Original scientific paper

https://doi.org/10.3336/gm.58.1.05

Pillai's conjecture for polynomials

Sebastian Heintze ; Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II, A-8010 Graz, Austria

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Abstract

In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation
p^n - q^m = f.
We prove that for any non-constant polynomial $f$ there are only finitely many quadruples $(n,m,\mathrm{deg}p,\mathrm{deg}q)$ consisting of integers $n,m\ge 2$ 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 $p,q$.

Keywords

Pillai problem, polynomials, $S$-units

Hrčak ID:

304391

URI

https://hrcak.srce.hr/304391

Publication date:

20.6.2023.

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