Glasnik Matematicki, Vol. 58, No. 1 (2023), 67-74. \( \)

PILLAI'S CONJECTURE FOR POLYNOMIALS

Sebastian Heintze

Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II, A-8010 Graz, Austria
e-mail:heintze@math.tugraz.at

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 \( p,q \).

2020 Mathematics Subject Classification.   11D61, 11D85

Key words and phrases.   Pillai problem, polynomials, \( S \)-units

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

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