# The congruence speed formula

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

## 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 ba, 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) ∈ ℕ0 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, pj = pj(V(a)), such that V(pj) = V(a).

### Keywords

• Tetration
• Exponentiation
• Integer sequence
• Congruence speed
• Modular arithmetic
• Dirichlet’s theorem
• Arithmetic progression
• Prime number

• 11A07
• 11N13

### References

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## 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.