A congruent number is a positive integer that is the area of a right triangle with three rational number sides. In equations, we are required to find rational positive numbers such that:

and (1)

The story of congruent numbers is a very old one, beginning with Diophantus. The Arabs and Fibonacci knew of the problem in the following form:

Find three rational numbers whose squares form an arithmetic progression with common difference .

This is equivalent to finding integers with such that which reduces to finding a right triangle with rational sides

with area . This is the congruent number problem for . The Arabs knew several examples of congruent numbers and Fermat stated that no square is a congruent numbers. Since we can scale triangles to assume that is square free, this is equivalent to saying that is not a congruent number.

As with many other problems in number theory, the proof of this statement had to wait four centuries for Fermat. The problem led Fermat to discover his method of infinite descent.

In more recent times, the problem has been fruitfully translated into one about Elliptic Curves. We perform a rational transformation of the defining equations (1) for a congruent number in the following way. Set and . A calculation shows that:

(2)

and . If , then and but then . Conversely, given satisfying (2), we find and and one can check that these numbers satisfy (1).

The projective closure of (2) defines an elliptic curve that we will call . We are interested in finding rational points on it that do not satisfy . I will prove that is a congruent number precisely when has positive rank.

The proof is an interesting use of Dirichlet’s Theorem on Arithmetic Progressions and some neat ideas about Elliptic Curves and their reductions modulo primes. I will essentially assume the material in Silverman’s first book and the aforementioned Dirichlet’s Theorem.