Diophantine equations with Euler polynomials

2013/12/13 by D. Kreso, Kreso, D., Cs. Rakaczki +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Primary 11D41 #Secondary 11B68 #math.NT #msc:11B68 #msc:11D41

paper · pdf · doi:10.48550/arxiv.1312.3907

to appear in Acta Arithmetica

arxiv created 2013/12/13 · arxiv updated 2013/12/16

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 - ⋯ + (-1)^x x^k=g(y), with g∈ ℚ[X] of degree at least 2 and k≥ 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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