Marco Ripà

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 3, Pages 245–260

DOI: 10.7546/nntdm.2020.26.3.245-260

**Full paper (PDF, 987 Kb)**

## Details

### Authors and affiliations

Marco Ripà

*sPIqr Society, World Intelligence Network
Rome, Italy*

### Abstract

Integer tetration, the iterated exponentiation * ^{b}a* for

*a*∈ ℕ − {0, 1}, is characterized by fascinating periodicity properties involving its rightmost figures, in any numeral system. Taking into account a radix-10 number system, in the book “La strana coda della serie

*n*^

*n*^ … ^

*n*” (2011), the author analyzed how many new stable digits are generated by every unitary increment of the hyperexponent

*b*, and he indicated this value as

*V*(

*a*) or “congruence speed” of

*a*≢ 0 (mod 10). A few conjectures about

*V*(

*a*) arose. If

*b*is sufficiently large, the congruence speed does not depend on

*b*, taking on a (strictly positive) unique value. We derive the formula that describes

*V*(

*a*) for every

*a*ending in 5. Moreover, we claim that

*V*(

*a*) = 1 for any

*a*(mod 25) ∈ {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and

*V*(

*a*) ≥ 2 otherwise. Finally, we show the size of the fundamental period

*P*for any of the remaining values of the congruence speed: if

*V*(

*a*) ≥ 2, then

*P*(

*V*(

*a*)) = 10

^{V(a)+1}.

### Keywords

- Number theory
- Power tower
- Tetration
- Hyperoperation
- Charmichael function
- Euler’s totient function
- Primitive root
- Exponentiation
- Integer sequence
- Congruence speed
- Modular arithmetic
- Stable digit
- Rightmost digit
- Cycle
- Periodicity

### 2010 Mathematics Subject Classification

- 11A07
- 11F33

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## Related papers

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*Notes on Number Theory and Discrete Mathematics*, 16(3), 18-24. - Ripà, M. (2021). The congruence speed formula.
*Notes on Number Theory and Discrete Mathematics*, 27(4), 43-61. - Ripà, M., & Onnis, L. (2022). Number of stable digits of any integer tetration.
*Notes on Number Theory and Discrete Mathematics*, 28(3), 441-457.

## Cite this paper

Ripà, M. (2020). On the constant congruence speed of tetration. *Notes on Number Theory and Discrete Mathematics*, 26 (3), 245-260, DOI: 10.7546/nntdm.2020.26.3.245-260.