Bounds on the number of squares in recurrence sequences: $y_{0}=b^{2}$ (I)
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912592096657408 |
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| author | Voutier, Paul M |
| author_facet | Voutier, Paul M |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on the number of squares in recurrence sequences: $y_{0}=b^{2}$ (I) Voutier, Paul M Number Theory 11B37, 11B39, 11J82 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. |
| title | Bounds on the number of squares in recurrence sequences: $y_{0}=b^{2}$ (I) |
| topic | Number Theory 11B37, 11B39, 11J82 |
| url | https://arxiv.org/abs/2502.14875 |