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Bounds on the number of squares in recurrence sequences: y0=b2 (II)

  • Paul M Voutier
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Abstract

For binary recurrence sequences, ( yk )k ∈ bbZ, arising from the solutions of generalised Pell equations, X2−dY2=c, where y0 is any positive square and c=−2ℓpm for an odd prime p and non-negative integers ℓ and m, we show that there are at most $4$ distinct squares with yk sufficiently large. From this result, we also show that there are at most $7$ distinct squares when y0=1,22,…,72, or once d exceeds an explicit lower bound, without any conditions on the size of such squares.

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Cite

BibTeXRIS
BibTeX
@misc{voutier2026bounds,
  title = {{Bounds on the number of squares in recurrence sequences: $y_0=b^2$ (II)}},
  author = {Paul M Voutier},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35158},
  eprint = {2610.35158},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35158},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Bounds on the number of squares in recurrence sequences: y_0=b^2 (II)
AU  - Paul M Voutier
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35158
UR  - https://arxiv.org/abs/2610.35158
AB  - For binary recurrence sequences, ( y_k )_k ∈ bbZ, arising from the solutions of generalised Pell equations, X^2−dY^2=c, where y_0 is any positive square and c=−2^ℓp^m for an odd prime p and non-negative integers ℓ 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,…,7^2, or once d exceeds an explicit lower bound, without any conditions on the size of such squares.
ER  -