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)y=(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.