r3
)= −
(p1s3
)
5
1.1. Congruent and Non-congruent Numbers
In his paper “Non-congruent numbers with arbitrarily many prime fac-
tors congruent to 3 modulo 8,” Iskra proved the existence of a new fam-
ily of non-congruent numbers containing arbitrarily many prime factors,
p1, p2, . . . , pt satisfying pi ≡ 3 (mod 8) for all 1 ≤ i ≤ t and(pjpi
)= −1 for
j < i [Isk96]. A thorough discussion of Iskra’s results, including two differ-
ent proofs of his main theorem (see Theorem 4.1), can be found in Chapter
4.
The results presented in this thesis broaden the current understanding
of congruent and non-congruent numbers by generating new families of both
types of these numbers. In Chapter 3, a method is provided for constructing
congruent numbers with three distinct prime factors of the form 8k + 3. A
family of such numbers is given for which the rank of their associated elliptic
curves equals two, the maximal rank for congruent number curves of this
type. These results were published in the journal Integers under the title
“On congruent numbers with three prime factors” [RSY11]. In Chapter 4,
Iskra’s work [Isk96] is discussed and a new, elegant method for generating
families of non-congruent numbers with arbitrarily many prime factors is
presented. This method is then applied to prove the existence of Iskra’s
family of non-congruent numbers. Chapter 5 offers an extension to Iskra’s
work and applies the method presented in Chapter 4 to construct infinitely
many distinct new families of non-congruent numbers with arbitrarily many
prime factors of the form 8k + 3. These results appeared in the paper
“Families of non-congruent numbers with arbitrarily many prime factors,”
which was recently published in the Journal of Number Theory [RSY13].
In Chapter 6, another collection of infinitely many new families of non-
congruent numbers is generated by utilizing the aforementioned method
once again. These numbers are a product of arbitrarily many primes, where
the first prime factor is of the form 8k + 1 and the remaining prime factors
are of the form 8k + 3. Before we prove the results of Chapters 3 through
6, a thorough discussion of the necessary background information must be
provided. In the next section, we recall various algebra and number theory
terminology. In Chapter 2, we describe the link between congruent numbers
and elliptic curves and present an introduction to the theory governing the
6
1.2. Algebra and Number Theory Preliminaries
properties of elliptic curves.
1.2 Algebra and Number Theory Preliminaries
1.2.1 Abstract Algebra
We begin by introducing some basic definitions involving binary algebraic
structures, denoted as 〈G, ∗〉, where G is a set and ∗ is a binary operation
on G. Note that the following definitions, taken from [Fra03], can be found
in most introductory abstract algebra textbooks.
Definition 1.2. A group 〈G, ∗〉 is a set G under a binary operation ∗ that
satisfies the following axioms:
1. (Associativity) For all a, b, c ∈ G, we have (a ∗ b) ∗ c = a ∗ (b ∗ c).
2. (Identity Element, e) There is an element e in G such that for all
g ∈ G, e ∗ g = g ∗ e = g.
3. (Inverse) For each a ∈ G, there exists an element a′ ∈ G such that
a ∗ a′ = a′ ∗ a = e.
Definition 1.3. An abelian group 〈G, ∗〉 is a group G with a commutative
binary operation ∗. This means that a ∗ b = b ∗ a for all a, b ∈ G.
Definition 1.4. Let 〈G, ∗〉 be a group and H be a non-empty subset of G.
Then H is called a subgroup of G if H is closed under the binary operation
∗ and 〈H, ∗〉 satisfies the three group axioms.
Definition 1.5. Let 〈G, ∗〉 and 〈G′, ∗′〉 be binary algebraic structures, where
G and G′ are groups. A map φ of G into G′ is a homomorphism if
φ(x ∗ y) = φ(x) ∗′ φ(y)
for all x, y ∈ G. A homomorphism that is one-to-one is called an injective
homomorphism, which is also known as a monomorphism.
7
1.2. Algebra and Number Theory Preliminaries
Definition 1.6. Let 〈G, ∗〉 and 〈G′, ∗′〉 be binary algebraic structures. An
isomorphism, also known as a bijective homomorphism, of G with G′ is a
one-to-one function φ mapping G onto G′ such that
φ(x ∗ y) = φ(x) ∗′ φ(y)
for all x, y ∈ G. If such a map φ exists, then G and G′ are isomorphic binary
structures, denoted by G ∼= G′.
Definition 1.7. Let G be a group and let a ∈ G. The element a generates
G and is a generator for G if G = {an|n ∈ Z} = 〈a〉. A group G is cyclic if
there exists an element a in G that generates G.
Definition 1.8. A finitely generated abelian group 〈G,+〉 is an abelian
group for which there exist finitely many elements g1, g2, . . . , gn ∈ G such
that every g ∈ G can be written as
g = a1g1 + a2g2 + . . .+ angn,
where a1, a2, . . . , an ∈ Z.
An important theorem which provides complete structural information
about finitely generated abelian groups is the fundamental theorem of finitely
generated abelian groups [Fra03].
Theorem 1.9 (Fundamental Theorem of Finitely Generated Abelian
Groups). Every finitely generated abelian group G is isomorphic to a direct
sum of cyclic groups in the form
G ∼= Zpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνss ⊕ Z⊕ Z⊕ · · · ⊕ Z,
where Z is an infinite cyclic group and Zpνii is a finite cyclic group with
prime-power order for 1 ≤ i ≤ s with i, s ∈ Z. Note that the primes, pi, are
not necessarily distinct and that the νi are positive integers.
8
1.2. Algebra and Number Theory Preliminaries
1.2.2 Linear Algebra
Next we recall some basic concepts and properties from linear algebra. This
information can be found in an introductory linear algebra textbook such
as [KH04].
Definition 1.10. A matrix A = [aij ] is called a square matrix of order n if
the number of rows and the number of columns are both equal to n.
Definition 1.11. Let A be a square matrix of order n.
1. The entries a11, a22, . . . , ann are called the diagonal entries of A.
2. A is said to be a diagonal matrix if all of the entries, except for the
diagonal entries, are zero.
3. A is an upper triangular matrix if all the entries below the diagonal
are zero. Similarly, A is a lower triangular matrix if all the entries
above the diagonal are zero.
Note that a product of upper triangular matrices is also an upper tri-
angular matrix. Similarly, any matrix that is a product of lower triangular
matrices is lower triangular.
Definition 1.12. The identity matrix of order n, denoted by In or I, is a
diagonal matrix whose diagonal entries are all equal to 1.
Definition 1.13. The zero matrix of order n, denoted by 0n or 0, is a
matrix whose entries are all equal to 0.
Definition 1.14. Let A = [aij ] be an m × n matrix. The transpose of A
is the n ×m matrix, denoted by AT , whose j-th column is taken from the
j-th row of A. In other words, [AT ]ij = [A]ji.
Definition 1.15. An n×n square matrix A is said to be invertible if there
exists a square matrix B of the same size such that
AB = In = BA.
The matrix B is called the inverse of A, and is denoted by A−1.
9
1.2. Algebra and Number Theory Preliminaries
Definition 1.16. For an m×n matrix A, the rank of A is defined to be the
maximal number of linearly independent column vectors (or row vectors) of
A. The rank of A is denoted by rank(A).
Definition 1.17. The determinant of a square n×n matrix A, denoted by
det(A), is a real-valued function that satisfies the following three properties:
1. The value of the determinant changes sign if any two rows or columns
within the matrix A are interchanged.
2. The determinant is linear. This means that if A = [a1, a2, . . . , an],
where the aj are column vectors of length n, then
det[a1, a2, . . . , bai+ca′i, . . . , an] = b·det[a1, a2, . . . , an]+c·det[a1, a2, . . . , a
′i, . . . , an].
Note that b and c are scalars and a′i is a column vector of length n.
3. The determinant of the identity matrix is 1, so det(In) = 1.
The determinant satisfies some important properties that can be sum-
marized by the following theorem [KH04, Theorems 2.2, 2.3 & 3.26].
Theorem 1.18. Let A be an n × n square matrix. Then the determinant
satisfies the following properties:
1. If A has two identical rows (or columns), which means that the rows
(or columns) of A form a linearly dependent set, then det(A) = 0.
2. The determinant remains unchanged if a scalar multiple of one row
is added to another row. Similarly, the determinant’s value does not
change when a scalar multiple of one column is added to another col-
umn.
3. The determinant of a triangular matrix is equal to the product of the
diagonal entries.
4. The matrix A is invertible if and only if det(A) 6= 0.
5. det(AT ) = det(A).
10
1.2. Algebra and Number Theory Preliminaries
6. det(A) 6= 0 if and only if rank(A) = n.
For a matrix subdivided into four separate blocks, the following identities
can be applied to compute its determinant [Mey00, p. 467, 475, and 483].
Proposition 1.19. If A and D are square matrices, then
det
([A B
0 D
])= det
([A 0
C D
])= det (A) det (D).
Proposition 1.20. If A and D are square matrices, then
det
([A B
C D
])=
det (A) det(D−CA−1B
), when A−1 exists,
det (D) det(A−BD−1C
), when D−1 exists.
Proposition 1.21. If B is an invertible n× n matrix, and if D and C are
n× k matrices, then
det(B + CDT
)= det(B) det(Ik + DTB−1C).
1.2.3 Number Theory
Now we introduce some terminology that is used in the field of number
theory. It is worthwhile to note that this information, taken from [Ros05],
can be found in most introductory number theory textbooks.
Definition 1.22. The greatest common divisor of two integers a and b,
which are not both 0, is the largest integer that divides both a and b.
Note that the greatest common divisor of a and b is written as gcd(a, b),
and that by definition gcd(0, 0) = 0. For integers a and b, the notation a|bindicates that a divides b and the notation a - b indicates that a does not
divide b.
Definition 1.23. The integers a1, a2, . . . , an are pairwise relatively prime
if, for each pair of integers ai and aj with i 6= j from the set, the greatest
common divisor of ai and aj is 1.
11
1.2. Algebra and Number Theory Preliminaries
Definition 1.24. Let m be a positive integer. If a and b are integers, we
say that a is congruent to b modulo m if m divides (a− b).
If a is congruent to b modulo m, we write this as a ≡ b (mod m), whereas
if a and b are incongruent modulo m, we denote this by a 6≡ b (mod m).
Definition 1.25. A congruence class modulo m is a set of integers that are
mutually congruent modulo m.
For instance, there are three congruence classes modulo 3; one class
contains all integers congruent to 0 modulo 3, another class contains all
integers congruent to 1 modulo 3, and the third class contains all integers
congruent to 2 modulo 3.
An important theorem which provides a method for solving systems of
linear congruences is the Chinese remainder theorem [Ros05, Theorem 4.12].
Theorem 1.26 (Chinese Remainder Theorem). If m1,m2, . . . ,ms are
pairwise relatively prime positive integers, then the system of congruences
x ≡ a1 (modm1),
x ≡ a2 (modm2),
...
x ≡ as (modms),
has a unique solution modulo M = m1m2 · · ·ms.
Next, we discuss quadratic residues, quadratic nonresidues, and Legen-
dre symbols.
Definition 1.27. If m is a positive integer, we say that the integer b is a
quadratic residue of m if gcd(b,m) = 1 and the congruence x2 ≡ b (mod m)
has a solution. If this congruence does not have a solution, then we say that
b is a quadratic nonresidue of m.
As an example, consider the number 5 and try to determine its quadratic
residues. To do this, we must compute the squares of the integers 1, 2, 3, and
12
1.2. Algebra and Number Theory Preliminaries
4. We find that 12 ≡ 42 ≡ 1 (mod 5) and 22 ≡ 32 ≡ 4 (mod 5). Therefore, 1
and 4 are quadratic residues of 5, whereas 2 and 3 are quadratic nonresidues
of 5.
Definition 1.28. Let p be an odd prime and a be an integer not divisible
by p. The Legendre symbol(ap
)is defined by
(a
p
)=
+1 if a is a quadratic residue of p,
−1 if a is a quadratic nonresidue of p.
For our previous example, the Legendre symbols for p = 5 and a = 1, 2,
3, and 4 are (1
5
)=
(4
5
)= +1 and
(2
5
)=
(3
5
)= −1.
Some important properties of Legendre symbols can be summarized by
the following theorem [Ros05, Theorems 11.4, 11.5, 11.6 & 11.7].
Theorem 1.29 (Properties of Legendre Symbols).
1. Let p be an odd prime and a and b be integers not divisible by p. Then
(i) if a ≡ b (mod p), then
(a
p
)=
(b
p
).
(ii)
(a
p
)(b
p
)=
(ab
p
).
(iii)
(a2
p
)= +1.
2. The Law of Quadratic Reciprocity.
If p and q are distinct odd primes, then(q
p
)=
(p
q
)(−1)
p−12· q−1
2 .
3. The First and Second Supplements to the Law of Quadratic
Reciprocity.
If p is an odd prime, then
13
1.2. Algebra and Number Theory Preliminaries
(i)
(−1
p
)=
+1 if p ≡ 1 (mod 4),
−1 if p ≡ 3 (mod 4).
(ii)
(2
p
)=
+1 if p ≡ 1, 7 (mod 8),
−1 if p ≡ 3, 5 (mod 8).
Next, we state Dirichlet’s theorem on primes in arithmetic progression
[Ros05, Theorem 3.3].
Theorem 1.30 (Dirichlet’s Theorem on Primes in Arithmetic Pro-
gression). Suppose that a and b are relatively prime positive integers. Then
the arithmetic progression an + b where n ∈ N+ contains infinitely many
primes.
Finally, we define a couple of terms that are commonly discussed when
studying p-adic number theory [Isk98, Ogg09].
Definition 1.31. Let p be a prime number and n be a non-zero rational
number. If n = pαn′, where n′ is a rational number whose prime-power
factorization does not contain p, then the p-adic valuation, vp(n), of the
non-zero rational number n is
vp(n) = α.
Note that vp(0) is defined to be equal to ∞.
Definition 1.32. The set of all places of the field Q is the set of primes in
Q, denoted by MQ = {∞, 2, 3, . . .}.
Having gained an understanding of some of the basic background infor-
mation in the fields of algebra and number theory, we are now ready to
discuss elliptic curves and the theory that governs their properties.
14
Chapter 2
Congruent Numbers and
Elliptic Curves
2.1 Introduction to Elliptic Curves
An elliptic curve is an algebraic curve which has a cubic equation of the
form
y2 = x3 + ax2 + bx+ c,
where a, b, c ∈ Q [ST92]. For such a cubic curve to be considered an elliptic
curve, it must have distinct roots, or equivalently, its discriminant, given by
the equation
D = −4a3c+ a2b2 + 18abc− 4b3 − 27c2,
must be non-zero. It is important to note that the above elliptic curve is
written in Weierstrass normal form and that any cubic equation with a
rational point can be converted to this simple form [ST92].
Since elliptic curves are cubic polynomials with distinct roots, they must
have either three real roots, or one real root and a pair of complex conjugate
roots. If a cubic curve has a double or triple root, it is not an elliptic curve.
Figures 2.1 and 2.2 provide examples of elliptic curves, whereas Figures 2.3
and 2.4 depict cubic curves with double and triple roots that are not elliptic
curves. Note that these plots were created with the aid of MapleTM13.
15
2.2. The Group Law and Mordell’s Theorem
Figure 2.1: Elliptic curvewith three real roots, y2 =(x− 1)(x− 2)(x+ 1).
Figure 2.2: Elliptic curvewith one real root, y2 =(x+ 2)(x2 − 2x+ 3).
Figure 2.3: Cubic curvewith a double root, y2 =x2(x+ 2).
Figure 2.4: Cubic curvewith a triple root, y2 =(x+ 1)3.
An important and well-known fact about elliptic curves is that the ra-
tional points on a given curve E form an abelian group, denoted by E(Q).
To discuss the structure of E(Q), we must first define the group law that
governs the set of rational points on our elliptic curve.
2.2 The Group Law and Mordell’s Theorem
Before we describe the group law, we must develop an understanding of
how points on elliptic curves are related to one another. Given two points
16
2.2. The Group Law and Mordell’s Theorem
on an elliptic curve, it is possible to find a third point on the curve by
applying a composition law known as the chord and tangent method [Hus04,
Joh09, SZ03, ST92, Sil09]. This technique allows us to map two points
P and Q on our elliptic curve to a third point P ∗ Q also on the curve.
Note that ∗ denotes the binary operator for the composition law. If P
and Q are distinct points then P ∗ Q is defined to be the third point of
intersection of the elliptic curve with the line passing through the points P
andQ [Hus04, Joh09, SZ03, ST92, Sil09]. Specifically, if P andQ are rational
points, then the line connecting the two points is a rational line. Therefore,
the point P ∗Q, which lies on the line, must also be rational [Hus04]. Figure
2.5 provides a geometric interpretation of this process for two distinct points
on the elliptic curve y2 = (x+ 2)(x2− 2x+ 3). If we only know one rational
point P on our elliptic curve, we can apply the same method to find another
rational point P ∗P on the curve [Hus04, Joh09, SZ03, ST92, Sil09]. In this
case, the line that passes through P lies tangent to the curve at that point;
this scenario is illustrated in Figure 2.6 for the curve y2 = (x+ 2)(x2− 2x+
3). Thus, the chord and tangent method provides us with a technique for
generating many rational points on a given elliptic curve.
