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Explicit formula for the solution of simultaneous Pell equations $ x^2-(a^2-1)y^2=1$, $ y^2-bz^2=1$


Author: Mihai Cipu
Journal: Proc. Amer. Math. Soc. 146 (2018), 983-992
MSC (2010): Primary 11D09; Secondary 11D25, 11D45, 11B37
DOI: https://doi.org/10.1090/proc/13802
Published electronically: October 23, 2017
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Abstract: For $ b$ an odd integer whose square-free part has at most two prime divisors, it is shown that the equations in the title have a common solution in positive integers precisely when $ b$ divides $ 4a^2-1$ and the quotient is a perfect square. The proof provides an explicit formula for the common solution, known to be unique. Similar results are obtained assuming the square-free part of $ b$ is even or has three prime divisors.


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Additional Information

Mihai Cipu
Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit nr. 5, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: Mihai.Cipu@imar.ro

DOI: https://doi.org/10.1090/proc/13802
Keywords: Pell equations, quartic Diophantine equations, Lucas/Lehmer sequences
Received by editor(s): February 9, 2017
Received by editor(s) in revised form: April 13, 2017, and April 26, 2017
Published electronically: October 23, 2017
Dedicated: Dedicated to Professor Maurice Mignotte on the occasion of his retirement
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

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