Diophantine equations with Euler polynomials

Volume 161 / 2013

Dijana Kreso, Csaba Rakaczki Acta Arithmetica 161 (2013), 267-281 MSC: Primary 11D41; Secondary 11B68. DOI: 10.4064/aa161-3-5


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.


  • Dijana KresoInstitut für Analysis und
    Computational Number Theory (Math A)
    Technische Universität Graz
    Steyrergasse 30/II
    8010 Graz, Austria
  • Csaba RakaczkiInstitute of Mathematics
    University of Miskolc
    H-3515 Miskolc Campus, Hungary

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