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ON A DIOPHANTINE EQUATION OF CASSELS

Published online by Cambridge University Press:  27 July 2005

F. LUCA  and
P. G. WALSH
F. LUCA
Affiliation:
Instituto de Matemáticas UNAM, Campus MoreliaAp. Postal 61-3 (Xangari)CP 58 089 Morelia, Michoacán, Mexico e-mail: [email protected]
P. G. WALSH
Affiliation:
Department of Mathematics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada, K1N-6N5 e-mail: [email protected]
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Abstract

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J.W.S. Cassels gave a solution to the problem of determining all instances of the sum of three consecutive cubes being a square. This amounts to finding all integer solutions to the Diophantine equation $y^2=3x(x^2+2)$. We describe an alternative approach to solving not only this equation, but any equation of the type $y^2=nx(x^2+2)$, with $n$ a natural number. Moreover, we provide an explicit upper bound for the number of solutions of such Diophantine equations. The method we present uses the ingenious work of Wilhelm Ljunggren, and a recent improvement by the authors.

Keywords

11D25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue 2 , May 2005 , pp. 303 - 307
Copyright
2005 Glasgow Mathematical Journal Trust