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arXiv · 2502.14875

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

Paul M Voutier
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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.

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Paul M Voutier. 2025-02-03. Bounds on the number of squares in recurrence sequences: $y_{0}=b^{2}$ (I). https://doi.org/10.1016/j.jnt.2025.08.003

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