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arXiv · 2201.10964

Pillai's conjecture for polynomials

Sebastian Heintze
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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 $.

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Sebastian Heintze. 2022-01-26. Pillai's conjecture for polynomials. https://doi.org/10.3336/gm.58.1.05

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