1.1. Elliptic curves 3

For instance, the number 6 is congruent, because the right triangle

with sides of length (a, b, c) = (3,4,5) has area equal to

3·4

2

= 6.

Similarly, the number 30 is the area of the right triangle with sides

(5,12,13); thus, 30 is a congruent number.

Figure 1. A right triangle of area 5 and rational sides.

The number 5 is congruent but there is no right triangle with

integer sides and area equal to 5. However, our definition allowed

rational sides, and the triangle with sides

(

3

2

,

20

3

,

41

6

)

has area exactly

5. We do not allow, however, triangles with irrational sides even if

the area is an integer. For example, the right triangle (1,2,

√

5) has

area 1, but that does not imply that 1 is a congruent number (in fact,

1 is not a congruent number, as we shall see below).

The congruent number problem is one of the oldest open problems

in number theory. For more than a millennium, mathematicians have

attempted to provide a characterization of all congruent numbers.

The oldest written record of the problem dates back to the early

Middle Ages, when it appeared in an Arab manuscript written before

972 (a later 10th century manuscript written by Mohammed Ben

Alcohain would go as far as to claim that the principal object of the

theory of rational right triangles is to find congruent numbers). It is

known that Leonardo Pisano, a.k.a. Fibonacci, was challenged around

1220 by Johannes of Palermo to find a rational right triangle of area