The congruence speed formula

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 Ripà's conjecture on the smallest $a$ such that $V(a)=n$. Moreover, for any $a\ne 1:a\not\equiv 0(\mathrm{mod}10)$, we show the existence of infinitely many prime numbers $p_j:=p_j(V(a))$ such that $V(p_j)=V(a)$.

19 pages, 2 tables

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• Number of stable digits of any integer tetration