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Nonconstant Congruence Speeds of Integer Tetration in Non-Squarefree Radices

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Marco Ripà ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

We further develop the theory of congruence speed for integer tetration in non-squarefree numeral systems, building on Appendix B of [M. Ripà and G. Di Pietro (2026), A compact notation for peculiar properties characterizing integer tetration, Notes on Number Theory and Discrete Mathematics, 32(3), 477–496]. For bases coprime to the radix, we establish an eventual floor-difference representation and determine the exact spectrum of attainable least periods. We construct explicit arithmetic progressions containing infinitely many prime bases with nonconstant congruence speed for every non-squarefree radix, thereby proving the coprime-infinitude conjecture posed in Appendix B. We also derive a uniform quadratic bound for the least coprime base with nonconstant congruence speed and establish exact natural densities of nonstabilizing bases, both among integers coprime to the radix and among those not divisible by its radical, together with the corresponding relative density among primes. Finally, we give necessary and sufficient conditions for the perfect-power congruence-speed identity (Equation (2.12) of the cited paper) to hold in arbitrary radices, determine the exact least eventual period when stabilization fails, and establish a separate criterion for the boundary case of the original parameter restriction.

Provenance statement

This manuscript is a substantially revised development of the earlier AI-generated research note deposited as hexagon:2610.00589v1 in October 2026. The author supplied the theoretical framework, results from his previous mathematical publications (2011–2026), independently developed observations on the periodic behavior of congruence speeds, and the LTE criterion publicly presented by Alper Ferudun in OEIS A390535 and A390576. All previous papers in this research line were developed and written entirely by human researchers, without AI-generated mathematical contributions. Through extensive interaction with GPT-6 (OpenAI), the author formulated research questions, supplied examples and counterexamples, challenged proposed hypotheses and boundary cases, and directed successive mathematical and editorial revisions. GPT-6 generated the principal new mathematical derivations and proposed proofs. The author subsequently conducted a detailed critical review of the entire manuscript, examining in depth its mathematical arguments and substantially refining the exposition. The earlier version (hexagon:2610.00589v1) was formally verified in Lean 4 by Aldo Roberto Pessolano, whereas the revised and expanded manuscript presented here has not yet undergone complete formal verification. The author assumes full responsibility for its contents.

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References

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  2. OEIS Foundation Inc., Smallest integer >1>1 whose radix-A013929(nn) congruence speed never stabilizes , Entry A390535 in The On-Line Encyclopedia of Integer Sequences , 2025 (updated 2026). Available at https://oeis.org/A390535.link
  3. OEIS Foundation Inc., Smallest integer greater than 11 and coprime to A013929(nn) whose radix-A013929(nn) congruence speed never stabilizes , Entry A390576 in The On-Line Encyclopedia of Integer Sequences , 2025 (updated 2026). Available at https://oeis.org/A390576.link
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  10. 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. Available at https://doi.org/10.7546/nntdm.2022.28.3.441-457.DOI
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Version history

  1. v1Submitted by Marco RipàInitial depositCurrentNov 04, 2026