Bounds on the number of squares in recurrence sequences: $y_0=b^2$ (II)

Abstract

For binary recurrence sequences, ${\left(y_k\right)}_{k\in \mathbb{Z}}$, arising from the solutions of generalised Pell equations, $X^2-d{Y}^2=c$, 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 $y_0=1,2^2,\dots ,7^2$, or once $d$ exceeds an explicit lower bound, without any conditions on the size of such squares.

arXiv commentpublished, with corrections

References in corpus (2)

• Bounds on the number of squares in recurrence sequences
• Bounds on the number of squares in recurrence sequences: $y_0=b^2$ (I)