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Diophantine equations with Euler polynomials

Dijana Kreso; Csaba Rakaczki — 2013

Acta Arithmetica

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) ∈ ℚ [x] for which the diophantine equation $- 1^{k} + 2^{k} - \dots + {(- 1)}^{x} x^{k} = g (y)$ with k ≥ 7 may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.

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