多项式的皮莱猜想

Pub Date : 2022-01-26 DOI:10.3336/gm.58.1.05

Sebastian Heintze

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引用次数: 0

摘要

本文研究了指数丢番图方程-17ex p^n - q^m = f上Pillai猜想的多项式版本。我们证明了对于任意非常多项式\( f \)，只有有限多个四元组\( (n,m,\deg p,\deg q) \)由整数\( n,m \geq 2 \)和非常多项式\( p,q \)组成，使得Pillai方程成立。此外，我们将给出一些例子，说明多项式仍然有无限多种可能性\( p,q \)。

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Pillai's conjecture for polynomials

In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation -17ex 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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