Diophantine equations with Euler polynomials

D. Kreso; Cs. Rakaczki

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arXiv · 1312.3907

Diophantine equations with Euler polynomials

D. Kreso·Cs. Rakaczki

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Abstract

In this paper we determine possible decompositions of Euler polynomials $E_k(x)$, i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation $$-1^k +2 ^k - \cdots + (-1)^{x} x^k=g(y),$$ with $g\in \mathbb{Q}[X]$ of degree at least $2$ and $k\geq 7$, has only finitely many integers solutions $x, y$ unless polynomial $g$ can be decomposed in ways that we list explicitly.

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BibTeX RIS

D. Kreso, Cs. Rakaczki. 2013-12-13. Diophantine equations with Euler polynomials. https://arxiv.org/abs/1312.3907

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