Jeśmanowicz’ conjecture for polynomials

Abstract

Let (a, b, c) be pairwise relatively prime integers such that $$a^2 + b^2 = c^2$$ . In 1956, Jeśmanowicz 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.