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

TL;DR

This paper establishes optimal bounds on the number of squares in specific recurrence sequences and applies these to solve certain Diophantine equations involving squares and prime powers.

Contribution

It provides the best possible bounds for the number of squares in recurrence sequences related to solutions of a class of quadratic Diophantine equations.

Findings

Derived sharp bounds for the number of squares in recurrence sequences.

Applied bounds to determine solutions of specific quadratic equations.

Established conditions under which solutions exist or are finite.

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.