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
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