OFFSET
2,1
COMMENTS
See A373387 for the definition of "constant congruence speed" in the decimal numeral system; i.e., assuming n = 10
Let ord_p(x) indicate the p-adic valuation of x (for any prime number p) and denote by V_n(m) the constant congruence speed of m in radix-n.
In particular, we have that for each odd prime p, than V_p(m) = ord_p(m^(p - 1) - 1) and V_2(m) = ord_2(m - 1) + ord_2(m + 1) - 1;
If n is squarefree and is not prime, V_n(m) depends on the residual class modulo n of m (i.e., by assuming n = 6, if m == 1,5 (mod 6), than V_6(m) = min(ord_2(m^2 - 1) + ord_3(m^2 - 1)); if m == 3 (mod 6), then V_6(m) = ord_2(m^2 - 1) - 1; if m == 2,4 (mod 6), than V_6(m) = ord_3(m^2 - 1); if m == 0 (mod 6), then V_6(m) does not exist).
In radix-n, for any selected integer n >= 2, the constant congruence speed exists for all integers m > 1 not divisible by n if and only if n is squarefree (see Appendix B of "A Compact Notation for Peculiar Properties Characterizing Integer Tetration", Version 3, in Links). If n is not squarefree, V_n(m) can exist or not and there are always infinitely many m := m(n), not divisible by n, whose constant congruence speed does not exist (while it still exists, for example, for all m := n^n - 1).
Let k be a positive integer. If n = 2^k or n = 3^k (e.g., 4, 8, 9, 16, 27, ...), then a(n) = n^n - 1 since for such n the only n-adic integer solutions of the fixed-point equation y^t = y (for any odd t >= 3) in the ring Z_n are 0, 1_n, and -1_n.
REFERENCES
Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.
LINKS
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024. See pp. 22-23, 27.
Marco Ripà and Gabriele Di Pietro, A Compact Notation for Peculiar Properties Characterizing Integer Tetration, Zenodo, 2025.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
Wikipedia, Tetration.
FORMULA
For p an odd prime, a(p) = min{m : ord_p(m^(p - 1) - 1) = p}.
For n >= 2, a(n) <= n^n - 1.
EXAMPLE
a(4) = 26 since 26 = 122_4 is the smallest integer (greater than 1) whose constant congruence speed exists and equals 4.
CROSSREFS
KEYWORD
more,hard,nonn
AUTHOR
Marco Ripà, Nov 12 2025
STATUS
approved