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On the Diophantine equation $1^k+2^k+\dotsb+x^k=y^n$

Published online by Cambridge University Press:  15 October 2004

Michael A. Bennett ,
Kálmán Győry  and
Ákos Pintér
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, [email protected]
Kálmán Győry
Affiliation:
Mathematical Institute, University of Debrecen, PO Box 12, 4010 Debrecen, [email protected], [email protected]
Ákos Pintér
Affiliation:
Mathematical Institute, University of Debrecen, PO Box 12, 4010 Debrecen, [email protected], [email protected]
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Abstract

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In this paper, we resolve a conjecture of Schäffer on the solvability of Diophantine equations of the shape $1^k + 2^k + \dotsb + x^k = y^n$, for $1 \leq k \leq 11$. Our method, which may, with a modicum of effort, be extended to higher values of k, combines a wide variety of techniques, classical and modern, in Diophantine analysis.

Keywords

Diophantine equationsBernoulli polynomials11D4111B68
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 6 , November 2004 , pp. 1417 - 1431
Copyright
Foundation Compositio Mathematica 2004