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arXiv · 1807.04116

Sharp bounds on the number of squares in recurrence sequences and solutions of $X^{2}-\left( a^{2}+b \right) Y^{4}=-b$

Paul M Voutier
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Abstract

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd \left( a^{2}, b \right)$ is squarefree and $X^{2}- \left( a^{2}+b \right) Y^{2}=-4$ has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations.

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Paul M Voutier. 2018-07-11. Sharp bounds on the number of squares in recurrence sequences and solutions of $X^{2}-\left( a^{2}+b \right) Y^{4}=-b$. https://doi.org/10.1007/s40993-026-00712-7

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