Marco Ripà and Luca Onnis

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 3, Pages 441–457

DOI: 10.7546/nntdm.2022.28.3.441-457

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## Details

### Authors and affiliations

**Marco Ripà**

*sPIqr Society, World Intelligence Network
Rome, Italy*

**Luca Onnis**

*Independent researcher
Cagliari, Italy*

### Abstract

In the present paper we provide a formula that allows to compute the number of stable digits of any integer tetration base . The number of stable digits, at the given height of the power tower, indicates how many of the last digits of the (generic) tetration are frozen. Our formula is exact for every tetration base which is not coprime to , although a maximum gap equal to digits (where denotes the constant congruence speed of ) can occur, in the worst-case scenario, between the upper and lower bound. In addition, for every which is not a multiple of , we show that corresponds to the -adic or -adic valuation of or , or even to the -adic order of , depending on the congruence class of modulo .

### Keywords

- Tetration
- Exponentiation
- Congruence speed
- Modular arithmetic
- Stable digits
*p*-adic valuation

### 2020 Mathematics Subject Classification

- 11A07
- 11A15

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### Manuscript history

- Received: 16 February 2022
- Revised: 29 June 2022
- Accepted: 21 July 2022
- Online First: 24 July 2022

## Related papers

- 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. (2021). The congruence speed formula.
*Notes on Number Theory and Discrete Mathematics*, 27(4), 43-61.