Pillai's conjecture for polynomials

Abstract

In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation -17ex p^n - q^m = f. We prove that for any non-constant polynomial \( f \) there are only finitely many quadruples \( (n,m,\deg p,\deg q) \) consisting of integers \( n,m \geq 2 \) and non-constant polynomials \( p,q \) such that Pillai's equation holds. Moreover, we will give some examples that there can still be infinitely many possibilities for the polynomials \( p,q \).