Bounds on the number of squares in recurrence sequences: $y_{0}=b^{2}$ (II)

Abstract

For binary recurrence sequences, $\left( y_{k} \right)_{k \in \bbZ}$, arising from the solutions of generalised Pell equations, $X^{2}-dY^{2}=c$, where $y_{0}$ is any positive square and $c=-2^{\ell}p^{m}$ for an odd prime $p$ and non-negative integers $\ell$ and $m$, we show that there are at most $4$ distinct squares with $y_{k}$ sufficiently large. From this result, we also show that there are at most $7$ distinct squares when $y_{0}=1,2^{2},\ldots,7^{2}$, or once $d$ exceeds an explicit lower bound, without any conditions on the size of such squares.