Diophantine equations with Euler polynomials
Volume 161 / 2013
Acta Arithmetica 161 (2013), 267-281
MSC: Primary 11D41; Secondary 11B68.
DOI: 10.4064/aa161-3-5
Abstract
We determine decomposition properties of Euler polynomials and using a strong result relating polynomial decomposition and diophantine equations in two separated variables, we characterize those $g(x)\in \mathbb {Q}[x]$ for which the diophantine equation $$-1^k +2 ^k - \cdots + (-1)^{x} x^k=g(y) \hskip 1em\ {\rm with}\ k\geq 7$$ may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.
