An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples

Abstract

Let m, n be positive integers such that m > n, gcd(m,n) = 1 and m not equivalent to n(mod2) . In 1956, L. Jesmanowicz conjectured that the equation (m(2)-n(2))(x) + (2mn)(y) = (m(2) + n(2))(z) has only the positive integer solution (x,y,z)=(2,2,2). This conjecture is still unsolved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent with some elementary methods, we prove that if mn equivalent to 2(mod 4) and m > 30.8n, then Jesmanowicz' conjecture is true.