Congruent Numbers and Heegner Points

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Mohammed Ben Alhocain, in an Arab manuscript of the tenth century, stated that the principal object of the theory of rational right triangles is to find a square which when increased or diminished by a certain number m becomes a square (see Dickson). In modern language, this object is to find a rational point of infinite order on the elliptic curve my²=x³-x. Heegner constructed (see also Monsky) such rational points in the case that m are primes congruent to 5, 7 modulo 8 or twice primes congruent to 6 modulo 8. We extend Heegner's result to integers m with many prime divisors.