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Diophantine Equations and Bernoulli Polynomials

Published online by Cambridge University Press:  04 December 2007

Yu. F. Bilu ,
B. Brindza ,
P. Kirschenhofer ,
Á. Pintér ,
R. F. Tichy  and
A. Schinzel
Yu. F. Bilu
Affiliation:
A2X, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France. E-mail: yuri@math.u-bordeaux.fr
B. Brindza
Affiliation:
Department of Mathematics, PO Box 12, H-4010, Debrecen, University of Debrecen, Hungary
P. Kirschenhofer
Affiliation:
Montanuniversität Leoben, Franz Josef-Str. 18, 8700 Leoben, Austria. E-mail: kirsch@unileoben.ac.at
Á. Pintér
Affiliation:
Department of Mathematics, PO Box 12, H-4010, Debrecen, University of Debrecen, Hungary. E-mail: apinter@math.klte.hu
R. F. Tichy
Affiliation:
Institut für Mathematik (A), Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria. E-mail: tichy@weyl.math.tu-graz.ac.at
A. Schinzel
Affiliation:
Mathematical Institute PAN, PO Box 137, 00-950 Warszawa, Poland. E-mail: schinzel@plearn.edu.pl
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Abstract

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Given m, n [ges ] 2, we prove that, for sufficiently large y, the sum 1n +···+ yn is not a product of m consecutive integers. We also prove that for m ≠ n we have 1m +···+ xm ≠ 1n +···+ yn, 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

Diophantine equationsBernoulli polynomialspower sumsproducts of consecutive integersindecomposable polynomials
Type
Research Article
Information
Compositio Mathematica , Volume 131 , Issue 2 , April 2002 , pp. 173 - 188
Copyright
© 2002 Kluwer Academic Publishers