Diophantine Equations and Bernoulli Polynomials

Authors: Bilu, Y.F.1; Brindza, B.2; Kirschenhofer, P.3; Pintér Á.4; Tichy, R.F.5; Schinzel, A.6

Source: Compositio Mathematica, Volume 131, Number 2, April 2002 , pp. 173-188(16)

Publisher: Springer

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Abstract:

Given m, n  2, we prove that, for sufficiently large y, the sum 1^n +···+ y^n is not a product of m consecutive integers. We also prove that for m ≠ n we have 1^m +···+ x^m ≠ 1^n +···+ y^n, provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are ‘almost’ indecomposable, a result of independent interest.

Keywords: Bernoulli polynomials; Diophantine equations; indecomposable polynomials; power sums; products of consecutive integers

Document Type: Regular Paper

Affiliations: 1: A2X, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France. E-mail: yuri@math.u-bordeaux.fr 2: Department of Mathematics, PO Box 12, H-4010, Debrecen, University of Debrecen, Hungary 3: Montanuniversität Leoben, Franz Josef-Str. 18, 8700 Leoben, Austria. E-mail: kirsch@unileoben.ac.at 4: Department of Mathematics, PO Box 12, H-4010, Debrecen, University of Debrecen, Hungary. E-mail: apinter@math.klte.hu 5: Institut für Mathematik (A), Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria. E-mail: tichy@weyl.math.tu-graz.ac.at 6: Mathematical Institute PAN, PO Box 137, 00-950 Warszawa, Poland. E-mail: schinzel@plearn.edu.pl

Publication date: April 1, 2002

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