Figure 2.5: The chord andtangent method applied to dis-tinct points P and Q on thecurve y2 = (x+2)(x2−2x+3).
Figure 2.6: The chord andtangent method applied to thepoint P on the curve y2 =(x+ 2)(x2 − 2x+ 3).
The set of rational points obtained by applying the chord and tangent
17
2.2. The Group Law and Mordell’s Theorem
method can be made into a group by introducing the concept of the point at
infinity. This point, denoted by O, is a rational point on every elliptic curve
and is the identity element in our group. By definition, the point at infinity
is an inflection point on our elliptic curve, and the tangent line to the curve
at that point is the line at infinity [ST92]. It is a well-known fact that a
line meets an elliptic curve at exactly three points [SZ03, ST92]; therefore,
we know that the line at infinity intersects the curve at the point O three
times, a vertical line intersects the curve at two points in the xy-plane and
once at the point O, and a non-vertical line intersects the curve at three
points in the xy-plane [ST92]. Consider a specific vertical line that meets
the elliptic curve E at the point P . By definition, the vertical line must also
pass through the point at infinity, so the third point of intersection between
the curve E and the line must be P ∗O = O ∗P [Joh09, ST92, Sil09] . Thus,
P ∗ O is the reflection of P about the x -axis [Joh09, ST92].
Using this information, we can now define the group law associated with
rational points on elliptic curves. Let + be the binary operator for the group
law and let P and Q be rational points on the elliptic curve E. Consider
the line through the points P and Q and find the third intersection point,
P ∗Q, of the line with the curve E. Next, draw a vertical line through the
point P ∗ Q. This line also meets the curve at the point at infinity, so the
third point of intersection between E and the vertical line is O ∗ (P ∗ Q);
we define this point to be equal to P + Q [Joh09, ST92, Sil09]. Therefore,
P + Q is the reflection of P ∗ Q about the x -axis. Figure 2.7 provides a
visual depiction of the group law being applied to the distinct points P and
Q on the curve y2 = (x+ 2)(x2 − 2x+ 3).
Let P , Q, and R be rational points on the elliptic curve E. The group
law operator + satisfies the following properties [Joh09, SZ03, ST92, Sil09]:
Commutative: P +Q = Q+ P for all P,Q ∈ E.
Closure: If P,Q ∈ E, then P +Q ∈ E.
Associative: (P +Q) +R = P + (Q+R) for all P,Q,R ∈ E.
Identity Element O: P + O = O + P = P for all P ∈ E.
18
2.2. The Group Law and Mordell’s Theorem
Inverse: P + (−P ) = (−P ) + P = O for all P ∈ E.
Note that −P = O ∗P , which means that it is the reflection of P about the
x -axis. Together, these five properties imply that E(Q) is an abelian group
under the binary operation +. A proof of this fact can be found in [Sil09].
Figure 2.7: The group law applied to points P and Q on the curve y2 =(x+ 2)(x2 − 2x+ 3).
This leads us to an important theorem by Mordell that offers a detailed
description of the structure of the group of rational points [Hem06, Hus04,
Joh09, ST92, Sil09].
Theorem 2.1 (Mordell’s Theorem). Let E be an elliptic curve over the
field of rational numbers. The group of rational points, E(Q), is a finitely
generated abelian group.
Proof. See Chapter 6 of [Hus04], Chapter III of [ST92], or Chapter VIII of
[Sil09].
As a result of the Mordell’s theorem, we can apply the fundamental
theorem of finitely generated abelian groups (see Theorem 1.9) to the group
of rational points. This allows us to write it as the following direct sum of
cyclic groups:
E(Q) ∼= Zpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνss ⊕ Z⊕ Z⊕ · · · ⊕ Z,
19
2.3. The Torsion Subgroup
where Z is an infinite cyclic group and Zpνii is a finite cyclic group with
prime-power order for 1 ≤ i ≤ s with i, s ∈ Z [Hem06, Hus04, Joh09, SZ03,
ST92, Sil09]. An equivalent way of writing the group of rational points is
E(Q) ∼= T ⊕F ,
where T ∼= Zpν11 ⊕ Zpν22 ⊕ · · · ⊕ Zpνss is the torsion part of E(Q) and F ∼=Z⊕Z⊕ · · · ⊕Z is the free part of E(Q). The number of copies of Z in F is
denoted by r and is called the rank, or the Mordell-Weil rank, of the elliptic
curve. We formally define the rank as follows [Joh09].
Definition 2.2. Let E(Q) be the group of rational points on the elliptic
curve E. The number of generators with infinite order in E(Q) is the rank
of the curve E denoted by r.
2.3 The Torsion Subgroup
In order to define the torsion subgroup, we must first explain what it means
for a point in E(Q) to have a particular order [ST92].
Definition 2.3. An element P = (x, y) in E(Q) is said to have order m if
mP = P + P + · · ·+ P︸ ︷︷ ︸m
= O,
but m′P 6= O for all 1 ≤ m′ < m, where m, m′ ∈ Z and O is the identity
element. If such an integer m exists, then point P is said to have finite
order ; otherwise P is said to have infinite order.
Note that the identity element, O, has order one and rational points,
(x, y), with y = 0 have order two [ST92]. The set of all points of finite order
in E(Q) forms a subgroup known as the torsion subgroup [Hus04, ST92,
Sil09].
Definition 2.4. The torsion subgroup, T , of E(Q) is the group consisting
of all the rational points of finite order on the elliptic curve E.
20
2.3. The Torsion Subgroup
Since the identity element, O, is a point on every elliptic curve, the
torsion subgroup always contains at least one rational point of finite order.
The rest of the points in the torsion subgroup can be found by applying the
following theorem [Joh09, ST92, Sil09].
Theorem 2.5 (Nagell-Lutz Theorem). Let y2 = f(x) = x3+ax2+bx+c
be an elliptic curve with a, b, c ∈ Z and let D be the the discriminant of f(x)
so
D = −4a3c+ a2b2 + 18abc− 4b3 − 27c3.
Let P = (x, y) be a rational point of finite order. Then x and y are integers
and either y = 0 or else y2 divides D.
Proof. See Chapter II of [ST92] or Chapter VIII of [Sil09].
Notice that this theorem is not an if-and-only-if statement. As a result,
it is possible to have points on the curve that are not of finite order, but
that do have integer coordinates with y2 dividing D [ST92]. To determine
whether a given point, P = (x, y) 6= O with y 6= 0, is finite, it is useful to
consider the duplication formula for the x-coordinate of P [ST92]:
x(2P ) =x4 − 2bx2 − 8cx+ b2 − 4ac
4y2.
If P = (x, y) 6= O is a rational point of finite order, then x and y are
integers and mP = O for some m ∈ Z. It follows that 2P also must have
finite order, so the x-coordinate of 2P , x(2P ), should have an integer value
too. Therefore, if we compute x(2P ) and find that it does not equal an
integer, we deduce that P is not a point of finite order [ST92]. Once all
of the points of finite order have been found, the following theorem can be
applied to determine the exact form of the torsion subgroup [Hem06, Hus04,
SZ03, ST92, Sil09].
Theorem 2.6 (Mazur’s Theorem). Let E be an elliptic curve defined over
Q. Then the torsion subgroup, T , of the group of rational points, E(Q), is
one of the following fifteen groups:
21
2.4. The Method of 2-Descent
1. A cyclic group of order N, ZN , with 1 ≤ N ≤ 10 or N = 12.
2. The product of a cyclic group of order two and a cyclic group of order
2N , Z2 ⊕ Z2N , with 1 ≤ N ≤ 4.
Proof. See [Maz77] or [Maz78].
2.4 The Method of 2-Descent
The method of 2-descent is an algorithm that is used for computing the
Mordell-Weil rank of an elliptic curve. Let us define the elliptic curve E:
y2 = x3 + ax2 + bx, where a, b ∈ Z and (x, y) is a rational point. In
addition, let Γ be the group of rational points on E. In order to compute
the rank of E, we must simultaneously consider another curve denoted by
E: y2 = x3 + ax2 + bx, where a = −2a, b = a2 − 4b, and (x, y) is a rational
point. Let Γ be the group of rational points on E [Joh09, ST92]. Define
Q∗ to be the multiplicative group of non-zero rational numbers, Q∗2 to be
the subgroup of squares in Q∗, and Q∗ = Q∗/Q∗2 to be the quotient group
consisting of square-free, non-zero rational numbers [ST92]. This allows the
following homomorphisms to be defined:
α : Γ −→ Q∗ and α : Γ −→ Q∗,
where
α(P ) =
1 (mod Q∗2) if P = O
b (mod Q∗2) if P = (0, 0)
x (mod Q∗2) if P = (x, y) with x 6= 0
and
α(P ) =
1 (mod Q∗2) if P = O
b (mod Q∗2) if P = (0, 0)
x (mod Q∗2) if P = (x, y) with x 6= 0,
for P = (x, y) ∈ Γ and P = (x, y) ∈ Γ [Joh09, ST92]. Furthermore, α(Γ) is a
subset of the square-free divisors of b, and α(Γ) is a subset of the square-free
22
2.4. The Method of 2-Descent
divisors of b [ST92]. The rank of E can be computed by using the equation
2r =|α(Γ)| |α(Γ)|
4,
where r is the rank of E, |α(Γ)| is the cardinality of α(Γ), and |α(Γ)| is
the cardinality of α(Γ) [Joh09, ST92]. Therefore, to calculate the rank,
we must first determine the elements in α(Γ) and α(Γ). Clearly, 1 and b
modulo Q∗2 are in α(Γ). To determine whether or not α(Γ) contains any
additional elements, we must consider all of the possible factors, b1, of b
with b1 6≡ 1, b (mod Q∗2), and b = b1b2. If the equation
N2 = b1M4 + aM2e2 + b2e
4, (2.1)
which is referred to as a torsor, has a solution (N,M, e) ∈ Z3 with M 6= 0,
e 6= 0, and gcd(M, e) = gcd(N, e) = gcd(b1, e) = gcd(b2,M) = gcd(M,N) =
1, then b1 modulo Q∗2 is an element of α(Γ) [Joh09, ST92]. Each equation
that we solve, produces a corresponding point on our elliptic curve of the
form
(x, y) =
(b1M
2
e2,b1MN
e3
)(2.2)
[ST92]. Similarly, working modulo Q∗2, α(Γ) contains 1, b, and all divisors
b1 of b satisfying the torsor equation
N2 = b1M4 + aM2e2 + b2e
4 (2.3)
for some (N,M, e) ∈ Z3 with M 6= 0 and e 6= 0. Note that a = −2a,
b = b1 b2, and b1 6≡ 1, b (mod Q∗2). In addition, we also require that the
gcd(M, e) = gcd(N, e) = gcd(b1, e) = gcd(b2,M) = gcd(M,N) = 1 [Joh09,
ST92].
Thus, the method of 2-descent provides us with a systematic procedure
for computing the Mordell-Weil rank of a given elliptic curve. However, if
b and b have many square-free divisors, then carrying out the method of 2-
descent can be a lengthy and tedious process. In addition, finding solutions
(N,M, e) ∈ Z3 that satisfy Equations (2.1) and (2.3) can be a challenging
23
2.5. The Relationship Between Elliptic Curves and Congruent Numbers
task, as there is no known method for determining whether equations of this
form are solvable [Joh09, ST92].
Nevertheless, if we can find a solution that satisfies a torsor equation,
we are guaranteed that there is corresponding rational point on our elliptic
curve. It is worthwhile to note that the rank of an elliptic curve is related
to the number of independent rational points of infinite order in the curve’s
group of rational points. An important theorem that can be used to show
that the points on an elliptic curve are independent is Silverman’s special-
ization theorem [Sil09, Theorem 20.3].
Theorem 2.7 (Silverman’s Specialization Theorem). Let C/K be a
curve and let E be an elliptic curve defined over the function field K(C)
such that j(E) 6∈ K, where j(E) is the j-invariant of the elliptic curve E.
Then the specialization map
σt : E(K(C))→ Et
is (well-defined) and injective for all but finitely many points t ∈ C(K).
(More generally, it is injective for all but finitely many points of⋃C(L),
where the union is over all fields L/K whose degree is bounded by a fixed
number.)
We can summarize this theorem as follows:
Suppose that there exists a family of elliptic curves, y2 = x3 − tx, in terms
of a parameter t and suppose that there is a finite set of points on these
curves also given in terms of t. If the points are independent in the group
of rational points for even a single value of t, then they are independent for
all rational values of t with at most finitely many possible exceptions.
2.5 The Relationship Between Elliptic Curves
and Congruent Numbers
Recall that in Chapter 1, we defined congruent numbers and non-congruent
numbers as follows:
24
2.5. The Relationship Between Elliptic Curves and Congruent Numbers
Definition 2.8. A positive integer n is a congruent number if it is equal to
the area of a right triangle with rational sides. Otherwise n is said to be a
non-congruent number.
Congruent numbers can be defined in an equivalent way by using elliptic
curves [DJS09, Hem06, Joh09, Kob93, NW93, RSY11, RSY13].
Lemma 2.9. A positive integer n is a congruent number if and only if the
rank of the elliptic curve
En: y2 = x(x2 − n2)
is positive. Otherwise, n is a non-congruent number. In other words, n is a
non-congruent number if and only if the rank of En is zero.
Note that the proof of the first if-and-only-if statement in Lemma 2.9 can
be found in Section 10 of Chapter 2 in [Hem06], or in Section 9 of Chapter
I in [Kob93].
By inspection, it is clear that the points (0, 0), (n, 0), and (−n, 0) are on
the curve y2 = x(x2−n2). These three points all have 0 as their y-coordinate,
so they are points of order two. Since O is a point of order one that lies on
every elliptic curve, we know that the torsion subgroup of En must contain
at least four elements. As it turns out, O, (0, 0), (n, 0), and (−n, 0) are the
only points of finite order on En; a proof of this fact can be found in Section
7 of Chapter 2 in [Hem06]. Since the torsion subgroup of En is composed
of one element of order one and three elements of order two, by Mazur’s
theorem (Theorem 2.6) we deduce that T ∼= Z2 ⊕ Z2. Thus, the group of
rational points on our elliptic curve En is isomorphic to Z2⊕Z2⊕Zr, where
r is the curve’s rank [Ser91]. Next we focus our attention on computing the
rank of elliptic curves En for various values of n, as the rank will allow us to
determine whether n is a congruent number or a non-congruent number. To
do this, we apply a technique known as the method of complete 2-descent.
25
2.6. The Method of Complete 2-Descent
2.6 The Method of Complete 2-Descent
The method of complete 2-descent is an algorithm that is used for computing
the rank of an elliptic curve. It considers pairs of quadratic equations and
determines whether or not they are solvable. For elliptic curves given by the
general Weierstrass equation y2 = (x−e1)(x−e2)(x−e3) with e1, e2, e3 ∈ Kwhere K is a number field, the process involved in carrying out the method of
complete 2-decent is described in Proposition 1.4 on page 315 of [Sil09]. For
curves of the form En : y2 = x(x2 − n2), the procedure can be summarized
by the following theorem [Ser91, Theorem 3.1].
Theorem 2.10 (Complete 2-Descent). Let
n = 2εp1p2 · · · pk
be a square-free positive integer with p1, p2, . . . , pk being primes that are not
equal to two, ε ∈ {0, 1}, and k ∈ N+. Let En be the elliptic curve over Qdefined by the equation
En : y2 = x(x2 − n2) = x(x− n)(x+ n),
and
S = {∞, 2, p1, . . . , pk}
be a finite subset of MQ, the set of all places of Q. In addition, define
Q(S, 2) := {c ∈ Q∗/Q∗2| vp(c) ≡ 0 (mod 2) ∀ p ∈MQ\S},
where vp(c) is the p-adic valuation of c. Then there exists an injective ho-
momorphism
b : En(Q)/2En(Q) ↪→ Q(S, 2)×Q(S, 2)
defined by
26
2.6. The Method of Complete 2-Descent
P = (x, y) 7→
(1, 1) if P = O
(−1,−n) if P = (0, 0)
(n, 2) if P = (n, 0)
(x, x− n) if P = (x, y) 6= O, (0, 0), (n, 0).
