Marco Ripà

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 4, Pages 43–61

DOI: 10.7546/nntdm.2021.27.4.43-61

**Full paper (PDF, 227 Kb)**

## Details

### Authors and affiliations

Marco Ripà

*sPIqr Society, World Intelligence Network
Rome, Italy
*

### Abstract

We solve a few open problems related to a peculiar property of the integer tetration * ^{b}a*, which is the constancy of its congruence speed for any sufficiently large

*b*=

*b*(

*a*). Assuming radix-10 (the well known decimal numeral system), we provide an explicit formula for the congruence speed

*V*(

*a*) ∈ ℕ

_{0}of any

*a*∈ ℕ − {0} that is not a multiple of 10. In particular, for any given

*n*∈ ℕ, we prove to be true Ripà’s conjecture on the smallest

*a*such that

*V*(

*a*) =

*n*. Moreover, for any

*a*≠ 1 ∶

*a*≢ 0 (mod 10), we show the existence of infinitely many prime numbers,

*p*=

_{j}*p*(

_{j}*V*(

*a*)), such that

*V*(

*p*) =

_{j}*V*(

*a*).

### Keywords

- Tetration
- Decadic number
- Exponentiation
- Integer sequence
- Congruence speed
- Modular arithmetic
- Radix-10
- Dirichlet’s theorem
- Arithmetic progression
- Prime number

### 2020 Mathematics Subject Classification

- 11A07
- 11N13

### References

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- Vassilev-Missana, M. (2010). Some results on infinite power towers.
*Notes on Number Theory and Discrete Mathematics*, 16(3), 18-24. - Ripà, M. (2020). On the constant congruence speed of tetration,
*Notes on Number Theory and Discrete Mathematics*, 26(3), 245–260. - 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. (2021). The congruence speed formula. *Notes on Number Theory and Discrete Mathematics*, 27(4), 43-61, DOI: 10.7546/nntdm.2021.27.4.43-61.