4 1. Introduction

n = 5, and Fibonacci found the triangle (

3

2

,

20

3

,

41

6

). We will explain a

method to find this triangle below. In 1225, Fibonacci wrote a more

general treatment about the congruent number problem, in which he

stated (without proof) that if n is a perfect square, then n cannot

be a congruent number. The proof of such a claim had to wait until

Pierre de Fermat (1601-1665) settled that the number 1 (and every

square number) is not a congruent number (a result that he showed

in order to prove the case n = 4 of Fermat’s last theorem).

The connection between the congruent number problem and el-

liptic curves is as follows:

Proposition 1.1.3. The number n 0 is congruent if and only if the

curve

y2

=

x3

−

n2x

has a point (x, y) with x, y ∈ Q and y = 0. More

precisely, there is a one-to-one correspondence Cn ←→ En between

the following two sets:

Cn = {(a, b, c) :

a2

+

b2

=

c2,

ab

2

= n}

En = {(x, y) :

y2

=

x3

−

n2x,

y = 0}.

Mutually inverse correspondences f : Cn → En and g : En → Cn are

given by

f((a, b, c)) =

nb

c − a

,

2n2

c − a

, g((x, y)) =

x2

−

n2

y

,

2nx

y

,

x2

+

n2

y

.

The reader can provide a proof (see Exercise 1.4.3). For example,

the curve E :

y2

=

x3

− 25x has a point (−4,6) that corresponds to

the triangle (

3

2

,

20

3

,

41

6

). But E has other points, such as (

1681

144

,

62279

1728

)

that corresponds to the triangle

1519

492

,

4920

1519

,

3344161

747348

which also has area equal to 5. See Figure 2.

Today, there are partial results toward the solution of the congru-

ent number problem, and strong results that rely heavily on famous

(and widely accepted) conjectures, but we do not have a full answer

yet. For instance, in 1975 (see [Ste75]), Stephens showed that the

Birch and Swinnerton-Dyer conjecture (which we will discuss in Sec-

tion 5.2) implies that any positive integer n ≡ 5, 6 or 7 mod 8 is a