If (b1, b2) ∈ Q(S, 2)×Q(S, 2)\{(1, 1), (−1,−n), (n, 2)}, then (b1, b2) ∈ image(b)
if and only if there exist (z1, z2, z3) ∈ Q∗ × Q∗ × Q such that the following
two equations simultaneously hold:
b1z21 − b2z22 = n, (2.4)
b1z21 − b1b2z23 = −n. (2.5)
In this case, (b1, b2) = b(P ) for
P = (b1z21 , b1b2z1z2z3) = (b2z
22 + n, b1b2z1z2z3).
Recall that En(Q) ∼= Z2⊕Z2⊕Zr, so En(Q) can be described as a set of
(r+ 2)-tuples. In addition, 2En(Q) is also a set of (r+ 2)-tuples. Note that
an arbitrary (r + 2)-tuple in En(Q) can be written as (a1, a2, a3, . . . , ar+2),
where a1, a2 ∈ Z2 and ai ∈ Z for all 3 ≤ i ≤ (r + 2). In 2En(Q), this
(r+2)-tuple becomes (2a1, 2a2, 2a3, . . . , 2ar+2). However, since the first two
components of the (r + 2)-tuple are in Z2, they reduce to zero. Therefore,
when we form the quotient group En(Q)/2En(Q), the first two components
remain unchanged and are isomorphic to Z2. Each of the other r components
is isomorphic to Z/2Z, which is equivalent to Z2. Thus,
En(Q)/2En(Q) ∼= (Z2)r+2,
so since Z2 is a group of order two, the total order of the above quotient
group is 2r+2 [Ser91]. This means that there are 2r+2 rational points on our
elliptic curve En: y2 = x(x2 − n2).In order to compute the rank, r, of the curve En, we need to recall from
Section 2.5 that the torsion subgroup of En(Q) contains four rational points
of finite order. By applying the homomorphism defined in Theorem 2.10,
27
2.6. The Method of Complete 2-Descent
we know that these four points, P = O, (0, 0), (n, 0), and (−n, 0), are re-
spectively mapped to (1, 1), (−1,−n), (n, 2), and (−n,−2n). Now we must
consider the set of pairs (b1, b2) 6∈ {(1, 1), (−1,−n), (n, 2), (−n,−2n)} for
which Equations (2.4) and (2.5) simultaneously have a solution. Determin-
ing whether or not a given pair, (b1, b2), belongs to this set can sometimes
be a difficult task. Therefore, we define B to be the upper bound for the
number pairs (b1, b2) that simultaneously solve Equations (2.4) and (2.5).
This enables us to use the following inequality to bound the rank of En
[Ser91]:
2r+2 ≤ B + 4
⇐⇒ 2r ≤ B + 4
4
⇐⇒ r ≤ log2
(B + 4
4
). (2.6)
If we are able to conclusively determine the solvability of Equations (2.4)
and (2.5) for all pairs, then the above inequality becomes a strict equality.
Clearly, the more pairs, (b1, b2), we find for which our system of two equa-
tions has a solution, the higher the rank our elliptic curve En is guaranteed
to have.
Note that for n to be a non-congruent number, we require that r =
0. Therefore, we need the bound, B, for the number of pairs (b1, b2) to
be equal to zero as well. This means that we need to show that our
system of two equations does not have a solution for any pair (b1, b2) 6∈{(1, 1), (−1,−n), (n, 2), (−n,−2n)}. To do this, it is beneficial to make use
of the following theorem, since it reduces the number of cases that need to
be considered by providing a list of criteria for which Equations (2.4) and
(2.5) cannot simultaneously be solved [Ser91, Theorem 3.3].
Theorem 2.11 (Unsolvability Conditions). Let
n = 2εp1p2 · · · pk
be a square-free positive integer with p1, p2, . . . , pk being primes that are not
28
2.6. The Method of Complete 2-Descent
equal to two, ε ∈ {0, 1}, and k ∈ N+. Define
R := {±2εpε11 pε22 · · · p
εkk |ε, ε1, ε2, . . . , εk ∈ {0, 1}}
and let (b1, b2) ∈ R × R. The system of equations given by Equations (2.4)
and (2.5) has no solution (z1, z2, z3) ∈ Q∗ ×Q∗ ×Q in the following cases:
1. b1 · b2 < 0,
2. 2 - n and 2|b1.
Once again, similar to the method of 2-descent, the method of complete
2-descent can be a long and tedious process to execute. In Section 4.1 of
Chapter 4, the method of complete 2-descent is applied to prove a theo-
rem of Iskra (see Theorem 4.1) that generates a family of non-congruent
numbers with arbitrarily many prime factors [Isk96]. A second approach
for proving Iskra’s theorem is also presented in Chapter 4. Unlike the
method of complete 2-descent using quadratic equations which involves a
series of lengthy and complex calculations, this new technique that I de-
veloped and describe in my paper [RSY13] offers a simple and elegant ap-
proach for determining whether a given square-free positive integer is non-
congruent. This method uses linear algebra in conjunction with a result of
Monsky [DJS09, HB94, Mon90] to compute the 2-Selmer rank of a congru-
ent number elliptic curve. Recall that the rank obtained by carrying out a
2-descent is called the Mordell-Weil rank. It is a known fact that an elliptic
curve’s Mordell-Weil rank must be less than or equal to its 2-Selmer rank
[DJS09, HB94, Mon90, Sil09]. Therefore, because congruent numbers have
elliptic curves with a positive Mordell-Weil rank, if a curve is found to have
a 2-Selmer rank of zero, it must correspond to a non-congruent number. For
a thorough explanation of the theory governing the relationship between the
Mordell-Weil rank and the 2-Selmer rank of an elliptic curve, see Chapter
X of [Sil09].
29
2.7. Monsky’s Formula for the 2-Selmer Rank
2.7 Monsky’s Formula for the 2-Selmer Rank
In order to bound the Mordell-Weil rank of the elliptic curve En : y2 =
x(x2−n2), we compute the curve’s 2-Selmer rank, s(n). To do this, we utilize
Monsky’s formula for the 2-Selmer rank given by Equation (2.7) [DJS09,
HB94, Mon90].
Let n be a square-free positive integer with odd prime factors P1, P2, . . . , Pt.
We define diagonal t×t matrices Dl = [di] for l ∈ {−2,−1, 2}, and the square
t× t matrix A = [aij ] by
dii =
0, if
(lPi
)= +1,
1, if(lPi
)= −1,
and
aij =
0, if
(PjPi
)= +1, j 6= i,
1, if(PjPi
)= −1, j 6= i,
aii =∑j:j 6=i
aij .
Then
s(n) =
{2t− rankF2(Mo), if n = P1P2 · · ·Pt,2t− rankF2(Me), if n = 2P1P2 · · ·Pt,
(2.7)
where Mo and Me are the 2t× 2t matrices:
Mo =
A + D2 D2
D2 A + D−2
, Me =
D2 A + D2
AT + D2 D−1
. (2.8)
We use the fundamental inequality
r(n) ≤ s(n), (2.9)
where r(n) is the Mordell-Weil rank of En.
Note that the inequality in Equation (2.9) is particularly useful for gen-
erating families of non-congruent numbers. For a given elliptic curve En, if
30
2.7. Monsky’s Formula for the 2-Selmer Rank
Monsky’s formula yields a 2-Selmer rank equal to zero, then the inequality
implies that the curve’s Mordell-Weil rank must also be zero. Hence, n is a
non-congruent number. We will use this technique to prove Iskra’s theorem
in Chapter 4, and to generate infinitely many new families of non-congruent
numbers in Chapters 5 and 6.
31
Chapter 3
A Family of Congruent
Numbers with Three Prime
Factors
The purpose of this chapter is to provide a method for constructing congru-
ent numbers with three prime factors of the form 8k + 3. A family of such
numbers is given for which the Mordell-Weil rank of their associated elliptic
curves equals two, the maximal rank for a congruent number curve of this
type [RSY11].
Recall that from Table 1.2 in Chapter 1, we know that p3 and p3q3
are non-congruent numbers. In addition to this, Kida noticed that 1419 =
3 · 11 · 43 is the only congruent number less that 4500 of the form p3q3r3
and that quite often a 2-descent shows that a number of the form p3q3r3 is
non-congruent [Kid93]. Other congruent numbers p3q3r3 less than 10, 000
include 4587 = 3·11·139, 4731 = 3·19·83, 6963 = 3·11·211, 7611 = 3·43·59,
and 9339 = 3 · 11 · 283 [RSY11]. The Magma code in Appendix A provides
verification that these five numbers are congruent. Our goal is to generate
a family of congruent numbers n = p3q3r3 for which we can prove that the
Mordell-Weil rank of
y2 = x(x2 − n2) (3.1)
is equal to two. We obtain this family by specializing a larger family used to
generate congruent numbers p3q3r3. Both of these families are conjecturally
infinite.
In Section 3.1, we state the main theorem of this chapter, give our
method of construction for congruent numbers p3q3r3, and provide the back-
32
3.1. Preliminary Results
ground material necessary for the proof of our theorem. In Section 3.2, we
prove our main theorem.
3.1 Preliminary Results
We begin by presenting the central theorem in this chapter.
Theorem 3.1. Suppose that the prime numbers q and r have the form
q = 3u4 + 3v4 − 2u2v2,
r = 3u4 + 3v4 + 2u2v2,
for non-zero integers u and v. Set n = 3qr. Then q ≡ r ≡ 3 (mod 8), n is a
congruent number, and the elliptic curve given by Equation (3.1) has a rank
of two.
Since the definition of a congruent integer can be immediately extended
to rational numbers, we can give the following lemma.
Lemma 3.2. Let v be a rational number with v /∈ (−∞,−1] ∪ [0, 1]. Then
v(v − 1)(v + 1) (3.2)
is a congruent number.
Proof. The restriction on v ensures that v(v − 1)(v + 1) is positive. If v is
an integer, then the congruent number v(v − 1)(v + 1) is a special case of
a formula in [Alt80]. It is sufficient to note that if n = v(v − 1)(v + 1) is a
rational number, then the elliptic curve given by (3.1) has the non-torsion
point
(x, y) =(−v(v − 1)2,−2v2(v − 1)2
).
This point is obtained by solving the torsor given by Equation (2.1):
N2 = b1M4 + aM2e2 + b2e
4.
33
3.1. Preliminary Results
Notice that for our elliptic curve y2 = x3 − n2x, we have a = 0 and b =
−n2 = b1b2. Therefore, the torsor reduces to
N2 = b1M4 + b2e
4.
By choosing b1 = −v(v − 1)2 and b2 = −n2
b1= v(v + 1)2, the torsor becomes
N2 = −v(v − 1)2M4 + v(v + 1)2e4.
Substituting M = 1 and e = 1 into the above equation and simplifying
yields
N2 = 4v2.
Hence N = ±2v, so by Equation (2.2), we know that the point
(x, y) =
(b1M
2
e2,b1MN
e3
)=(−v(v − 1)2,−2v2(v − 1)2
)(3.3)
is on our curve. Note that if we choose N = −2v, then we obtain the point(−v(v − 1)2, 2v2(v − 1)2
), which is simply the inverse of the point given by
Equation (3.3). Recall that elliptic curves of the form y2 = x3−n2x have the
following four torsion points: O, (0, 0), (n, 0), and (−n, 0). Clearly, the point
given by Equation (3.3) does not correspond to any of these four points, as
for v /∈ (−∞− 1] ∪ [0, 1], the y-coordinate of the point in Equation (3.3) is
not equal to zero. Thus,(−v(v − 1)2,−2v2(v − 1)2
)is a non-torsion point
on the curve (3.1). This indicates that the rank of the curve must be at
least one, so v(v − 1)(v + 1) is a congruent number.
Lemma 3.3. Suppose that the prime numbers p3, q3, and r3 satisfy
q3 = p3a2 − 16b2,
r3 = p3a2 + 16b2,
for integers a and b. Then n = p3q3r3 is a congruent number.
34
3.1. Preliminary Results
Proof. Put v = p3a2/16b2 into Equation (3.2) to give the congruent number
p3a2/16b2(p3a
2/16b2 − 1)(p3a2/16b2 + 1).
This number is positive if we impose the restrictions stated in Lemma 3.2.
Since congruent numbers scaled by squares are still congruent, we multiply
by 212b6/a2 to obtain the stated congruent number p3q3r3.
Lemma 3.4. If
n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) (3.4)
for a rational number z 6= 0,±1, then the rank of the elliptic curve given by
Equation (3.1) is at least two with at most finitely many exceptions.
Proof. To obtain the formula for n stated in (3.4), we begin by considering
Equation (3.2). We would like to solve the torsor given by Equation (2.1)
corresponding to the elliptic curve y2 = x(x2−n2) with n = v(v−1)(v+ 1).
For our congruent number elliptic curve, we have a = 0 and b = −n2 = b1b2,
so the torsor reduces to
N2 = b1M4 + b2e
4.
We choose b1 = v(v − 1)(v + 1)2, so b2 = −n2
b1= −v(v − 1) and the torsor
becomes
N2 = v(v − 1)(v + 1)2M4 − v(v − 1)e4.
Substituting M = 1 and e = 1 into the above equation and simplifying
yields
N2 = v2(v − 1)(v + 2).
Next we make a change of variable and set N = w ·v. This causes the above
equation to reduce to
w2 = (v − 1)(v + 2).
Now we would like to make a substitution for v such that (v − 1)(v + 2)
35
3.1. Preliminary Results
becomes a perfect square. By using Maple’s parametrization command, we
deduce that v =(−2−t22t−1
)transforms the right-hand of the above equation
into a square. The Maple code used to carry out this computation can be
found in Appendix B. Substituting this value for v into Equation (3.2) yields
v(v − 1)(v + 1) =−(2 + t2)(3 + t2 − 2t)(t+ 1)2
(2t− 1)3. (3.5)
We want this number to have the form 3q3r3. As a result, we must make a
substitution for t that produces a factor of 3 and changes the denominator
in Equation (3.5) into a perfect square. Substituting t = −3z2+12 into (3.5)
yields the desired result:
3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2)(z2 − 1)2
64z6.
Scaling this equation by squares gives us
3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2),
which is Equation (3.4). Thus, to summarize, we obtain Equation (3.4) by
substituting
v =3z4 − 2z2 + 3
4z2
into Equation (3.2) and then scaling by a factor of
(z2 − 1)2
64z6
to remove the squares. Note that the restriction z 6= 0,±1 ensures that
v > 1.
Our next step is to verify that n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) is
a congruent number. To do this, we will show that for this value of n, the
elliptic curve (3.1) over Q(z) possesses the two points
(x1, y1) =(−9(3 + 3z4 − 2z2
)(z2 − 1)2, 36(3 + 3z4 − 2z2)2z(z2 − 1)) (3.6)
36
3.1. Preliminary Results
and
(x2, y2) =
(3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2)
4z2,
9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1)
8z3
). (3.7)
By Lemma 3.2, we know that the point (x, y) =(−v(v − 1)2,−2v2(v − 1)2
)lies on the curve y2 = x3 − n2x with n = v(v − 1)(v + 1). Substituting
v = 3z4−2z2+34z2
into the x-coordinate of the point and scaling it by a factor
of (z2−1)264z6
yields
−9(3 + 3z4 − 2z2)(z2 − 1)2.
The corresponding y-coordinate, 36(3 + 3z4 − 2z2)2z(z2 − 1), can be found
by substituting x = −9(3 + 3z4 − 2z2(z2 − 1)2 into y2 = x3 − n2x where
n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) and then solving for y. Thus,
(x1, y1) =(−9(3 + 3z4 − 2z2
)(z2 − 1)2, 36(3 + 3z4 − 2z2)2z(z2 − 1))
is a point on our curve.
A second point on our curve can be found by using the solution to the
torsor that we solved at the beginning of the proof with b1 = v(v−1)(v+1)2.
Recall that each solvable torsor corresponds to a point on our elliptic curve.
This point is given by Equation (2.2) and for b1 = v(v − 1)(v + 1)2, M = 1,
and e = 1, its x-coordinate reduces to
v(v − 1)(v + 1)2.
Substituting v = 3z4−2z2+34z2
and scaling this coordinate by a factor of (z2−1)264z6
yields3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2)
4z2.
Setting x = 3(3+3z4+2z2)2(3+3z4−2z2)4z2
and solving y2 = x3 − n2x with n =
37
3.1. Preliminary Results
3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) for y gives
y =9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1)
8z3.
Thus, a second point on our elliptic curve is
(x2, y2) =
(3(3 + 3z4 + 2z2)2(3 + 3z4 − 2z2)
4z2,
9(3 + 3z4 − 2z2)2(3 + 3z4 + 2z2)2(z2 + 1)
8z3
).
Finally, we must show that the two points, (x1, y1) and (x2, y2), are
independent. To do this, we apply Silverman’s specialization theorem (See
Theorem 2.7). If z = 2, then Equation (3.4) yields the congruent number
n = 7611 = 3 · 43 · 59, while (3.6) and (3.7) give two points on y2 =
x(x2 − 76112), namely
(x1, y1) = (−3483, 399384)
and
(x2, y2) =
(449049
16,289636605
64
).
