Je{\'s}manowicz' conjecture for polynomials

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.