Diophantine equations with Euler polynomials

Dijana Kreso; Csaba Rakaczki

Acta Arithmetica, Tome 161 (2013), p. 267-281 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 2013-01-01

Zbl 06238629

EUDML-ID : urn:eudml:doc:279149

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     author = {Dijana Kreso and Csaba Rakaczki},
     title = {Diophantine equations with Euler polynomials},
     journal = {Acta Arithmetica},
     volume = {161},
     year = {2013},
     pages = {267-281},
     zbl = {06238629},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-3-5}
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Dijana Kreso; Csaba Rakaczki. Diophantine equations with Euler polynomials. Acta Arithmetica, Tome 161 (2013) pp. 267-281. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-3-5/