The Magma code in Appendix A confirms that these two non-torsion points
are independent in the group of rational points on y2 = x(x2 − 76112). By
Silverman’s specialization theorem, the two points given by (3.6) and (3.7)
are independent over Q (z) and are therefore independent for all but finitely
many values of the rational number z. Thus, for n = 3(3 + 3z4 − 2z2)(3 +
3z4 +2z2), the rank of the curve given by Equation (3.1) is at least two with
at most finitely many exceptions.
Lemma 3.5. Let
n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2)
38
3.1. Preliminary Results
for a rational number z 6= 0,±1. If (3+3z4−2z2) = p3c2 and (3+3z4+2z2) =
q3d2 for distinct primes p3 and q3 different from 3, and rational numbers c
and d, then the rank of the congruent number curve given by Equation (3.1)
is at least two.
Proof. If we impose the restrictions that (3 + 3z4 − 2z2) = p3c2 and (3 +
3z4 +2z2) = q3d2 for distinct primes p3 and q3 different from 3, and rational
numbers z, c, and d, then an argument using the method of 2-descent shows
that the points given by Equations (3.6) and (3.7) are always independent.
Consider the x-coordinates of the points (x1, y1) and (x2, y2) modulo Q∗2
and notice that
x1 = −9p3c2(z2 − 1)2 ≡ −p3 (mod Q∗2)
and
x2 =3(q3d
2)2p3c2
4z2≡ 3p3 (mod Q∗2).
Clearly −p3 and 3p3 are not congruent modulo Q∗2, so they are two unique
elements in α(Γ). Hence, for (3+3z4−2z2) = p3c2 and (3+3z4+2z2) = q3d
2,
the points (x1, y1) and (x2, y2) are always independent.
In order to bound the rank, r(n), of the congruent curves in our theorem,
we need to make use of Monsky’s formula for s(n), the 2-Selmer rank, which
was introduced in Section 2.7.
Lemma 3.6. If n = p3q3r3, then s(n) ≤ 2.
Proof. We calculate s(n) using Equations (2.7) and (2.8) with P1 = p3,
P2 = q3 and P3 = r3 for all possible choices of values for the Legendre
symbols(p3q3
),(p3r3
), and
(q3r3
). We record the results for all eight cases in
Table 3.1 and provide the Maple code that was used to obtain the results in
Appendix B.
39
3.1. Preliminary Results
Table 3.1: Values of s(n) for n = p3q3r3(p3q3
) (p3r3
) (q3r3
)s(n)
+1 +1 +1 0
+1 +1 −1 0
+1 −1 +1 2
+1 −1 −1 0
−1 +1 +1 0
−1 +1 −1 2
−1 −1 +1 0
−1 −1 −1 0
Remark 3.7. In the proof of Lemma 3.6, the six cases where s(n) = 0
are related by permutation of the primes p3, q3, and r3. The cases where
s(n) = 2 are similarly related.
We recall Schinzel’s hypothesis H [SS58], which states that if a finite
product Q(x) =m∏i=1
fi(x) of polynomials fi(x) ∈ Z [x] has no fixed divisors,
then all of the fi(x) are simultaneously prime, for infinitely many integral
values of x. From this hypothesis we deduce that for any fixed prime p3 the
two forms
p3a2 − 16b2 and p3a
2 + 16b2 (3.8)
assume prime values infinitely often. To ensure that these two forms result
in q3 and r3 being prime numbers, a must be odd. By Lemma 3.3, the
number n = p3q3r3 is guaranteed to be congruent. All of the examples
of congruent numbers mentioned in the introduction have p3 = 3, but we
can generate examples for any fixed prime p3 by using Equation (3.8). For
40
3.2. Proof of the Main Theorem
example if p3 = 43 then using (3.8) with a = 9 and b = 1 yields the value
n = p3q3r3 = 43 · 3467 · 3499,
which by Lemma 3.3 is a congruent number.
3.2 Proof of the Main Theorem
We now provide the proof of Theorem 3.1.
Proof. If the formulas for q and r given in the statement of our theorem
assume prime values, then u and v must have opposite parity. Without loss
of generality, suppose that u = 2h+ 1 and v = 2j, with j, h ∈ Z and j 6= 0.
Then
q = 3u4 + 3v4 − 2u2v2
= 3(2h+ 1)4 + 3(2j)4 − 2(2h+ 1)2(2j)2
= 8(6h4 + 12h3 + 9h2 + 3h+ 6j4 − 4j2h2 − 4j2h− j2) + 3
≡ 3 (mod 8).
A similar argument shows that r is also congruent to 3 modulo 8. Thus,
q ≡ r ≡ 3 (mod 8). From Lemma 3.4, we know that the curve y2 = x(x2−n2)with n = 3(3 + 3z4 − 2z2)(3 + 3z4 + 2z2) has rank at least two for all but
finitely many values of the rational number z. Hence, setting z = u/v and
scaling by v8 shows that n = 3qr is a congruent number. By Lemma 3.5,
the curve (3.1) with n = 3qr has rank at least two. However, Lemma 3.6
shows that s(n) ≤ 2, and since the rank is bounded above by s(n), the rank
is at most two. Thus, the rank equals two and the theorem is proved.
Example 3.8. A few smaller congruent numbers whose associated congru-
ent number curves have rank two and are generated by the formulas in our
theorem include 7611 = 3 · 43 · 59, 1021683291 = 3 · 13219 · 25763, and
2700420027 = 3 · 30203 · 29803.
41
Chapter 4
Iskra’s Family of
Non-congruent Numbers
This chapter focuses on a theorem proven by Iskra that describes a family of
non-congruent numbers with arbitrarily many prime factors. The theorem,
which appeared in Iskra’s paper “Non-congruent numbers with arbitrarily
many prime factors congruent to 3 modulo 8” [Isk96], provides an answer
to the following question posed by Kida [Kid93]:
Can we find an infinite set of primes of the form 8k + 3 with k ∈ Zsuch that for any product n of primes in the set, the elliptic curve
y2 = x(x2 − n2) has a rank of zero?
By applying the method of complete 2-descent, Iskra proved that the curve
y2 = x(x2 − n2) has a rank of zero for infinitely many values of n whose
prime factors are congruent to 3 modulo 8 and satisfy a certain pattern of
Legendre symbols. This family of non-congruent numbers is described by
Iskra’s theorem [Isk96].
Theorem 4.1 (Iskra’s Theorem). Let p1, p2, . . . , pt be distinct primes
such that pi ≡ 3 (mod 8) and(pjpi
)= −1 for j < i. Then the product
n = p1p2 · · · pt is a non-congruent number.
In this chapter, we prove Iskra’s theorem using two different approaches.
In Section 4.1 we prove the theorem by using the method of complete 2-
descent; this approach is based on Iskra’s proof [Isk96]. However, for the
sake of clarity, and to provide a thorough example of the method of com-
plete 2-descent, we include additional details. Following this, in Section 4.2,
we present a new technique for generating non-congruent numbers. This
42
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
method uses linear algebra in conjunction with Monsky’s formula for the
2-Selmer rank to provide a simple and elegant proof of Iskra’s theorem. We
conclude this section by verifying that for any value of t, there always exists
a collection of primes satisfying the conditions of Theorem 4.1.
4.1 The Proof of Iskra’s Theorem Using the
Method of Complete 2-Descent
In order to apply the method of complete 2-descent to prove Theorem 4.1,
we need to use various properties of Legendre symbols and congruences.
Remark 4.2. Since all of the primes in Theorem 4.1 are congruent to 3
modulo 8, we can make the following useful simplifications [Isk96].
1. Since pi ≡ 3 (mod 8), by 3(i) and 3(ii) in Theorem 1.29 we deduce
that (−1
pi
)= −1 and
(2
pi
)= −1.
2. Because(pjpi
)= −1 for j < i, it follows that
(pjpi
)= +1 if i < j.
Explanation: Since pi and pj are distinct primes that are congruent
to 3 modulo 8, the above fact can easily be deduced from the law of
quadratic reciprocity that was stated in Theorem 1.29.
3. Let n = nipi. Then (nipi
)= (−1)i−1.
Explanation: Since n = p1p2 · · · pt = nipi, we have ni = p1 · · · pi−1pi+1 · · · pt.This means that (
nipi
)=
(p1 · · · pi−1pi+1 · · · pt
pi
).
43
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
By assumption, the primes are all distinct, so we can apply 1(ii) The-
orem 1.29 to rewrite the Legendre symbol on the right-hand side of the
above equation as a product of (t− 1) Legendre symbols:(p1 · · · pi−1pi+1 · · · pt
pi
)=
(p1pi
)(p2pi
)· · ·(pi−1pi
)(pi+1
pi
)· · ·(ptpi
).
Recall that(pjpi
)= −1 if j < i and that
(pjpi
)= +1 if i < j. As a
result, the above equation reduces to(nipi
)= (−1)(−1) · · · (−1)(+1) · · · (+1) = (−1)i−1.
4. Let b be a divisor of n, and define
b′ =
b
piif pi|b,
b if pi - b.
Let k = |{j : pj |b and j < i}|. Then(b′
pi
)= (−1)k.
Explanation: Since b is a divisor of n = p1p2 · · · pt, we have b =
pa1pa2 · · · paf where ah ∈ {1, 2, . . . , t} for all h = 1, 2, . . . , f , and the
ahs are all distinct. Because b′ does not have pi as a factor and the
primes pah are distinct, we can apply 1(ii) Theorem 1.29 to get(b′
pi
)=
(pa1pa2 · · · paf
pi
)=
(pa1pi
)(pa2pi
)· · ·(pafpi
).
We know that(pahpi
)= −1 if ah < i and that
(pahpi
)= +1 if ah > i.
By definition, there are k primes that divide b and satisfy the property
that ah < i, so k of the Legendre symbols on the right-hand side of the
above equation have a value of −1 and the remainder of the Legendre
44
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
symbols have a value of +1. Thus,(b′
pi
)= (−1)k.
We are now ready to apply the method of complete 2-descent to prove
Iskra’s theorem.
Proof. To prove that n = p1p2 · · · pt is a non-congruent number, we apply
Theorems 2.10 and 2.11 to show that for all pairs (b1, b2) 6∈ {(1, 1), (−1,−n),
(n, 2), (−n,−2n)} with bi ∈ {±2εpε11 pε22 · · · p
εkk |ε, ε1, ε2, . . . , εk ∈ {0, 1}}, Equa-
tions (2.4) and (2.5) cannot simultaneously be solved. By the unsolvability
conditions stated in Theorem 2.11, we know that Equations (2.4) and (2.5)
do not have a solution when b1 · b2 < 0 or when 2 - n and 2|b1. Since
n = p1p2 · · · pt with each of the pi ≡ 3 (mod 8), it follows that 2 - n. There-
fore, we only need to consider pairs (b1, b2) for which b1 · b2 > 0 and 2 - b1.We split these restrictions into four separate cases and verify that in each of
these cases, there does not exist a pair (b1, b2) that simultaneously satisfies
Equations (2.4) and (2.5).
Case 1: b2 > 0 and 2 - b2Define
s = min{i : pi|b1 or pi|b2}.
If s exists, then ps|b1, or ps|b2, or both of these division statements hold.
Consider(b′1ps
). By Property 4 in Remark 4.2, we know that
(b′1ps
)= (−1)k,
where k = |{j : pj |b1 and j < s}|. However, ps is by definition the prime
with the smallest subscript that divides b1 or b2. Therefore, there cannot
exist an integer j with j < s such that pj |b1. As a result, we conclude that
the set {j : pj |b1 and j < s} is empty and that k = 0. It follows that(b′1ps
)= (−1)0 = +1. (4.1)
45
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
By using an analogous argument, we can also deduce that(b′2ps
)= +1. (4.2)
According to the definition of s, we now need to consider three separate
subcases.
Subcase 1: ps|b1 and ps|b2Consider Equation (2.5). Dividing both sides by ps yields
b1psz21 −
b1psb2z
23 =−nps.
By using Properties 3 and 4 from Remark 4.2, we can replace nps
by ns andb1ps
by b′1. Therefore, the equation becomes
b′1z21 − b′1b2z23 = −ns.
Consider this equation modulo ps. Since b2 contains a factor of ps, we know
that b′1b2z23 ≡ 0 (mod ps). As a result, our equation reduces to
b′1z21 ≡ −ns (mod ps).
Multiplying both sides of this congruence by b′1 yields
(b′1z1)2 ≡ −nsb′1 (mod ps).
Clearly ns and b′1 are not divisible by ps, so nsb′1 is not divisible by ps.
Therefore, it follows that the left-hand side of the congruence is not divisible
by ps either. We can now apply 1(i) Theorem 1.29 to write((b′1z1)
2
ps
)=
(−nsb′1ps
).
Since ps does not divide (b′1z1)2, ps does not divide (b′1z1), so by 1(iii) The-
orem 1.29, we know that((b′1z1)
2
ps
)= +1. This allows us to conclude that
46
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
(−nsb′1ps
)= +1. (4.3)
If we consider the Legendre symbol in Equation (4.3) and apply 1(ii) Theo-
rem 1.29, we can write(−nsb′1ps
)=
(−1
ps
)(nsps
)(b′1ps
).
Note that(−1ps
)= −1 by Property 1 in Remark 4.2,
(nsps
)= (−1)s−1 by
Property 3 in Remark 4.2, and(b′1ps
)= +1 by Equation (4.1). Thus,
(−nsb′1ps
)= (−1)(−1)s−1(+1) = (−1)s. (4.4)
Now consider the following equation:
b1b2z23 − b2z22 = 2n. (4.5)
This equation can be obtained by subtracting Equation (2.5), b1z21−b1b2z23 =
−n, from Equation (2.4), b1z21−b2z22 = n. If we divide both sides of Equation
(4.5) by ps, we can write
b1b2psz23 −
b2psz22 = 2
n
ps.
By applying Properties 3 and 4 from Remark 4.2, we can replace nps
by ns
and b2ps
by b′2. Therefore, the equation becomes
b1b′2z
23 − b′2z22 = 2ns.
Consider this equation modulo ps. Since b1 contains a factor of ps, we know
that b1b′2z
23 ≡ 0 (mod ps). As a result, our equation reduces to
−b′2z22 ≡ 2ns (mod ps).
47
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
Multiplying both sides of this congruence by −b′2 yields
(b′2z2)2 ≡ −2nsb
′2 (mod ps).
Clearly 2, ns, and b′2 are not divisible by ps, so 2nsb′2 is not divisible by ps.
Therefore, the left-hand side of the congruence is not divisible by ps either.
We can now apply 1(i) Theorem 1.29 to write((b′2z2)
2
ps
)=
(−2nsb
′2
ps
).
Since ps does not divide (b′2z2)2, it follows that ps does not divide (b′2z2), so
by 1(iii) Theorem 1.29, we know that((b′2z2)
2
ps
)= +1. This enables us to
conclude that (−2nsb
′2
ps
)= +1. (4.6)
If we consider the Legendre symbol in Equation (4.6) and apply 1(ii) Theo-
rem 1.29, we can write(−2nsb
′2
ps
)=
(−1
ps
)(2
ps
)(nsps
)(b′2ps
).
Since(−1ps
)= −1 and
(2ps
)= −1 by Property 1 in Remark 4.2,
(nsps
)=
(−1)s−1 by Property 3 in Remark 4.2, and(b′2ps
)= +1 by Equation (4.2),
the above product of Legendre symbols simplifies to(−2nsb
′2
ps
)= (−1)(−1)(−1)s−1(+1) = (−1)s−1. (4.7)
Now we compare Equations (4.4) and (4.7). Because(−nsb′1ps
)= (−1)s and(
−2nsb′2ps
)= (−1)s−1, it follows that for every s, one of these two Legendre
symbols must have a value of −1. This is a contradiction, since by Equations
(4.3) and (4.6), we know that both of these Legendre symbols have a value
of +1. Thus, we conclude that in the case where ps|b1 and ps|b2, there is no
solution that simultaneously solves Equations (2.4) and (2.5).
48
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
Subcase 2: ps|b1 and ps - b2Consider the following equation:
2b1z21 − b2z22 − b1b2z23 = 0. (4.8)
This equation can be obtained by adding Equation (2.4), b1z21−b2z22 = n, to
Equation (2.5), b1z21 − b1b2z23 = −n. Dividing both sides of Equation (4.8)
by ps yields
2b1psz21 −
b2z22
ps− b1psb2z
23 = 0.
By applying Property 4 from Remark 4.2, we can replace b1ps
by b′1. Therefore,
the equation becomes
2b′1z21 −
b2z22
ps− b′1b2z23 = 0.
Recall that if a prime divides a product, then it must divide at least one of
its factors. By assumption, we know that ps does not divide b2. Therefore,
ps must divide z2. If z2 contains a factor of ps, then b2z2z2ps≡ 0 (mod ps).
