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 $m\not\equiv n\mathrm{mod}2$. In 1956, L. Jeśmanowicz \cite{Jes} 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 problem is not yet solved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent \cite{Lau} with some elementary methods, we prove that if $mn\equiv 2\mathrm{mod}4$ and $m>30.8n$, then Jeśmanowicz' conjecture is true.