The congruence speed formula

Marco Ripà, sPIqr Society, World Intelligence Network
2021 Notes on Number Theory and Discrete Mathematics  
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) ∈ ℕ_0 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
more » ... 10), we show the existence of infinitely many prime numbers, p_j = p_j(V(a)), such that V(p_j) = V(a).
doi:10.7546/nntdm.2021.27.4.43-61 fatcat:h5qpmxxl3nc3vdpaecah5q2hkm