The congruence speed formula

TL;DR

This paper derives an explicit formula for the congruence speed of integer tetration in base 10, proves Ripà's conjecture on minimal values, and shows infinitely many primes share the same congruence speed as given integers.

Contribution

It provides a new explicit formula for the congruence speed of tetration, proves Ripà's conjecture, and establishes the existence of infinitely many primes with the same congruence speed.

Findings

Explicit formula for congruence speed V(a) in base 10.

Proof of Ripà's conjecture on minimal a for given V(a).

Existence of infinitely many primes with the same congruence speed as a given non-multiple of 10.

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 $V(a) \in \mathbb{N}_0$ of any $a \in \mathbb{N}-\{0\}$ that is not a multiple of $10$. In particular, for any given $n \in \mathbb{N}$, we prove to be true Rip\`a's conjecture on the smallest $a$ such that $V(a)=n$. Moreover, for any $a \neq 1 : a \not\equiv 0 \pmod {10}$, we show the existence of infinitely many prime numbers $p_j:=p_j(V(a))$ such that $V(p_j)=V(a)$.