English

The congruence speed formula

Number Theory 2022-08-05 v1

Abstract

We solve a few open problems related to a peculiar property of the integer tetration ba{^{b}a}, which is the constancy of its congruence speed for any sufficiently large b=b(a)b=b(a). Assuming radix-1010 (the well-known decimal numeral system), we provide an explicit formula for the congruence speed V(a)N0V(a) \in \mathbb{N}_0 of any aN{0}a \in \mathbb{N}-\{0\} that is not a multiple of 1010. In particular, for any given nNn \in \mathbb{N}, we prove to be true Rip\`a's conjecture on the smallest aa such that V(a)=nV(a)=n. Moreover, for any a1:a≢0(mod10)a \neq 1 : a \not\equiv 0 \pmod {10}, we show the existence of infinitely many prime numbers pj:=pj(V(a))p_j:=p_j(V(a)) such that V(pj)=V(a)V(p_j)=V(a).

Keywords

Cite

@article{arxiv.2208.02622,
  title  = {The congruence speed formula},
  author = {Marco Ripà},
  journal= {arXiv preprint arXiv:2208.02622},
  year   = {2022}
}

Comments

19 pages, 2 tables

R2 v1 2026-06-25T01:28:40.184Z