Je{\'s}manowicz' conjecture for polynomials

TL;DR

This paper proves a polynomial analogue of Je{\'s}manowicz' conjecture, which originally concerns integer solutions to a specific exponential equation involving Pythagorean triples.

Contribution

The paper introduces and proves a polynomial version of Je{\'s}manowicz' conjecture, extending the classical number theory problem to polynomial equations.

Findings

Polynomial analogue of Je{\'s}manowicz' conjecture proven

Unique solution in polynomial case is the quadratic form

Extends classical conjecture to polynomial domain

Abstract

Let $(a,b,c)$ be pairwise relatively prime integers such that $a^2 + b^2 = c^2$. In 1956, Je{\'s}manowicz conjectured that the only solution of $a^x + b^y = c^z$ in positive integers is $(x,y,z)=(2,2,2)$. In this note we prove a polynomial analogue of this conjecture.