As a result, the above equation reduces to
2b′1z21 − b′1b2z23 ≡ 0 (mod ps).
Rearranging this congruence and multiplying both sides by 2b′1 yields
(2b′1)2z21 ≡ 2(b′1)
2b2z23 (mod ps).
Note that ps cannot divide both z1 and z2. If it did, then p2s would be
a common factor of the left-hand side of Equation (2.4), so it would also
be a common divisor of the right-hand side, n. However, by assumption
n only contains a single factor of ps. Therefore, p2s cannot divide n, so ps
cannot divide both z1 and z2. We already deduced that ps is a factor of
z2, so it follows that ps cannot divide z1. Since ps does not divide 2, b′1, or
z1, we conclude that ps is not a common factor of the left-hand side of the
above congruence. This means that ps must not be a common factor of the
49
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
right-hand side either. We can now apply 1(i) Theorem 1.29 to write((2b′1z1)
2
ps
)=
(2b2(b
′1z3)
2
ps
).
By using 1(ii) and 1(iii) from Theorem 1.29, we can rewrite the Legendre
symbol on the right-hand side of the equation as(2b2(b
′1z3)
2
ps
)=
(2b2ps
)((b′1z3)
2
ps
)=
(2b2ps
)(+1).
Also, by 1(iii) Theorem 1.29, we know that((2b′1z1)
2
ps
)= +1. Utilizing these
simplifications allows us to conclude that(2b2ps
)= +1. (4.9)
If we consider the Legendre symbol in Equation (4.9) and apply 1(ii) Theo-
rem 1.29, we can write (2b2ps
)=
(2
ps
)(b2ps
).
Since ps does not divide b2, by Property 4 in Remark 4.2, we can replace b2
by b′2. In addition, by Property 1 in Remark 4.2, we know that(
2ps
)= −1,
and by Equation (4.2), we know that(b′2ps
)= +1. Therefore,
(2b2ps
)= (−1)(+1) = −1.
However, this result contradicts Equation (4.9). Thus, we conclude that
when ps|b1 and ps - b2, there does not exist a solution that simultaneously
satisfies Equations (2.4) and (2.5).
50
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
Subcase 3: ps - b1 and ps|b2Consider Equation (4.8) and divide both sides of it by ps to get
2b1z21
ps− b2psz22 − b1
b2psz23 = 0.
By applying Property 4 from Remark 4.2, we can replace b2ps
by b′2, which
yields2b1z
21
ps− b′2z22 − b1b′2z23 = 0.
Since ps divides 2b1z21 , it must divide at least one of its factors. By our
initial assumptions, we know that ps does not divide 2 or b1. Therefore, ps
must divide z1. If z1 contains a factor of ps, then 2b1z1z1ps≡ 0 (mod ps). As
a result, the above equation reduces to
−b′2z22 − b1b′2z23 = 0 (mod ps).
Rearranging this congruence and multiplying both sides by −b′2 yields
b′22 z22 = −b1b′22 z23 (mod ps).
By using the same argument as in Subcase 2, we conclude that ps cannot
divide both z1 and z2. Since we deduced that z1 is divisible by ps, it follows
that ps does not divide z2. Because b′2 and z2 are not divisible by ps, we
conclude that ps is not a factor of the left-hand side of the above congruence.
Consequently, we know that ps must not divide the right-hand side either.
As a result, we can apply 1(i) Theorem 1.29 to write((b′2z2)
2
ps
)=
(−b1(b′2z3)2
ps
).
By using 1(ii) and 1(iii) from Theorem 1.29, we can rewrite the Legendre
symbol on the right-hand side of the equation as(−b1(b′2z3)2
ps
)=
(−b1ps
)((b′2z3)
2
ps
)=
(−b1ps
)(+1).
51
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
In addition, by 1(iii) Theorem 1.29, we know that((b′2z2)
2
ps
)= +1. Applying
these simplifications allows us to conclude that(−b1ps
)= +1. (4.10)
Notice that if we apply 1(ii) Theorem 1.29 to the Legendre symbol in Equa-
tion (4.10), we obtain (−b1ps
)=
(−1
ps
)(b1ps
).
Since ps does not divide b1, by Property 4 in Remark 4.2, we can replace b1
by b′1. In addition, by Property 1 in Remark 4.2, we know that(−1ps
)= −1,
and by Equation (4.1), we know that(b′1ps
)= +1. Therefore,
(−b1ps
)= (−1)(+1) = −1.
However, this result contradicts Equation (4.10). Thus, we conclude that if
ps - b1 and ps|b2, there does not exist a solution that simultaneously solves
Equations (2.4) and (2.5).
None of the three subcases of Case 1 provide a pair, (b1, b2), for which
Equations (2.4) and (2.5) are simultaneously solvable. Therefore, we deduce
that s = min{i : pi|b1 or pi|b2} does not exist, which means that no prime
divides b1 or b2. As a result, since b1 · b2 > 0 and b2 > 0, we conclude
that (b1, b2) = (1, 1), which is a contradiction to our initial assumption that
(b1, b2) 6∈ {(1, 1), (−1,−n), (n, 2), (−n,−2n)}.
We now consider Case 2 where b2 > 0 and 2|b2, Case 3 where b2 < 0
and 2 - b2, and Case 4 where b2 < 0 and 2|b2. In his paper [Isk96], Iskra
utilized arguments similar to those presented in Case 1 to verify that the
remaining three cases do not yield pairs for which Equations (2.4) and (2.5)
are solvable. However, carrying out this process for each of the cases is
52
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
not only lengthy and tedious, but also unnecessary. By making use of the
properties of groups and our results from Case 1, we can easily verify that
that Cases 2, 3, and 4 do not yield any pairs either.
Case 2: b2 > 0 and 2|b2By way of contradiction, suppose there exists a pair (b1, b2) with b2 > 0 and
2|b2 for which Equations (2.4) and (2.5) are simultaneously solvable. Since
the set of points δ = {(1, 1), (−1,−n), (n, 2), (−n,−2n), (b1, b2)} generates a
finite subgroup of Q∗/Q∗2 × Q∗/Q∗2, by closure the pair (b1, b2) · (n, 2) =
(nb1, 2b2) must also belong to the group. By assumption 2|b2, so we can
write b2 = 2b∗2, where b∗2 ∈ Q∗/Q∗2 and 2 - b∗2. If we set b∗1 = nb1, then we
have (nb1, 2b2) = (b∗1, 4b∗2). Because (b∗1, 4b
∗2) ∈ Q∗/Q∗2 ×Q∗/Q∗2, it follows
that 4b∗2 is equivalent to b∗2. Notice that the pair (b∗1, b∗2) has b∗2 > 0 and
2 - b∗2. This means that the pair (b∗1, b∗2) has the same properties as the one
described in Case 1. However, from Case 1, we know that there does not
exist a solution that simultaneously solves Equations (2.4) and (2.5) for pairs
of this form. Therefore, (b∗1, b∗2) cannot belong to the group generated by the
set δ. This is a contradiction, so our initial assumption that (b1, b2) with
b2 > 0 and 2|b2 is an element of δ must be incorrect. Thus, we conclude that
there does not exist a pair (b1, b2) with b2 > 0 and 2|b2 for which Equations
(2.4) and (2.5) are solvable.
Case 3: b2 < 0 and 2 - b2By way of contradiction, assume there exists a pair (b1, b2) with b2 < 0 and
2 - b2 for which Equations (2.4) and (2.5) are simultaneously solvable. Since
the set of points δ = {(1, 1), (−1,−n), (n, 2), (−n,−2n), (b1, b2)} generates a
finite subgroup of Q∗/Q∗2×Q∗/Q∗2, by closure the pair (b1, b2) · (−1,−n) =
(−b1,−nb2) must also be an element of the group. Let b∗1 = −b1 and b∗2 =
−nb2, so (−b1,−nb2) = (b∗1, b∗2). Since 2 - n and 2 - b2, it follows that
2 - (−nb2). In addition, because b2 < 0 and n > 0, we know that −nb2 > 0.
Hence, b∗2 > 0 and 2 - b∗2, which means that the pair (b∗1, b∗2) has the same
properties as the one described in Case 1. By the same argument as in Case
2, we conclude that (b∗1, b∗2) cannot belong to the group generated by the set
53
4.1. The Proof of Iskra’s Theorem Using the Method of Complete 2-Descent
δ. Hence, there does not exist a pair (b1, b2) with b2 < 0 and 2 - b2 for which
Equations (2.4) and (2.5) have a solution.
Case 4: b2 < 0 and 2|b2By way of contradiction suppose that there exists a pair (b1, b2) with b2 < 0
and 2|b2 for which Equations (2.4) and (2.5) are simultaneously solvable.
Since the set of points δ = {(1, 1), (−1,−n), (n, 2), (−n,−2n), (b1, b2)} gen-
erates a finite subgroup of Q∗/Q∗2 × Q∗/Q∗2, by closure the pair (b1, b2) ·(−n,−2n) = (−nb1,−2nb2) must also belong to the group. By assump-
tion 2|b2, so we can write b2 = 2b~2 , where b~2 ∈ Q∗/Q∗2 and 2 - b~2 . Be-
cause (−nb1,−2nb2) = (−nb1,−4nb~2 ) ∈ Q∗/Q∗2 × Q∗/Q∗2, it follows that
−4nb~2 is equivalent to −nb~2 . If we set b∗1 = −nb1 and b∗2 = −nb~2 , we have
(−nb1,−nb~2 ) = (b∗1, b∗2). Clearly b~2 < 0, since b2 < 0. By using this fact
and noting that n > 0, we deduce that −nb~2 = b∗2 > 0. In addition, because
2 - n and 2 - b~2 , we know that 2 - b∗2. Therefore, the pair (b∗1, b∗2) has b∗2 > 0
and 2 - b∗2, which means that it has the same properties as the one described
in Case 1. By using the same argument as in Case 2, we deduce that (b∗1, b∗2)
cannot belong to the group generated by the set δ. Thus, we conclude that
there does not exist a pair (b1, b2) with b2 < 0 and 2|b2 for which Equations
(2.4) and (2.5) are solvable.
None of the four cases yields a pair (b1, b2) 6∈ {(1, 1), (−1,−n), (n, 2),
(−n,−2n)} with bi ∈ {±2εpε11 pε22 · · · p
εkk |ε, ε1, ε2, . . . , εk ∈ {0, 1}} for which
Equations (2.4) and (2.5) simultaneously have a solution. Therefore, our
bound, B, for the number of such pairs equals zero, so the inequality in
Equation (2.6) becomes
r ≤ log2
(0 + 4
4
)= 0.
Since the rank, r, must be a non-negative integer, the inequality implies that
r = 0. Hence, for n = p1p2 · · · pt, where p1, p2, . . . , pt are distinct primes with
pi ≡ 3 (mod 8) and(pjpi
)= −1 for j < i, the elliptic curve y2 = x(x2 − n2)
has a rank of zero. Thus, n is a non-congruent number.
54
4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .
4.2 The Proof of Iskra’s Theorem Using
Monsky’s Formula for the 2-Selmer Rank
We now provide a proof of Iskra’s theorem using Monsky’s formula for the
2-Selmer rank.
Proof. By applying Equation (2.8), we can define the Monsky matrix, Mo,
for numbers of the form n = p1p2 · · · pt with pi ≡ 3 (mod 8) for all i and(pjpi
)= −1 for j < i.
Mo =
1 0 0 . . . . . . . . . 0 1 0 0 . . . . . . . . . 0
1 2 0 . . . . . . . . . 0 0 1 0 . . . . . . . . . 0
1 1 3. . .
... 0 0 1. . .
......
.... . .
. . .. . .
......
.... . .
. . .. . .
......
.... . .
. . .. . .
......
.... . .
. . .. . .
......
.... . . t− 1 0
......
. . . 1 0
1 1 . . . . . . . . . 1 t 0 0 . . . . . . . . . 0 1
1 0 0 . . . . . . . . . 0 0 0 0 . . . . . . . . . 0
0 1 0 . . . . . . . . . 0 1 1 0 . . . . . . . . . 0
0 0 1. . .
... 1 1 2. . .
......
.... . .
. . .. . .
......
.... . .
. . .. . .
......
.... . .
. . .. . .
......
.... . .
. . .. . .
......
.... . . 1 0
......
. . . t− 2 0
0 0 . . . . . . . . . 0 1 1 1 . . . . . . . . . 1 t− 1
.
Note that Mo is a 2t × 2t matrix. We now execute a series of column and
row interchanges on this matrix. These operations vary slightly depending
on whether t is even or odd. If t is even, we begin by exchanging columns
1 and t, 2 and (t − 1), 3 and (t − 2), . . . , t2 and t+22 . We then interchange
columns (t + 1) and 2t, (t + 2) and (2t − 1), (t + 3) and (2t − 2), . . . , 3t2and 3t+2
2 . Following this, we exchange rows 1 and t, 2 and (t − 1), 3 and
55
4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .
(t−2), . . . , t2 and t+22 , and rows (t+1) and 2t, (t+2) and (2t−1), (t+3) and
(2t − 2), . . . , 3t2 and 3t+22 . If t is odd, we exchange columns 1 and t, 2 and
(t−1), 3 and (t−2), . . . , t−12 and t+32 , and columns (t+1) and 2t, (t+2) and
(2t− 1), (t+ 3) and (2t− 2), . . . , 3t−12 and 3t+32 . However, columns t+1
2 and3t+12 are left in their original positions. Similarly, in regards to the exchange
of the rows, we carry out the same procedure as for columns and interchange
all of the rows except for rows t+12 and 3t+1
2 , which remain in their original
positions. Upon carrying out these column and row operations, we obtain
the following matrix:
M′o =
t 1 1 . . . . . . 1 1 1 0 0 . . . . . . . . . 0
0 t− 1 1 . . . . . . 1 1 0 1 0 . . . . . . . . . 0
0 0 t− 2. . .
...... 0 0 1
. . ....
......
. . .. . .
. . ....
......
.... . .
. . .. . .
......
.... . . 3 1 1
......
. . .. . .
. . ....
...... 0 2 1
......
. . . 1 0
0 0 . . . . . . 0 0 1 0 0 . . . . . . . . . 0 1
1 0 0 . . . . . . . . . 0 t− 1 1 1 . . . . . . 1 1
0 1 0 . . . . . . . . . 0 0 t− 2 1 . . . . . . 1 1
0 0 1. . .
... 0 0 t− 3. . .
......
......
. . .. . .
. . ....
......
. . .. . .
. . ....
......
.... . .
. . .. . .
......
.... . . 2 1 1
......
. . . 1 0...
... 0 1 1
0 0 . . . . . . . . . 0 1 0 0 . . . . . . 0 0 0
.
Note that if t is even, we must carry out t column interchanges and t row
interchanges to obtain the matrix M′o. By Property 1 in Definition 1.17, we
obtain
det(Mo) = (−1)2tdet(M′o) = det(M′
o).
Similarly, if t is odd, we must perform (t− 1) column exchanges and (t− 1)
56
4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .
row exchanges. By applying Property 1 in Definition 1.17, we obtain
det(Mo) = (−1)2t−2det(M′o) = det(M′
o).
Also, we can write the matrix M′o as
M′o =
[U It
It U− It
],
where It is the t× t identity matrix and U is a t× t matrix of the form
U =
t 1 1 . . . . . . 1 1
0 t− 1 1 . . . . . . 1 1
0 0 t− 2. . .
......
......
. . .. . .
. . ....
......
.... . . 3 1 1
...... 0 2 1
0 0 . . . . . . 0 0 1
.
We perform row interchanges on M′o t times to obtain the matrix
N =
[It U− It
U It
].
By Property 1 in Definition 1.17, det(M′o) = (−1)tdet(N). Applying the
formula for computing block determinants given in Proposition 1.20 yields
det(N) = det(It)det(It −UI −1t (U− It))
= det(It −U(U− It)).
Notice that U(U− It) is a product of two upper triangular matrices, so
by the statement following Definition 1.11, we know that U(U−It) must be
an upper triangular matrix. Each diagonal entry in the matrix U(U − It)
is equal to the product of two consecutive integers, so the diagonal entries
must be even and hence congruent to 0 modulo 2. Therefore, It −U(U −
57
4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .
It) is an upper triangular matrix with diagonal entries all congruent to 1
modulo 2. By Property 3 in Theorem 1.18, it follows that the determinant
of It −U(U− It) is congruent to 1 modulo 2. Hence,
det(Mo) = det(M′o)
= (−1)tdet(N)
= (−1)tdet(It −U(U− It))
≡ 1 (mod 2).
Since det(Mo) 6≡ 0 (mod 2), by Property 6 in Theorem 1.18, we know
that rankF2(Mo) = 2t. Therefore, by Equation (2.7), we deduce that s(n) =
0. Thus, the inequality in Equation (2.9) implies that r(n) = 0, so n is a
non-congruent number.
