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Glasnik matematički, Vol. 58 No. 1, 2023.

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 \( (n,m,\deg p,\deg q) \) 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 \( p,q \).

Keywords

Pillai problem, polynomials, \( S \)-units

Hrčak ID:

304391

URI

https://hrcak.srce.hr/304391

Publication date:

20.6.2023.

Visits: 525 *

Publication date: 30 June 2023

Volume: Vol 58

Issue: Svezak 1

Pages: 67-74

DOI: 10.3336/gm.58.1.05

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: Pillai problem, polynomials, \( S \)-units

Pillai's conjecture for polynomials

Sebastian Heintze[1]

Email: heintze@math.tugraz.at

 

[1] Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II, A-8010 Graz, Austria

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 \( (n,m,\deg p,\deg q) \) 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 \( p,q \).

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