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A computational approach for solving $y^2=1^k+2^k+\dotsb+x^k$

Authors: M. J. Jacobson Jr., Á. Pintér and P. G. Walsh
Journal: Math. Comp. 72 (2003), 2099-2110
MSC (2000): Primary 11D25, 11J86
Published electronically: May 1, 2003
MathSciNet review: 1986826
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Abstract: We present a computational approach for finding all integral solutions of the equation $y^2=1^k+2^k+\dotsb+x^k$ for even values of $k$. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for $2\le k\le 70$ assuming the Generalized Riemann Hypothesis, and for $2\le k\le 58$ unconditionally.

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Additional Information

M. J. Jacobson Jr.
Affiliation: Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4 Canada

Á. Pintér
Affiliation: Institute for Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary

P. G. Walsh
Affiliation: Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, Ontario, K1N 6N5 Canada

Keywords: Diophantine equations, elliptic curves, quadratic fields
Received by editor(s): June 5, 2001
Published electronically: May 1, 2003
Additional Notes: The first and third authors are supported by the Natural Sciences and Engineering Research Council of Canada
The second author is supported by the Hungarian National Foundation for Scientific Research, grants T29330, F34891, and FKFP-066-2001
Article copyright: © Copyright 2003 American Mathematical Society