Finally, we verify that for any value of t, there always exists a collection
of primes satisfying the conditions of Theorem 4.1.
Corollary 4.3. Let Ht denote the collection of positive integers with prime
factorization p1p2 · · · pt, where the pi are distinct primes of the form 8k + 3
satisfying(pjpi
)= −1 for all 1 ≤ j < i ≤ t. For any value of t, the set Ht is
non-empty and, in fact, contains infinitely many elements.
Proof. We need to use Dirichlet’s theorem on primes in arithmetic progres-
sion (See Theorem 1.30) to verify that this is true. The case where t = 1 is
obviously true, since by Dirichlet’s theorem on primes in arithmetic progres-
sion, there are infinitely many primes of the form 8k + 3. We use induction
on t, and assume that the result is true up to (t − 1) for t > 1. Now we
must show that the set Ht is infinite. By the induction hypothesis, we know
that there exist integers p1p2 · · · pt−1, where the pi are distinct primes of the
form 8k + 3 satisfying(pjpi
)= −1 for all 1 ≤ j < i < t. We would like to
choose a prime pt satisfying pt ≡ 3 (mod 8),
pt ≡ 1 (mod pj) for each 1 ≤ j < t,(4.11)
58
4.2. The Proof of Iskra’s Theorem Using Monsky’s Formula. . .
and append this prime onto the end of the product p1p2 · · · pt−1. The Chi-
nese remainder theorem (See Theorem 1.26) guarantees that the system of
congruences given by (4.11) has a solution. By applying this theorem in
conjunction with Dirichlet’s theorem on primes in arithmetic progression,
we are able to conclude that there exist infinitely many primes pt satisfying
the system of congruences in (4.11). Note that since pt ≡ 1 (mod pj), we
know that pt is a quadratic residue modulo pj for each 1 ≤ j < t. Hence,
by applying Property 2 from Remark 4.2, it follows that(pjpt
)= −1 for all
1 ≤ j < t. Thus, the set Ht contains infinitely many elements.
59
Chapter 5
Families of Non-congruent
Numbers with Arbitrarily
Many Prime Factors of the
Form 8k + 3
In this chapter, we provide an extension to Iskra’s work and generate in-
finitely many distinct new families of non-congruent numbers with arbitrar-
ily many prime factors of the form 8k+3. In order to construct these families,
we utilize the technique involving Monsky’s formula for the 2-Selmer rank
that was presented in Section 4.2.
In Section 5.1, we present the main theorem of this chapter and use lin-
ear algebra to establish necessary conditions to construct the new families
of non-congruent numbers. Following this in Section 5.2, we prove our main
theorem and then conclude with a couple of supplementary corollaries. The
first corollary provides verification that the sets described by our main the-
orem are non-empty, and the second modifies our main theorem in such a
way that it yields congruent numbers.
5.1 Preliminary Results Involving the
Generation of Non-congruent Numbers
We begin by stating the main result of this chapter.
Theorem 5.1. Let m be a fixed non-negative even integer and let t be any
positive integer satisfying t ≥ m. Let Nm denote the set of positive integers
60
5.1. Preliminary Results Involving the Generation of Non-congruent Numbers
with prime factorization p1p2 · · · pt, where p1, p2, . . . , pt are distinct primes
of the form 8k + 3 such that
(pjpi
)=
−1 if 1 ≤ j < i and (j, i) 6= (1,m),
+1 if 1 ≤ j < i and (j, i) = (1,m).(5.1)
If n ∈ Nm, then n is non-congruent. Moreover for m > 0, the sets Nm are
pairwise disjoint.
For convenience we define three matrices that will be used in our con-
struction of non-congruent numbers.
Definition 5.2. For a positive integer r, we define the matrices U, Q, and
A by
U = Ur =
r − 1 1 1 · · · 1 1
0 r − 2 1 · · · 1 1
0 0. . .
. . ....
......
.... . . 2 1 1
0 0 · · · 0 1 1
0 0 · · · 0 0 0
,
Q = Qr =
1 0 0 · · · 0 1
0 0 0 · · · 0 0
0 0 0 · · · 0 0...
......
......
0 0 0 · · · 0 0
1 0 0 · · · 0 1
,
and
A = Ar =
r − 2 1 1 · · · 1 0
0 r − 2 1 · · · 1 1
0 0. . .
. . ....
......
.... . . 2 1 1
0 0 · · · 0 1 1
1 0 · · · 0 0 1
.
61
5.1. Preliminary Results Involving the Generation of Non-congruent Numbers
As usual, I = Ir denotes the r × r identity matrix and 0 = 0r denotes the
r × r zero matrix.
Our first lemma is a direct calculation.
Lemma 5.3. With Q defined as in Definition 5.2, we have
Q2 = 2Q ≡ 0r (mod 2).
The next lemma establishes an identity involving U.
Lemma 5.4. With U defined as in Definition 5.2, we have
U(U + I) ≡ 0r (mod 2).
Proof. We apply mathematical induction on r. The lemma is true when
r = 1 since U =[
0]
and U + I =[
1]. Now assume that
Ur−1 (Ur−1 + Ir−1) ≡ 0r−1 (mod 2).
We can write
Ur =
r − 1 1 · · · 1
0...
0
Ur−1
.Using block multiplication we see that
Ur(Ur+Ir) =
r − 1 1 · · · 1
0...
0
Ur−1
r 1 · · · 10...
0
Ur−1 + Ir−1
,
62
5.1. Preliminary Results Involving the Generation of Non-congruent Numbers
which for some 1× (r − 1) matrix W, simplifies tor(r − 1) W
0...
0
Ur−1 (Ur−1 + Ir−1)
≡
0 W
0...
0
0r−1
(mod 2)
by the induction hypothesis. It remains to calculate W. We see that
W = [r − 1][
1 1 · · · 1]
+[
1 1 · · · 1]
[Ur−1 + Ir−1]
=[r − 1 r − 1 · · · r − 1
]+[
1 1 · · · 1]
r − 1 1 1 · · · 1 1
0 r − 2 1 · · · 1 1
0 0. . .
. . ....
......
.... . . 3 1 1
...... · · · 0 2 1
0 0 · · · 0 0 1
=[r − 1 r − 1 · · · r − 1
]+[r − 1 r − 1 · · · r − 1
]≡[
0 0 · · · 0](mod 2).
The proofs of the next two lemmas use direct calculation.
Lemma 5.5. With U and Q as given in Definition 5.2, we have
UQ =
r 0 0 · · · 0 r
1 0 0 · · · 0 1
1 0 0 · · · 0 1...
......
......
1 0 0 · · · 0 1
0 0 0 · · · 0 0
.
63
5.1. Preliminary Results Involving the Generation of Non-congruent Numbers
Lemma 5.6. With U and Q as given in Definition 5.2, we have
Q(U + I) ≡
r 1 1 · · · 1 0
0 0 0 · · · 0 0
0 0 0 · · · 0 0...
......
......
0 0 0 · · · 0 0
r 1 1 · · · 1 0
(mod 2).
We now prove a lemma that establishes an identity involving A.
Lemma 5.7. With A as given in Definition 5.2, we have
A(A + I) ≡
0 1 1 · · · 1 r
1 0 0 · · · 0 1
1 0 0 · · · 0 1...
......
......
1 0 0 · · · 0 1
r 1 1 · · · 1 0
(mod 2).
Proof. From Definition 5.2 we have
A(A + I) ≡ (U + Q)(U + I + Q) ≡ U(U + I) + UQ + Q(U + I) + Q2(mod 2).
Applying Lemmas 5.3, 5.4, 5.5, and 5.6 yields the desired result.
The next lemma provides the starting point for our families of non-
congruent numbers.
Lemma 5.8. With A = Ar as given in Definition 5.2, r even, and T defined
by
T =
[I A
A + I I
],
we have det(T) ≡ 1 (mod 2).
64
5.1. Preliminary Results Involving the Generation of Non-congruent Numbers
Proof. Recalling Lemma 5.7, we have
A(A + I) ≡
0 1 1 · · · 1 r
1 0 0 · · · 0 1
1 0 0 · · · 0 1...
......
......
1 0 0 · · · 0 1
r 1 1 · · · 1 0
≡
0 1 1 · · · 1 0
1 0 0 · · · 0 1
1 0 0 · · · 0 1...
......
......
1 0 0 · · · 0 1
0 1 1 · · · 1 0
≡ CDT (mod 2),
where
C =
1 0
0 1...
...
0 1
1 0
and DT =
[0 1 · · · 1 0
1 0 · · · 0 1
].
By using this fact and applying Proposition 1.20, we deduce that
det(T) = det (I−A(A + I))
≡ det(I−CDT
)(mod 2).
Furthermore, by applying Proposition 1.21 with B = Ir for r even, we are
able to determine that
65
5.1. Preliminary Results Involving the Generation of Non-congruent Numbers
det(T) ≡ det(I2 −DTC
)(mod 2)
≡ det
[
1 0
0 1
]−
[0 1 · · · 1 0
1 0 · · · 0 1
]
1 0
0 1...
...
0 1
1 0
(mod 2)
≡ det
([1 0
0 1
]−
[0 r − 2
2 0
])(mod 2)
≡ det
([1 0
0 1
])(mod 2)
≡ 1 (mod 2).
Our final lemma is a crucial step in producing families of non-congruent
numbers with arbitrarily many prime factors.
Lemma 5.9. Let m be a fixed non-negative even integer and let t be any
positive integer satisfying t ≥ m. Suppose that the matrix M = M2t is given
by
M =
[U + I I
I U
],
with
U =
[U11 U12
0 U22
].
U11 is a (t−m)× (t−m) (possibly empty) matrix given by
U11 =
t− 1 1 1 · · · 1
0 t− 2 1 · · · 1
0 0. . .
. . ....
......
. . .. . . 1
0 0 · · · 0 m
,
66
5.1. Preliminary Results Involving the Generation of Non-congruent Numbers
U12 is a (t −m) × m (possibly empty) matrix with all of its entries equal
to 1, and U22 is a (possibly empty) m×m matrix of integers with
det
([I U22
U22 + I I
])≡ 1 (mod 2). (5.2)
Then det(M) ≡ 1 (mod 2). We note that by convention the empty matrix
has determinant 1 and if U22 is empty then U22 + I0 is equal to the empty
matrix.
Proof. After performing t row interchanges on M, we obtain the matrix N
given by
N =
[I U
U + I I
].
By Property 1 in Definition 1.17, it follows that
det(M) = (−1)t det(N). (5.3)
Applying the formula for block determinants given in Proposition 1.20 yields
det(N) = det(It) det(It −UI−1t (U + It)) = det(It −U(U + It)). (5.4)
Meanwhile, since U11 and (U11 + It−m) are upper triangular matrices, it
follows that U11(U11 + It−m) is an upper triangular matrix. Each of the
diagonal entries in U11(U11+It−m) is equal to the product of two consecutive
integers, so by Property 3 in Theorem 1.18 we have
det (U11(U11 + It−m)) ≡ 0 (mod 2). (5.5)
Therefore,
67
5.2. Proof of the Main Theorem
It−U(U + It) = It−
[U11 U12
0 U22
][U11 + It−m U12
0 U22 + Im
]
≡ It −
[U11(U11 + It−m) ∗
0 U22(U22 + Im)
](mod 2)
≡
[It−m −U11(U11 + It−m) ∗
0 Im −U22(U22 + Im)
](mod 2).
Finally, by applying Equations (5.3) and (5.4), and Proposition 1.19, we
deduce that
det(M) = (−1)t det (N)
≡ det (It−U(U + It)) (mod 2)
≡ det(It−m −U11(U11 + It−m)) det(Im −U22(U22 + Im)) (mod 2).
Equation (5.5) implies that det(It−m −U11(U11 + It−m)) ≡ 1 (mod 2) and
by Equation (5.2), we know that det(Im − U22(U22 + Im)) ≡ 1 (mod 2).
Thus, we conclude that
det(M) ≡ 1 (mod 2).
5.2 Proof of the Main Theorem
We are now prepared to provide the proof of Theorem 5.1.
Proof. We apply Lemma 5.9 to generate our families of non-congruent num-
bers. For the choice of prime factors with Legendre symbols as specified in
68
5.2. Proof of the Main Theorem
our theorem, the Monsky matrix given by Equation (2.8) becomes
Mo =
2 0 · · · · · · · · · 0 1 0 · · · · · · 0 1 0 0 · · · · · · · · · · · · · · · · · · · · · 0
1 2 0 · · · · · · · · · · · · · · · · · · · · · 0 0 1 0 · · · · · · · · · · · · · · · · · · · · · 0
1 1 3 0 · · · · · · · · · · · · · · · · · · 0 0 0 1. . .
...
1 1 1 4 0...
. . .. . .
. . ....
......
.... . .
......
. . .. . .
. . ....
1 1 1 · · · 1 m− 1 0 · · · · · · · · · 0...
. . .. . .
. . ....
0 1 1 · · · · · · 1 m− 1 0 · · · · · · 0...
. . .. . .
. . ....
1 1 1 · · · · · · · · · 1 m+ 1 0 · · · 0...
. . .. . .
. . ....
......
.... . .
......
. . .. . .
. . ....
......
... t− 1 0...
. . .. . . 0
1 1 1 · · · · · · · · · · · · · · · · · · 1 t 0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1
1 0 0 · · · · · · · · · · · · · · · · · · · · · 0 1 0 · · · · · · · · · 0 1 0 · · · · · · 0
0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 1 1 0 · · · · · · · · · · · · · · · · · · · · · 0
0 0 1. . .
... 1 1 2 0 · · · · · · · · · · · · · · · · · · 0...
. . .. . .
. . .... 1 1 1 3 0
.... . .
. . .. . .
......
......
. . ....
.... . .
. . .. . .
... 1 1 1 · · · 1 m− 2 0 · · · · · · · · · 0...
. . .. . .
. . .... 0 1 1 · · · · · · 1 m− 2 0 · · · · · · 0
.... . .
. . .. . .
... 1 1 1 · · · · · · · · · 1 m 0 · · · 0...
. . .. . .
. . .......
......
. . ....
.... . .
. . . 0...
...... t− 2 0
0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1 1 1 1 · · · · · · · · · · · · · · · · · · 1 t− 1
,
Note that Mo is a 2t × 2t matrix. We now apply a series of column and
row interchanges to the matrix Mo. These column and row exchanges are
described in the proof of Iskra’s theorem presented in Section 4.2. Upon
executing these operations, we obtain a matrix of the form
M =
[U + I I
I U
],
69
5.2. Proof of the Main Theorem
where
U =
[U11 U12
0 U22
],
with U11 and U12 as given in Lemma 5.9. The matrix U22 is the empty
matrix if m = 0, while U22 is equal to Am if m > 0. Lemma 5.8 shows
that the conditions of Lemma 5.9 are fulfilled with these choices. Applying
Lemma 5.9, we deduce that
det(M) ≡ 1(mod 2).
In addition, we know by the explanation presented in the proof of Iskra’s
theorem in Section 4.2 that for all values of t satisfying the requirements of
Theorem 5.1
det(Mo) = det(M) ≡ 1(mod 2).
Thus, by Property 6 in Theorem 1.18, the rank of Mo is equal to 2t. It
follows from (2.7) that s(n) = 0 if n ∈ Nm and by inequality (2.9) that the
rank of (3.1) is equal to zero. Hence n is non-congruent.
We note that N0 is the family of non-congruent numbers in Iskra’s theo-
rem (see Theorem 4.1) and that N2 ⊆ N0 by permuting the first two primes
in any n ∈ N2. We now prove that all other sets Nm are new. Assume that
the positive integer n satisfies
n ∈ Nm ∩Nm′ ,
for even integers m and m′ with m′ > m ≥ 4. Suppose that the prime
factorization of the integer n, satisfying (5.1) is given by
n = p1p2 · · · pt ∈ Nm
and that a permutation π of the prime factors pi of n results in
n = q1q2 · · · qt ∈ Nm′ ,
70
5.2. Proof of the Main Theorem
where the qi are the prime factors of n and
(qjqi
)=
−1 if 1 ≤ j < i and (j, i) 6= (1,m′),
+1 if 1 ≤ j < i and (j, i) = (1,m′).(5.6)
Let k denote the largest subscript for which pk is not fixed by the permuta-
tion π. Clearly k ≥ 2. If k = 2 then q1 = p2 and q2 = p1 so that
(q1q2
)= +1,
contradicting (5.6) as m′ > m ≥ 4. If k = 3, then the ordered set {q1, q2, q3}is one of the ordered sets {p3, p1, p2}, {p3, p2, p1}, {p1, p3, p2}, or {p2, p3, p1}.Considering these choices in order leads to the Legendre symbol values(
q1q2
)= +1,
(q1q2
)= +1,
(q2q3
)= +1, and
(q2q3
)= +1,
each of which contradicts (5.6) and the inequality m′ > m ≥ 4. Therefore,
k ≥ 4. By the definition of k we know that pk = qj for some j satisfying
1 ≤ j < k. If pk = q1 then as
{p1, p2, . . . , pk−1} = {q2, q3, . . . , qk} (5.7)
and (q1qi
)= −1 (5.8)
for 2 ≤ i ≤ k and i 6= m′, we conclude by (5.7), (5.8), and the inequality
k ≥ 4 that the symbol
(pkp`
)has a value of −1 for at least two values of `
satisfying 1 ≤ ` ≤ k− 1. This contradicts (5.1). If qk = p1 then we obtain a
contradiction in a similar manner. Therefore, pk = qj for some j satisfying
2 ≤ j ≤ k− 1. We also have that qk = pi for some i satisfying 2 ≤ i ≤ k− 1.
