On the constant congruence speed of tetration

Marco Ripà
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 245—260
DOI: 10.7546/nntdm.2020.26.3.245-260
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Authors and affiliations

Marco Ripà
sPIqr Society, World Intelligence Network
Rome, Italy

Abstract

Integer tetration, the iterated exponentiation ^ba for a ∈ ℕ − {0, 1}, is characterized by fascinating periodicity properties involving its rightmost figures, in any numeral system. Taking into account a radix-10 number system, in the book “La strana coda della serie n ^ n ^ … ^ n” (2011), the author analyzed how many new stable digits are generated by every unitary increment of the hyperexponent b, and he indicated this value as V(a) or “congruence speed” of a ≢ 0 (mod 10). A few conjectures about V(a) arose. If b is sufficiently large, the congruence speed does not depend on b, taking on a (strictly positive) unique value. We derive the formula that describes V(a) for every a ending in 5. Moreover, we claim that V(a) = 1 for any a (mod 25) ∈ {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and V(a) ≥ 2 otherwise. Finally, we show the size of the fundamental period P for any of the remaining values of the congruence speed: if V(a) ≥ 2, then P(V(a)) = 10^V(a)+1.

Keywords

• Number theory
• Power tower
• Tetration
• Hyperoperation
• Charmichael function
• Euler’s totient function
• Primitive root
• Exponentiation
• Integer sequence
• Congruence speed
• Modular arithmetic
• Stable digit
• Rightmost digit
• Cycle
• Periodicity

2010 Mathematics Subject Classification

• 11A07
• 11F33

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