A compact notation for peculiar properties characterizing integer tetration

Marco Ripà and Gabriele Di Pietro
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 477–496
DOI: 10.7546/nntdm.2026.32.3.477-496
Full paper (PDF, 310 Kb)

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Authors and affiliations

Marco Ripà
Independent Researcher, Rome, Italy

Gabriele Di Pietro
Independent Researcher, Roseto degli Abruzzi, Italy

Abstract

By initially working in the decimal numeral system, we introduce a compact notation to express the congruence speed of an integer tetration base , along with the cycle of the rightmost non-stable digits of for unit increments of . The resulting discrete function provides a useful tool for efficiently computing the exact number of frozen digits that characterize the right tail of each nontrivial integer tetration. We also establish an improved upper bound for the minimum hyperexponent that guarantees the constancy of the congruence speed of for all heights . Moreover, we prove that the minimum between the constant congruence speeds of any two integers greater than , whose product is not divisible by , is always less than or equal to the constant congruence speed of their product. Additionally, within the decimal numeral system, we give examples of infinitely many perfect powers whose degree matches their constant congruence speed at every height above , emphasizing the peculiar recurrence relations of hyper-. Finally, Appendix B generalizes the decimal framework described above to all squarefree numeral systems, showing that only in such systems the congruence speed stabilizes to a fixed (positive) value for all not divisible by the radical of the radix. Furthermore, we derive compact formulas for all prime-radix numeral systems and for the composite squarefree senary case.

Keywords

• Congruence speed
• Decimal number system
• Integer factorization
• Modular arithmetic
• Phase shift
• Tetration

2020 Mathematics Subject Classification

• Primary: 11A07, 11N75
• Secondary: 11B50, 13A05

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Manuscript history

• Received: 20 May 2025
• Revised: 19 February 2026
• Accepted: 7 October 2026
• Online First: 15 October 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).