An application of Baker's method to the Je\'smanowicz' conjecture on primitive Pythagorean triples

TL;DR

This paper proves Je'smanowicz' conjecture for certain primitive Pythagorean triples by combining bounds on linear forms in logarithms with elementary methods, under specific conditions on m and n.

Contribution

It establishes the conjecture's validity for triples where mn ≡ 2 mod 4 and m > 30.8 n, using a novel combination of analytical and elementary techniques.

Findings

Proves Je'smanowicz' conjecture under specified conditions.

Uses bounds on linear forms in logarithms to derive results.

Identifies a new range of triples satisfying the conjecture.

Abstract

Let $m$, $n$ be positive integers such that $m>n$, $\gcd(m,n)=1$ and $m \not\equiv n \bmod 2$. In 1956, L. Je\'smanowicz \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 $mn \equiv 2 \bmod 4$ and $m > 30.8 n$, then Je\'smanowicz' conjecture is true.