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

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.

Mathematically same as published version. Comments always welcome!