Diophantine equations with Euler polynomials

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 $k\ge 7$, has only finitely many integers solutions $x,y$ unless polynomial $g$ can be decomposed in ways that we list explicitly.

arXiv commentto appear in Acta Arithmetica