From (5.6) we must have (qjqk
)= −1,
so that (pkpi
)= −1,
71
5.2. Proof of the Main Theorem
which contradicts (5.1). Thus, the sets Nm and Nm′ are distinct.
A similar argument shows that for m ≥ 4, the integers in the sets Nm
are different from those in Iskra’s family of non-congruent numbers, which
was described in Theorem 4.1.
Next, we prove a corollary that provides verification that the sets Nm
described in Theorem 5.1 are non-empty.
Corollary 5.10. Let Nm denote the set of positive integers defined in the
statement of Theorem 5.1. For any value of m, the set Nm is non-empty
and, in fact, contains infinitely many elements.
Proof. Recall Corollary 4.3 and consider a positive integer of the form
p1p2 · · · pm−1 whose prime factors fulfill the conditions of Theorem 4.1. Ap-
pend a prime pm onto the end of this product that satisfies the following
system of congruences:pm ≡ 3 (mod 8),
pm ≡ −1 (mod p1),
pm ≡ 1 (mod pj) for each 1 < j < m.
(5.9)
By applying the Chinese remainder theorem and Dirichlet’s theorem on
primes in arithmetic progression (See Theorems 1.26 and 1.30), we deduce
that there exist infinitely many primes pm that satisfy the system of con-
gruences in (5.9). Since pm ≡ −1 (mod p1), it follows that pm is a quadratic
nonresidue modulo p1. Therefore,(p1pm
)= +1. Conversely,
(pjpm
)= −1 for
each 1 < j < m, because the third congruence in (5.9) indicates that pm is
a quadratic residue modulo pj . Thus, (5.1) is satisfied, so we conclude that
the sets Nm contain infinitely many elements.
It is worthwhile to note that we proved this corollary without imposing
the restrictions t ≥ m and m is even from the statement of Theorem 5.1.
72
5.2. Proof of the Main Theorem
Finally, we conclude this chapter by presenting a corollary that offers
evidence that congruent numbers whose prime factors are of the form 8k+3
and satisfy Equation (5.1) exist whenever m is odd and m ≥ 3.
Corollary 5.11. Let m ≥ 3 be a fixed odd integer and let t be any positive
integer satisfying t ≥ m. Let Nm denote the set of positive integers with
prime factorization p1p2 · · · pt, where p1, p2, . . . , pt are distinct primes of the
form 8k + 3 satisfying Equation (5.1). If n ∈ Nm, then n is congruent.
Proof. We recall Lemma 3.2 which states that for any rational number v /∈(−∞,−1] ∪ [0, 1], the form v(v − 1)(v + 1), properly scaled to an integer
by squares of rational numbers, produces a congruent number. Let m ≥ 3
be odd, and assume that p2, p3, . . . , pm−1 are distinct prime numbers of the
form 8k + 3 satisfying(pjpi
)= −1 if 1 ≤ j < i. Define the integer b by b =
p2p3 · · · pm−1. Since b is a product of an odd number of primes of the form
8k + 3, it follows that b ≡ 3 (mod 8). Let v =bx2
16y2for positive integers x
and y. Scaling by squares yields the congruent number
(bx2 − 16y2)b(bx2 + 16y2).
Schinzel’s hypothesis H [SS58] states that if a finite product Q(x) =m∏i=1
fi(x) of polynomials fi(x) ∈ Z [x] has no fixed divisors, then all of the
fi(x) are simultaneously prime, for infinitely many integral values of x. From
this hypothesis we deduce that the two forms
bx2 − 16y2 and bx2 + 16y2
assume prime values infinitely often. Notice that bx2− 16y2 and bx2 + 16y2
only attain prime values if x is odd. Since b ≡ 3 (mod 8), it is easy to verify
that bx2− 16y2 and bx2 + 16y2 are primes of the form 8k+ 3 by using basic
properties of congruences.
Furthermore, we can prove that the product (bx2 − 16y2)b(bx2 + 16y2)
where b = p2p3 · · · pm−1 satisfies the conditions given by (5.1) in Theorem
5.1. To do this, we must show that for any prime divisor p of b, the following
73
5.2. Proof of the Main Theorem
three equations hold:(bx2 − 16y2
p
)= −1,
(bx2 − 16y2
bx2 + 16y2
)= +1, and
(p
bx2 + 16y2
)= −1.
We begin by considering the Legendre symbol(bx2−16y2
p
)first. Since p|b,
we know that bx2 ≡ 0 (mod p). Therefore,(bx2 − 16y2
p
)=
(−16y2
p
).
Note that p - (−16y2). By way of contradiction, suppose that p|(−16y2).
Clearly, p cannot divide 2 as p is a prime of the form 8k + 3. Therefore,
we must have p|y. This means that p is a common factor of both b and
y, so it follows that p|(bx2 + 16y2). However, since p 6= (bx2 + 16y2), we
cannot have p|(bx2 + 16y2) as this is a contradiction to our assumption that
(bx2 + 16y2) is prime. Therefore, p - y so p - (−16y2). As a result, we can
use 1(ii) Theorem 1.29 to write(−16y2
p
)=
(−1
p
)((4y)2
p
).
Finally, we apply 1(iii) and 3(i) from Theorem 1.29 to conclude that(bx2 − 16y2
p
)= −1.
Next, we verify that(bx2−16y2bx2+16y2
)= +1. Clearly, bx2 ≡ −16y2 (mod (bx2+
16y2)). Therefore,(bx2 − 16y2
bx2 + 16y2
)=
(−16y2 − 16y2
bx2 + 16y2
)=
(−2(16y2)
bx2 + 16y2
).
Note that (bx2 + 16y2) - (−25y2), because if it did, the prime (bx2 + 16y2)
would have to divide 2 or y. However, since 2 and y are both less than
(bx2+16y2), it is impossible for (bx2+16y2) to divide −2(16y2). By making
use of this fact and applying 1(ii) Theorem 1.29, we obtain the following
74
5.2. Proof of the Main Theorem
equation:(−2(16y2)
bx2 + 16y2
)=
(−1
bx2 + 16y2
)(2
bx2 + 16y2
)((4y)2
bx2 + 16y2
).
Since (bx2 + 16y2) is a prime of the form 8k+ 3, by applying 1(iii), 3(i), and
3(ii) from Theorem 1.29, we can conclude that(bx2 − 16y2
bx2 + 16y2
)= +1.
The third and final Legendre symbol that we need to consider is(
pbx2+16y2
).
Since p and (bx2 + 16y2) are primes of the form 8k + 3, it follows from the
law of quadratic reciprocity that(p
bx2 + 16y2
)= −
(bx2 + 16y2
p
).
In addition, because p|b we know that bx2 ≡ 0 (mod p). Therefore,
−(bx2 + 16y2
p
)= −
(16y2
p
).
By 1(iii) Theorem 1.29, it is clear that(16y2
p
)= +1. This enables us to
conclude that (p
bx2 + 16y2
)= −1.
Therefore, (bx2−16y2)b(bx2+16y2) with b = p2p3 · · · pm−1 is a congruent
number that satisfies the conditions given by (5.1) in Theorem 5.1. Thus,
when m is odd and m ≥ 3, we cannot generate families of non-congruent
numbers.
75
Chapter 6
Families of Non-congruent
Numbers with One Prime
Factor of the Form 8k + 1 and
Arbitrarily Many Prime
Factors of the Form 8k + 3
Chapter 6 focuses on the generation of families of non-congruent numbers
with arbitrarily many prime factors. However, unlike the non-congruent
numbers presented in the previous two chapters whose prime divisors are
only of the form 8k+3, the numbers described in this chapter are a product
of primes belonging to two different congruence classes modulo 8; the non-
congruent numbers in these new families contain a single prime factor of the
form 8k+ 1 and at least one prime factor of the form 8k+ 3. It is important
to note that these families of non-congruent numbers are an extension of
work done by Lagrange involving non-congruent numbers with two or three
prime factors [Lag75]. Lagrange’s non-congruent numbers are listed in Table
1.2 and have the form n = pq with(pq
)= −1, or n = pqr with
(pq
)=
−(pr
), where p ≡ 1 (mod 8) and q ≡ r ≡ 3 (mod 8). To construct these
new families, we utilize the technique introduced in Section 4.2 and used to
prove our main theorem in Chapter 5.
In Section 6.1 we state and prove the main theorem for this chapter, and
in Section 6.2 we discuss and prove a supporting corollary.
76
6.1. Proof of the Main Theorem
6.1 Proof of the Main Theorem
We begin by stating the main theorem of this chapter.
Theorem 6.1. Let m be a fixed positive integer and let t be any integer
satisfying t ≥ m. Let Sm denote the set of positive integers with prime fac-
torization pq1q2 · · · qt, where p is a prime of the form 8k+1 and q1, q2, . . . , qt
are distinct primes of the form 8k + 3 such that(p
qi
)=
{−1 if i = m,
+1 if i 6= m,
and (qjqi
)= −1 if j < i.
If n ∈ Sm, then n is non-congruent. Moreover for different m, the sets Sm
are pairwise disjoint.
77
6.1. Proof of the Main Theorem
Proof. Going directly to the Monsky matrix we have
Mo =
1 0 · · · · · · · · · 0 1 0 · · · · · · 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · 0
0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 1 0 · · · · · · · · · · · · · · · · · · · · · 0
0 1 2 0 · · · · · · · · · · · · · · · · · · 0 0 0 1. . .
...
0 1 1 3 0...
. . .. . .
. . ....
......
.... . .
......
. . .. . .
. . ....
0 1 1 · · · 1 m− 1 0 · · · · · · · · · 0...
. . .. . .
. . ....
1 1 1 · · · · · · 1 m+ 1 0 · · · · · · 0...
. . .. . .
. . ....
0 1 1 · · · · · · · · · 1 m+ 1 0 · · · 0...
. . .. . .
. . ....
......
.... . .
......
. . .. . .
. . ....
......
... t− 2 0...
. . .. . . 0
0 1 1 · · · · · · · · · · · · · · · · · · 1 t− 1 0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1
0 0 0 · · · · · · · · · · · · · · · · · · · · · 0 1 0 · · · · · · · · · 0 1 0 · · · · · · 0
0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · 0
0 0 1. . .
... 0 1 1 0 · · · · · · · · · · · · · · · · · · 0...
. . .. . .
. . .... 0 1 1 2 0
.... . .
. . .. . .
......
......
. . ....
.... . .
. . .. . .
... 0 1 1 · · · 1 m− 2 0 · · · · · · · · · 0...
. . .. . .
. . .... 1 1 1 · · · · · · 1 m 0 · · · · · · 0
.... . .
. . .. . .
... 0 1 1 · · · · · · · · · 1 m 0 · · · 0...
. . .. . .
. . ....
......
.... . .
......
. . .. . . 0
......
... t− 3 0
0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1 0 1 1 · · · · · · · · · · · · · · · · · · 1 t− 2
.
Note that each block in Mo is a t × t matrix. We start by applying a
sequence of elementary row and column operations on Mo. Specifically, we
add column 1 to column (t+ 1) and then subtract column t+ (m+ 1) from
column (t + 1). This is followed by adding row 1 to row (t + 1) and then
subtracting row t+ (m+ 1) from row (t+ 1). We obtain a matrix M′o given
78
6.1. Proof of the Main Theorem
below.
M′o =
1 0 · · · · · · · · · 0 1 0 · · · · · · 0 1 0 0 · · · · · · · · · · · · · · · · · · · · · 0
0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 1 0 · · · · · · · · · · · · · · · · · · · · · 0
0 1 2 0 · · · · · · · · · · · · · · · · · · 0 0 0 1. . .
...
0 1 1 3 0...
. . .. . .
. . ....
......
.... . .
......
. . .. . .
. . ....
0 1 1 · · · 1 m− 1 0 · · · · · · · · · 0...
. . .. . .
. . ....
1 1 1 · · · · · · 1 m+ 1 0 · · · · · · 0...
. . .. . .
. . ....
0 1 1 · · · · · · · · · 1 m+ 1 0 · · · 0...
. . .. . .
. . ....
......
.... . .
......
. . .. . .
. . ....
......
... t− 2 0...
. . .. . . 0
0 1 1 · · · · · · · · · · · · · · · · · · 1 t− 1 0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1
1 0 0 · · · · · · · · · · · · · · · · · · · · · 0 m −1 · · · · · · · · · −1 1−m 0 · · · · · · 0
0 1 0 · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · 0
0 0 1. . .
... 0 1 1 0 · · · · · · · · · · · · · · · · · · 0
.... . .
. . .. . .
... 0 1 1 2 0
.... . .
. . .. . .
......
......
. . ....
.... . .
. . .. . .
... 0 1 1 · · · 1 m− 2 0 · · · · · · · · · 0
.... . .
. . .. . .
... 1−m 1 1 · · · · · · 1 m 0 · · · · · · 0
.... . .
. . .. . .
... −1 1 1 · · · · · · · · · 1 m 0 · · · 0
.... . .
. . .. . .
......
......
. . ....
.... . .
. . . 0...
...... t− 3 0
0 · · · · · · · · · · · · · · · · · · · · · · · · 0 1 −1 1 1 · · · · · · · · · · · · · · · · · · 1 t− 2
.
We write this matrix in the form
M′o =
[U It
It V
],
79
6.1. Proof of the Main Theorem
where
U =
1 0 0 · · · · · · 0 1 0 · · · · · · · · · · · · 0
0 1 0 · · · · · · · · · 0...
...
0 1 2. . .
......
......
.... . .
. . .. . .
......
......
.... . .
. . .. . .
......
...
0 1 · · · · · · 1 m− 1 0...
...
1 1 · · · · · · · · · 1 m+ 1 0 · · · · · · · · · · · · 0
0 1 · · · · · · · · · · · · 1 m+ 1 0 · · · · · · · · · 0...
...... 1 m+ 2
. . ....
......
......
. . .. . .
. . ....
......
......
. . .. . .
. . ....
......
......
. . . t− 2 0
0 1 · · · · · · · · · · · · 1 1 · · · · · · · · · 1 t− 1
=
[U11 U12
U21 U22
],
and
V =
m −1 −1 · · · · · · −1 1−m 0 · · · · · · · · · · · · 0
0 0 0 · · · · · · · · · 0...
...
0 1 1 0...
......
0 1 1 2. . .
......
......
.... . .
. . .. . .
......
...
0 1 · · · · · · 1 m− 2 0...
...
1−m 1 · · · · · · · · · 1 m 0 · · · · · · · · · · · · 0
−1 1 · · · · · · · · · · · · 1 m 0 · · · · · · · · · 0...
...... 1 m+ 1
. . ....
......
......
. . .. . .
. . ....
......
......
. . .. . .
. . ....
......
......
. . . t− 3 0
−1 1 · · · · · · · · · · · · 1 1 · · · · · · · · · 1 t− 2
=
[V11 V12
V21 V22
].
80
6.1. Proof of the Main Theorem
The matrix resulting from performing t row interchanges on M′o is
N =
[It V
U It
].
Note that by Property 1 in Definition 1.17 and Property 2 in Theorem 1.18,
det(Mo) = det(M′o) = (−1)t det(N). (6.1)
In addition, by the formula for block determinants given in Proposition 1.20,
det(N) = det
([It V
U It
])= det(It) det(It−UI−1t V) = det(It−UV). (6.2)
Consider
U11V11 =
1 0 0 · · · · · · 0 1
0 1 0 · · · · · · · · · 0
0 1 2. . .
......
.... . .
. . .. . .
......
.... . .
. . .. . .
...
0 1 · · · · · · 1 m− 1 0
1 1 · · · · · · · · · 1 m+ 1
m −1 −1 · · · · · · −1 1−m0 0 0 · · · · · · · · · 0
0 1 1 0...
0 1 1 2. . .
......
.... . .
. . .. . .
...
0 1 · · · · · · 1 m− 2 0
1−m 1 · · · · · · · · · 1 m
=
m+ (1−m) 0 0 · · · · · · 0 (1−m) +m
0 0 0 · · · · · · 0 0
0 2 2. . .
...
... 4 4 6. . .
...
.... . .
. . ....
