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A √3 growth criterion for weighted reciprocal series

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by twinheterodox ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

We prove that ∑n≥11/(an2an+1)\sum_{n\ge1}1/(a_n^2a_{n+1}) is irrational for every strictly increasing sequence of positive integers satisfying lim sup⁡n→∞log⁡an/(3)n=∞\limsup_{n\to\infty}\log a_n/(\sqrt{3})^n=\infty. This lowers the sufficient growth base from 22 to 3\sqrt{3} in the criterion of Barreto, Kang, Kim, Kovač and Zhang (Bull. Lond. Math. Soc. 58 (2026), e70457, Theorem 3) for weights (2,1)(2,1). Irregular growth is allowed. The result also extends to nondecreasing sequences and positive integer numerators subject to polynomial size bounds.

Author comments: 6 pages. Includes a general criterion with positive integer numerators. Source and build instructions: https://github.com/twinheterodox/weighted-reciprocal-growth

Provenance statement

This work was developed with OpenAI Codex using GPT-6-Astra max. I directed the investigation, discussed the mathematical ideas, and reviewed and edited the exposition. Two cooperating Codex research sessions developed and cross-reviewed the argument. A separate mathematical audit of the underlying proof, completed on 13 November 2026, found no error or missing hypothesis. My review covers the problem, the main result and parts of the reasoning; I have not independently verified the complete technical proof. The audit is not formal verification.

Tools used

OpenAI
CodexVersion GPT-6-Astra max

References

  1. K. Barreto, J. Kang, S.-h. Kim, V. Kovač and S. Zhang, Irrationality of rapidly converging series: A problem of Erdős and Graham , Bull.\ Lond.\ Math.\ Soc.\ 58 (2026), no. 8, e70457. doi:10.1112/blms.70457 (https://doi.org/10.1112/blms.70457). arXiv:2601.21442v3 (https://arxiv.org/abs/2601.21442v3).arXivDOI
  2. P. Erdős, Some problems and results on the irrationality of the sum of infinite series , J.\ Math.\ Sci.\ 10 (1975), 1--7. Full text (https://www.renyi.hu/~p_erdos/1976-44.pdf).link
  3. P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory , Monographies de L'Enseignement Mathématique, vol. 28, Université de Genève, L'Enseignement Mathématique, Geneva, 1980. Full text (https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf).link

Version history

  1. v1 · 6 pagesSubmitted by twinheterodox6 pages. Includes a general criterion with positive integer numerators. Source and build instructions: https://github.com/twinheterodox/weighted-reciprocal-growthCurrentNov 13, 2026