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

Abstract

We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, $\left( y_{k} \right)_{k=-\infty}^{\infty}$, arising from the solutions of generalised negative Pell equations, $X^{2}-dY^{2}=c$, where $-c$ and $y_{0}$ are any positive squares. We show that there are at most $2$ distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most $5$ distinct squares when $y_{0}=b^{2}$ for infinitely many values of $b$, including all $1 \leq b \leq 24$, as well as once $d$ exceeds an explicit lower bound, without any conditions on the size of such squares.