THE FERMAT–MERSENNE PROBLEM ON PYTHAGOREAN TRIPLES

Abstract In the 17th century, a letter from Fermat to Mersenne considered the problem of finding the Pythagorean triples in which the hypotenuse and the sum of the other two sides are squares. We call this question the Fermat–Mersenne problem. Number-theoretic solutions of the problem are known, but they are complicated. We solve the Fermat–Mersenne problem by using a geometric approach that provides new insight. The argument involves stereographically projecting points onto an elliptic curve and applying complex multiplication on that curve.