1.1. Elliptic curves 5

Figure 2. Two rational points on the curve

y2

=

x3

− 25x.

congruent number. For example, n = 157 ≡ 5 mod 8 must be a con-

gruent number and, indeed, Don Zagier has exhibited a right triangle

(a, b, c) whose area equals 157. The hypotenuse of the simplest such

triangle is:

c =

2244035177043369699245575130906674863160948472041

8912332268928859588025535178967163570016480830

.

In Example 5.2.7 we will see an application of the conjecture of Birch

and Swinnerton-Dyer to find a rational point P on

y2

=

x3

−

1572x,

which corresponds to a right triangle of area 157 via the correspon-

dence in Proposition 1.1.3.

The best known result on the congruent number problem is due

to J. Tunnell:

Theorem 1.1.4 (Tunnell, 1983, [Tun83]). If n is an odd square-

free positive integer and n is the area of a right triangle with rational

sides, then the following cardinalities are equal:

#{(x, y, z) ∈

Z3

: n =

2x2

+

y2

+

32z2}

=

1

2

(

# (x, y, z) ∈

Z3

: n =

2x2

+

y2

+

8z2

)