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Practical solution of the Diophantine equation $ y^2 = x(x+2^ap^b)(x-2^ap^b)$


Authors: Konstantinos Draziotis and Dimitrios Poulakis
Journal: Math. Comp. 75 (2006), 1585-1593
MSC (2000): Primary 11Y50; Secondary 11D25, 11G05
DOI: https://doi.org/10.1090/S0025-5718-06-01841-2
Published electronically: March 29, 2006
MathSciNet review: 2219047
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Abstract: Let $ p$ be an odd prime and $ a$, $ b$ positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation $ y^2 = x(x+2^ap^b)(x-2^ap^b)$ can be easily reduced to the resolution of the unit equation $ u+\sqrt{2}v = 1$ over $ \mathbb{Q}(\sqrt{2},\sqrt{p})$. The solutions of the latter equation are given by Wildanger's algorithm.


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Additional Information

Konstantinos Draziotis
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: drazioti@math.auth.gr

Dimitrios Poulakis
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: poulakis@math.auth.gr

DOI: https://doi.org/10.1090/S0025-5718-06-01841-2
Received by editor(s): May 27, 2005
Received by editor(s) in revised form: June 18, 2005
Published electronically: March 29, 2006
Additional Notes: The research of the first author was supported by the Hellenic State Scholarships Foundation, I.K.Y
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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