Explicit formula for the solution of simultaneous Pell equations 𝑥²-(𝑎²-1)𝑦²=1, 𝑦²-𝑏𝑧²=1

Cipu, Mihai

doi:10.1090/proc/13802

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

Proc. Amer. Math. Soc. 146 (2018), 983-992

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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