6 1. Introduction

and, if n is even,

#{(x, y, z) ∈

Z3

:

n

2

=

4x2

+

y2

+

32z2}

=

1

2

#{(x, y, z) ∈

Z3

:

n

2

=

4x2

+

y2

+

8z2}

.

Moreover, if the Birch and Swinnerton-Dyer conjecture is true, then,

conversely, these equalities imply that n is a congruent number.

For example, for n = 2 we have

n

2

= 1 =

4x2

+

y2

+

32z2

if and

only if x = z = 0 and y = ±1, so the left-hand side of the appropriate

equation in Tunnell’s theorem is equal to 2. However, the right-hand

side is equal to 1 and the equality does not hold. Hence, 2 is not a

congruent number.

For a complete historical overview of the congruent number prob-

lem, see [Dic05], Ch. XVI. The book [Kob93] contains a thorough

modern treatment of the problem. The reader may also find useful an

expository paper [Con08] on the congruent number problem, written

by Keith Conrad. Another neat exposition, more computational in

nature (using Sage), appears in [Ste08], Section 6.5.3.

Example 1.1.5 (Fermat’s last theorem). Let n ≥ 3. Are there any

solutions to

xn

+

yn

=

zn

in integers x, y, z with xyz = 0? The

answer is no. In 1637, Pierre de Fermat wrote in the margin of a

book (Diophantus’ Arithmetica; see Figure 9 in Section 5.5) that he

had found a marvellous proof, but the margin was too small to contain

it. Since then, many mathematicians tried in vain to demonstrate (or

disprove!) this claim. A proof was finally found in 1995 by Andrew

Wiles ([Wil95]). We shall discuss the proof in some more detail

in Section 5.5. For now, we will outline the basic structure of the

argument.

First, it is easy to show that, to prove the theorem, it suﬃces to

show the cases n = 4 and n = p ≥ 3, a prime. It is not diﬃcult to

show that

x4 +y4

=

z4

has no non-trivial solutions in Z (this was first

shown by Fermat). Now, suppose that p ≥ 3 and a, b, c are integers

with abc = 0 and ap + bp = cp. Gerhard Frey conjectured that if such

a triple of integers exists, then the elliptic curve

E :

y2

= x(x −

ap)(x

+

bp)