0. . . 0
m+ (m+ 1)(1−m) ∗ · · · · · · · · · ∗ (1−m) +m(m+ 1)
.
81
6.1. Proof of the Main Theorem
Notice that all of the diagonal entries in the matrix U11V11 except for the
two corner ones are equal to the product of two consecutive integers, so they
are congruent to 0 modulo 2. Moreover all of the entries of U11V11 except
for the corner entries in the first and last row are even, which means that
they are congruent to 0 modulo 2. We note that the entries denoted by ∗are of the form
−1 + (m− 2) + (m+ 1),
hence are even. We reduce U11V11 modulo 2 to obtain
U11V11 ≡
1 0 · · · 0 1
0 0 · · · 0 0
......
......
......
......
0...
... 0
−m2 +m+ 1 0 · · · 0 m2 + 1
(mod 2).
Further reduction modulo 2 yields
U11V11 ≡
1 0 · · · 0 1
0 0 0...
. . ....
0 0 0
1 0 · · · 0 1
(mod 2) if m is even,
1 0 · · · 0 1
0 0 0...
. . ....
0 0 0
1 0 · · · 0 0
(mod 2) if m is odd.
82
6.1. Proof of the Main Theorem
Returning to the matrices U and V, we notice that all of the entries in U12
and V12 are equal to zero. In addition,
U22V22 =
m+ 1 0 · · · · · · · · · 0
1 m+ 2. . .
......
. . .. . .
. . ....
.... . .
. . .. . .
......
. . . t− 2 0
1 · · · · · · · · · 1 t− 1
m 0 · · · · · · · · · 0
1 m+ 1. . .
......
. . .. . .
. . ....
.... . .
. . .. . .
......
. . . t− 3 0
1 · · · · · · · · · 1 t− 2
.
Since U22V22 is a product of two lower triangular matrices, it is lower tri-
angular. Each diagonal entry in the matrix U22V22 is equal to the product
of two consecutive integers, hence is congruent to 0 modulo 2. Therefore,
It −UV = It −
U11 U12
U21 U22
V11 V12
V21 V22
= It −
U11V11 0
∗ U22V22
.
83
6.1. Proof of the Main Theorem
If m is even, then
It −UV ≡
0 0 0 · · · 0 1 0 . . . . . . . . . . . . 0
0 1 0 · · · · · · 0...
...
0 0 1. . .
......
......
. . .. . .
. . ....
......
0. . . 1 0
......
1 0 · · · · · · 0 0 0 . . . . . . . . . . . . 0
1 0 · · · · · · · · · 0
∗ . . .. . .
......
. . .. . .
. . ....
∗...
. . .. . .
. . ....
.... . .
. . . 0
∗ · · · · · · · · · ∗ 1
(mod 2).
By using Equations (6.1) and (6.2), and Proposition 1.19, we deduce that
det(Mo) = (−1)t det(N) = (−1)t det(It −UV)
≡ (−1)t det
0 0 0 · · · 0 1
0 1 0 · · · · · · 0
0 0 1. . .
......
. . .. . .
. . ....
0. . . 1 0
1 0 · · · · · · 0 0
(mod 2).
Finally, by applying Property 1 of Definition 1.17 and exchanging the first
and last rows of the matrix whose determinant we are trying to compute,
84
6.1. Proof of the Main Theorem
we have
det(Mo) ≡ −det
1 0 · · · · · · 0
0 1. . .
......
. . .. . .
. . ....
.... . . 1 0
0 · · · · · · 0 1
≡ 1 (mod 2).
If m is odd, then
It −UV ≡
0 0 0 · · · 0 1 0 . . . . . . . . . . . . 0
0 1 0 · · · · · · 0...
...
0 0 1. . .
......
......
. . .. . .
. . ....
......
0. . . 1 0
......
1 0 · · · · · · 0 1 0 . . . . . . . . . . . . 0
1 0 · · · · · · · · · 0
∗ . . .. . .
......
. . .. . .
. . ....
∗...
. . .. . .
. . ....
.... . .
. . . 0
∗ · · · · · · · · · ∗ 1
(mod 2).
By combining Equations (6.1) and (6.2), and using Proposition 1.19, we
85
6.1. Proof of the Main Theorem
deduce that
det(Mo) = (−1)t det(N) = (−1)t det(It −UV)
≡ (−1)t det
0 0 0 · · · 0 1
0 1 0 · · · · · · 0
0 0 1. . .
......
. . .. . .
. . ....
0. . . 1 0
1 0 · · · · · · 0 1
(mod 2).
Continuing by subtracting the first row of this matrix from the last row
yields
det(Mo) ≡ (−1)t det
0 0 0 · · · 0 1
0 1 0 · · · · · · 0
0 0 1. . .
......
. . .. . .
. . ....
0. . . 1 0
1 0 · · · · · · 0 0
(mod 2).
This matrix is the same as the one we obtained in the case where m was
even. As a result, we deduce that
det(Mo) ≡ 1 (mod 2)
when m is odd. Thus, the matrix Mo has full rank for all m. We apply
Equations (2.7) and (2.9) to conclude that n is a non-congruent number.
Next we show that for different m, the sets Sm are pairwise disjoint.
Suppose that for some positive integer n the two sets Sm and Sm′ satisfy
n ∈ Sm ∩ Sm′ ,
86
6.2. A Supporting Corollary
where we may assume that m > m′ ≥ 1. Let π denote a permutation of the
prime factors qi of n and suppose that
pq1q2 · · · qt ∈ Sm and pπ(q1)π(q2) · · ·π(qt) = pq′1q′2 · · · q′t ∈ Sm′ .
By definition of the sets Sm and Sm′ , we deduce that
q′m′ = qm.
As m > m′ ≥ 1, we conclude that
{q1, q2, . . . , qm−1} ⊆{q′1, q
′2, . . . , q
′m′−1
}is impossible. Therefore, for some integer j with 1 ≤ j ≤ m− 1, we have
qj ∈{q′m′+1, q
′m′+2, . . . , q
′t
}.
It follows that (q′m′
qj
)= −1,
or (qmqj
)= −1,
contradicting the definition of Sm. Thus, the sets Sm and Sm′ are distinct.
This completes the proof of the theorem.
6.2 A Supporting Corollary
By applying Dirichlet’s theorem on primes in arithmetic progression and
using a similar argument to the one presented in Corollary 5.10, we can
deduce that the sets Sm are non-empty and can verify that it is possible to
form sequences
p, q1, q2, . . .
87
6.2. A Supporting Corollary
of prime numbers satisfying the hypotheses of Theorem 6.1. In addition, re-
call that a sequence of primes {pi} that satisfies the conditions of Theorem
4.1 has the additional property that any product of primes chosen from this
sequence is non-congruent. The families of non-congruent numbers gener-
ated by Theorem 6.1 have a property similar to this, as they also give rise
to a sequence of integers such that any product of them is non-congruent.
This leads to the following corollary.
Corollary 6.2. Let {p, q1, q2, . . . , qm, qm+1, . . .} be a sequence of prime num-
bers satisfying the hypotheses of Theorem 6.1. Any product of integers from
the set
{pq1q2 · · · qm, qm+1, qm+2, . . .}
is non-congruent.
Proof. Let w be a product of integers belonging to the set
{pq1q2 · · · qm, qm+1, qm+2, . . .}.
If this product does not contain the integer factor pq1q2 · · · qm, then it is non-
congruent by Theorem 4.1. If it does contain pq1q2 · · · qm, then Theorem 6.1
implies that w is a non-congruent number.
88
Chapter 7
Conclusion and Future Work
7.1 Conclusion
The focus of this thesis was the construction of new families of congruent
and non-congruent numbers. In order to generate these families of num-
bers, techniques that did not rely upon the currently unproven Birch and
Swinnerton-Dyer conjecture were used.
In Chapter 3, a method was provided for constructing congruent numbers
with three prime factors of the form 8k + 3. A family of such numbers
was given for which the rank of their associated elliptic curves equals two,
the maximal rank for congruent number curves of this type. In order to
compute the rank, both the method of 2-descent and Monsky’s formula for
the 2-Selmer rank were applied. We showed that the rank of the curves
was at least two by solving torsors to find two independent points on the
corresponding elliptic curves. Furthermore, we applied Monsky’s formula to
deduce that the upper bound for the rank of the elliptic curves corresponding
to numbers with three prime factors of the form 8k + 3 was two. Together,
these results proved the existence of a family of congruent numbers with
associated elliptic curves of rank two.
Chapter 4 focused on an important result by Iskra that describes a fam-
ily of non-congruent numbers with arbitrarily many prime factors of the
form 8k + 3. Since a new method for generating non-congruent numbers
with arbitrarily many prime factors is presented in Section 4.2, this chap-
ter arguably contains the most valuable information in the thesis. This
new method utilizes linear algebra and employs Monsky’s formula for the
2-Selmer rank. Unlike the method of 2-descent which uses quadratic equa-
tions and involves a series of lengthy and complex calculations, Monsky’s
89
7.2. Future Work
formula offers a simple and elegant approach for determining whether a given
square-free positive integer is non-congruent. We demonstrated the beauty
of this method by applying it to prove Iskra’s theorem in Section 4.2. If
we compare Iskra’s original proof contained in Section 4.1 to the proof we
provided in Section 4.2, it becomes obvious that the new method is not only
less mathematically complex than the method of complete 2-descent, but it
is also more efficient at generating families of non-congruent numbers.
In Chapters 5 and 6 the method described in Section 4.2 was used to
generate new families of non-congruent numbers. Specifically, Chapter 5
provided an important extension to Iskra’s work by proving the existence
of infinitely many distinct new families of non-congruent numbers with ar-
bitrarily many prime factors of the form 8k + 3. Chapter 6 expanded upon
results by Lagrange to generate families of non-congruent numbers whose
prime factors belonged to two different congruence classes modulo 8; these
integers are a product of a single prime of the form 8k + 1 and at least one
prime of the form 8k+3, and have distinct prime factors satisfying a specific
pattern of Legendre symbols.
7.2 Future Work
The field of study involving congruent and non-congruent numbers has con-
siderable potential for future research work. The open problems in this area
of mathematics are diverse and vary in their level of difficulty. Of utmost
importance would be the discovery of a proof that verifies the validity of the
Birch and Swinnerton-Dyer conjecture. Proving this significant conjecture
would have extensive implications, including the verification of Tunnell’s
theorem. This, in turn, would provide an answer to the congruent number
problem by establishing a complete classification of congruent numbers. Al-
though the Birch and Swinnerton-Dyer conjecture is widely believed to be
true, finding a proof for this conjecture is an immensely difficult and poten-
tially even impossible task. Therefore, it is important to look for alternative
solutions to the congruent number problem.
Of particular interest is the search for families of congruent and non-
90
7.2. Future Work
congruent numbers with arbitrarily many prime factors. I believe that the
method for generating non-congruent numbers introduced in Section 4.2
could be applied to prove the existence of infinitely many more families of
non-congruent numbers with arbitrarily many prime factors. Specifically,
it may be possible to extend some of the results stated in Table 1.2 by
following the approach used to extend Lagrange’s results in Chapter 6. It is
worthwhile to note that since Monsky’s formula for the 2-Selmer rank only
provides an upper bound for the Mordell-Weil rank, the method described
in Section 4.2 cannot be used to verify the existence of families of congruent
numbers. Nevertheless, as illustrated in Chapter 3, the bound on the rank
provided by Monsky’s formula can be a useful tool in determining the precise
value of the rank for a specific family of congruent numbers. Additional
research could be done on this topic to generate other families of congruent
numbers with a specified rank. As Johnstone notes in her thesis [Joh09],
congruent number elliptic curves with a provable rank greater than two are
quite rare. Therefore, another topic of interest is the search for families of
congruent number elliptic curves with moderate or high rank. Thus, the
field of congruent and non-congruent numbers has an immense potential for
future research work and exciting new mathematical discoveries.
91
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95
Appendix A
Magma Code
The calculations in this section were carried out by using Magma Version
2.19-4, which can be found online at
http://magma.maths.usyd.edu.au/calc/.
A.1 Elliptic Curve Calculations
To verify that the numbers of the form p3q3r3 mentioned at the beginning
of Chapter 3 are congruent, we use Magma to compute the rank of their
corresponding elliptic curves. Recall that we know that the number n is
congruent if the elliptic curve En: y2 = x(x2−n2) has a rank that is greater
than or equal to one.
First consider the number n = 4587 = 3 ·11 ·139. Notice that the elliptic
curve y2 = x3 − 45872x has a rank of two, which means that 4587 is a
congruent number.
Input:
E:=EllipticCurve([-4587^2,0]);
Rank(E);
AnalyticRank(E);
Output:
Warning: rank computed (1) is only a lower bound
(It may still be correct, though)
1
2 47.276
96
A.1. Elliptic Curve Calculations
Similarly, the number n = 4731 = 3 · 19 · 83 is congruent, as its corre-
sponding elliptic curve also has a rank of two.
Input:
E:=EllipticCurve([-4731^2,0]);
Rank(E);
AnalyticRank(E);
Output:
Warning: rank computed (1) is only a lower bound
(It may still be correct, though)
1
2 42.726
In addition, the elliptic curve y2 = x3− 69632x has a rank of two, which
indicates that n = 6963 = 3 · 11 · 211 is a congruent number.
Input:
E:=EllipticCurve([-6963^2,0]);
Rank(E);
AnalyticRank(E);
Output:
Warning: rank computed (1) is only a lower bound
(It may still be correct, though)
1
2 52.336
The number n = 7611 = 3 · 43 · 59 is also congruent, as the elliptic curve
y2 = x3 − 76112x has a positive rank equal to two.
Input:
E:=EllipticCurve([-7611^2,0]);
Rank(E);
AnalyticRank(E);
Output:
97
A.1. Elliptic Curve Calculations
2
2 7.8098
Finally, n = 9339 = 3 · 11 · 283 is a congruent number, because its
corresponding elliptic curve has a rank equal to two.
Input:
E:=EllipticCurve([-9339^2,0]);
Rank(E);
AnalyticRank(E);
Output:
Warning: rank computed (1) is only a lower bound
(It may still be correct, though)
1
2 42.069
We use the following Magma code to provide verification that the two
points, (x1, y1) and (x2, y2) on the curve y2 = x(x2 − 76112) in the proof of
Lemma 3.4 are linearly independent.
Input:
E:=EllipticCurve([-7611^2,0]);
P:=E![-3483,399384];
Q:=E![449049/16,289636605/64];
S:=[P,Q];
IsLinearlyIndependent(S);
Output:
true
98
Appendix B
Maple Code
The calculations in this section were carried out with MapleTM13. Note that
the symbol > indicates Maple input and the text centred under each line of
input code is Maple output.
B.1 Parametrization and 2-Selmer Rank
Computations
We used the following Maple code to determine the parametrization for v
in terms of t in the proof of Lemma 3.4.
The Maple code used to determine the values for the 2-Selmer rank listed
in Table 3.1 is given below.
Recall that for primes p3 and q3 that are congruent to 3 modulo 8, the
law of quadratic reciprocity implies that(p3q3
)= −
(q3p3
). In the following,
the block matrices D2 and D−2 within the matrix Mo described in Equation
(2.8) are denoted by D2 and Dneg2, respectively.
99
B.1. Parametrization and 2-Selmer Rank Computations
Case 1:(p3q3
)= +1,
(p3r3
)= +1, and
(q3r3
)= +1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to find that
s(n) = 0.
100
B.1. Parametrization and 2-Selmer Rank Computations
Case 2:(p3q3
)= +1,
(p3r3
)= +1, and
(q3r3
)= −1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to deduce that
s(n) = 0.
101
B.1. Parametrization and 2-Selmer Rank Computations
Case 3:(p3q3
)= +1,
(p3r3
)= −1, and
(q3r3
)= +1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 4 to find that
s(n) = 2.
102
B.1. Parametrization and 2-Selmer Rank Computations
Case 4:(p3q3
)= +1,
(p3r3
)= −1, and
(q3r3
)= −1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to deduce that
s(n) = 0.
103
B.1. Parametrization and 2-Selmer Rank Computations
Case 5:(p3q3
)= −1,
(p3r3
)= +1, and
(q3r3
)= +1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to find that
s(n) = 0.
104
B.1. Parametrization and 2-Selmer Rank Computations
Case 6:(p3q3
)= −1,
(p3r3
)= +1, and
(q3r3
)= −1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 4 to deduce that
s(n) = 2.
105
B.1. Parametrization and 2-Selmer Rank Computations
Case 7:(p3q3
)= −1,
(p3r3
)= −1, and
(q3r3
)= +1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to find that
s(n) = 0.
106
B.1. Parametrization and 2-Selmer Rank Computations
Case 8:(p3q3
)= −1,
(p3r3
)= −1, and
(q3r3
)= −1
We apply Equation (2.7) with t = 3 and rankF2(Mo) = 6 to deduce that
s(n) = 0.
107