A Computational Approach for Solving$y^{2} = 1^{k} + 2^{k} + \cdot \cdot \cdot + x^k

We present a computational approach for finding all integral solutions of the equation$y^{2} = 1^{k} + 2^{k} + \cdot\cdot\cdot + x^{k}$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, w...

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Bibliographic Details
Published in:Mathematics of computation 2003-10, Vol.72 (244), p.2099-2110
Main Authors: M. J. Jacobson, Jr, Pintér, Á., Walsh, P. G.
Format: Article
Language:English
Subjects:
Algorithms
Curves
Decimals
Diophantine equation
Integers
Mathematical tables
Mathematical theorems
Polynomials
Trivial solutions
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Summary:We present a computational approach for finding all integral solutions of the equation$y^{2} = 1^{k} + 2^{k} + \cdot\cdot\cdot + x^{k}$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$2 \leq k \leq 70$assuming the Generalized Riemann Hypothesis, and for$2 \leq k \leq 58$unconditionally.
ISSN:0025-5718
1088-6842