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Nonconstant congruence speeds of integer tetration: period spectra, prime bases, explicit bounds, and densities

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

Contributed by Marco Ripà ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

Building on earlier human research into the constant congruence speed of integer tetration, including Appendix B of Ripà and Di Pietro (2026), and on a lifting-the-exponent (LTE) criterion publicly presented by Alper Ferudun for OEIS A390535 and A390576, this model-generated note proposes an explicit account of nonconstant speeds in non-squarefree numeral systems. Its principal results are an eventual floor-difference representation, an exact spectrum of attainable least periods, and explicit arithmetic progressions containing infinitely many prime bases whose congruence speed never becomes constant for each non-squarefree radix. The latter supplies a proposed proof of the coprime-infinitude assertion left conjectural in Appendix B of the earlier human-authored article. We also derive bounds for the least such base and exact natural densities, both among coprime bases and among all bases not divisible by the radical of the radix, together with relative prime densities. The mathematical derivations and manuscript are primarily GPT-6-generated. Ripà supplied the theoretical setting, earlier results and specialist guidance, and subsequently examined selected deductions and historical test cases. The manuscript has not received comprehensive independent human proof verification.

Provenance statement

This manuscript was produced primarily by GPT-6 (OpenAI) through an interactive research dialogue directed by me, Marco Ripà. I supplied the mathematical context, my previously published theory of tetration congruence speeds, my earlier knowledge of eventual periodicity and rational mean speeds, and the LTE criterion publicly presented by Alper Ferudun on OEIS A390535 and A390576. GPT-6 generated the proposed theorems, mathematical arguments, proofs, computational checks, and LaTeX manuscript. The initial research note was generated on October 25, 2026; a subsequent expansion, also generated by GPT-6 in response to my questions, included density results for bases not necessarily coprime to the radix. Preliminary versions were shared on ResearchGate. On October 27, I directed a further review, identified connections with my earlier published results (including my 2021 use of Dirichlet's theorem), and supplied selected test cases informed by unpublished mathematical investigations predating this dialogue. GPT-6 carried out additional analyses and computational comparisons and prepared the present manuscript for Hexagon. My earlier mathematical papers, including my sole-authored 2021 paper, my 2022 paper with Luca Onnis, and my 2026 paper with Gabriele Di Pietro, were developed and written entirely by human researchers, without any use of artificial intelligence. I have discussed and examined selected claims in the present manuscript, but the proofs have not undergone comprehensive independent human verification.

Tools used

OpenAI
ChatGPTVersion 6

References

  1. M. Ripà and G. Di Pietro, A compact notation for peculiar properties characterizing integer tetration , Notes on Number Theory and Discrete Mathematics 32 (2026), no. 3, 477--496, especially Appendix B. https://doi.org/10.7546/nntdm.2026.32.3.477-496.DOI
  2. OEIS Foundation, A390576: Smallest integer greater than 1 and coprime to A013929(n) whose radix-A013929(n) congruence speed never stabilizes , especially Alper Ferudun's exact LTE criterion and PARI implementation, October 2026. https://oeis.org/A390576.link
  3. OEIS Foundation, A390535: Smallest integer >1>1 whose radix-A013929(n) congruence speed never stabilizes , especially Alper Ferudun's October 2026 contribution. https://oeis.org/A390535.link
  4. M. Ripà, Predicting the smallest nontrivial natural number whose congruence speed never stabilizes in a given numeral system , MathOverflow question 504857 (11 December 2025). https://mathoverflow.net/questions/504857.link
  5. M. Ripà, Evaluation of GPT-5 on an Advanced Extension of Kashihara's Problem , Zenodo, August 2025. https://doi.org/10.5281/zenodo.16919232.DOI
  6. M. Ripà, La strana coda della serie n n ... n , UNI Service, Trento, 2011. ISBN 978-88-6178-789-6.
  7. M. Ripà, On the constant congruence speed of tetration , Notes on Number Theory and Discrete Mathematics 26 (2020), no. 3, 245--260. https://doi.org/10.7546/nntdm.2020.26.3.245-260.DOI
  8. M. Ripà, The congruence speed formula , Notes on Number Theory and Discrete Mathematics 27 (2021), no. 4, 43--61. https://doi.org/10.7546/nntdm.2021.27.4.43-61. ArXiv version, Section 4 and Theorem 4.6: https://arxiv.org/abs/2208.02622.DOI
  9. M. Ripà and L. Onnis, Number of stable digits of any integer tetration , Notes on Number Theory and Discrete Mathematics 28 (2022), no. 3, 441--457. https://doi.org/10.7546/nntdm.2022.28.3.441-457.DOI

Version history

  1. v1Submitted by Marco RipàInitial depositCurrentOct 27, 2026