A note on the exceptional solutions of Jeśmanowicz’ conjecture concerning primitive Pythagorean triples

Abstract

Let $(m,\ n)$ be fixed positive integers such that $m>n,\ \gcd (m,\ n)=1$ and $mn\equiv 0 \pmod 2$. Then the triple $(m^2-n^2,\ 2mn,\ m^2+n^2)$ is called a primitive Pythagorean triple. In 1956, Jeśmanowicz (Wiadom Math 1(2):196–202, 1955/1956 ) 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 solved yet. A solution $(x,\ y,\ z)$ of this equation is called exceptional if $(x,\ y,\ z)\ne (2,\ 2,\ 2)$. In this paper, using Baker’s method, we prove that if $m>\max \{10^{127550},\ n^{5127},\ n^{(\log n)^2}\}$, then the above equation has no exception solutions $(x,\ y,\ z)$ with $x\equiv y\equiv 0 \pmod 2$. By this conclusion, we can deduce that if m, n satisfy the above condition, and $m\equiv 3 \pmod 4$ or $n\equiv 3 \pmod 4$, then Jeśmanowicz’ conjecture is true.