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PreprintCC BY 4.0arXiv

An Asymptotic for E. P. Stein Numbers

  • Cameron Rampell
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Abstract

Epstein's put-or-take-a-square game is an impartial combinatorial game in which a player may, in each turn, add or subtract the largest square not exceeding the current integer. We study E. P. Stein numbers, the positions from which both legal moves lead directly to positive squares, and prove the following asymptotic formula for the counting function S2(X) of these positions: S2(X)=1/π2X1/4log X+O(X1/4).

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Cite

BibTeXRIS
BibTeX
@misc{rampell2026asymptotic,
  title = {{An Asymptotic for E. P. Stein Numbers}},
  author = {Cameron Rampell},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05898},
  eprint = {2611.05898},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05898},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An Asymptotic for E. P. Stein Numbers
AU  - Cameron Rampell
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05898
UR  - https://arxiv.org/abs/2611.05898
AB  - Epstein's put-or-take-a-square game is an impartial combinatorial game in which a player may, in each turn, add or subtract the largest square not exceeding the current integer. We study E. P. Stein numbers, the positions from which both legal moves lead directly to positive squares, and prove the following asymptotic formula for the counting function S_2(X) of these positions: S_2(X)=1/π^2X^1/4log X+O(X^1/4).
ER  -