Pillai's conjecture for polynomials

TL;DR

This paper investigates a polynomial analogue of Pillai's conjecture, proving finiteness results for solutions involving non-constant polynomials and providing examples of infinite solutions under certain conditions.

Contribution

It establishes the finiteness of solutions for polynomial Pillai's equation with non-constant polynomials, extending classical number theory conjectures into polynomial settings.

Findings

Finiteness of solutions for non-constant polynomial cases

Existence of infinitely many solutions in specific polynomial cases

Examples illustrating the diversity of solutions

Abstract

In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation \begin{equation*} p^n - q^m = f. \end{equation*} We prove that for any non-constant polynomial $ f $ there are only finitely many vectors $ (n,m,\mathrm{deg}\ p,\mathrm{deg}\ q) $ with 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 $.