An open problem on Jeśmanowicz' conjecture concerning primitive Pythagorean triples

Abstract

Let m > 31 be an even integer with gcd(m, 31) = 1. In this paper, using some elementary methods, we prove that the equation (m2 −312)x +(62m) = (m2 +312)z has only the positive integer solution (x, y, z) = (2, 2, 2). This result resolves an open problem raised by T. Miyazaki (Acta Arith. 186 (2018), 1–36) about Jeśmanowicz’ conjecture concerning primitive Pythagorean triples.