1. Introduction. A rational right triangle is a right triangle whose sides are all positive rational numbers. Such a triangle is denoted {a, b, c} where a and b are the legs, and c is the hypotenuse. Throughout this paper, a squarefree integer is understood to be a positive integer which is not divisible by the square of an integer greater than 1. A congruent number is a square-free integer which is the area of a rational right triangle. A square-free integer N is a congruent number if and only if the elliptic curve N y 2 = (x 2 − 1)x has positive rank. For details, see Koblitz [7].In the spirit of Euclid's proof of the infinitude of prime numbers, one can also show that there are infinitely many congruent numbers as follows: If there were only finitely many of them, say N 1 , . . . , N r , all greater than 1, then consider N = N 1 · · · N r . Elementary number theory shows that sqf(N 3 − N ), the square-free part of N 3 − N, cannot be 1. Moreover, it is a congruent number which cannot be any of the N i 's. Indeed, if it were N 1 , say, let M = N 2 · · · N r and d = gcd(N 1 , M ). Writing N 1 = dn and M = dm with gcd(m, n) = 1, one sees that sqf(m) sqf(N 2 1 M 2 − 1) = d. This last equality implies that sqf(N 2 1 M 2 − 1) divides d, and hence M, but at the same time, since it divides N 2 1 M 2 − 1, it must be 1, and this is impossible.Chahal [2] established that the residue classes of 1, 2, 3, 5, 6, 7 modulo 8 contain infinitely many congruent numbers. Bennett [1] extended Chahal's result by showing that if a and m are positive integers such that gcd(a, m) is square-free, then the residue class of a modulo m contains infinitely many congruent numbers.In this paper we prove the following: