OFFSET
1,1
COMMENTS
10-adic expansion of the iterated exponential 1984^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n>=9, 1984^^n(mod 10^8) == 98703616.
1984^^n, for any n>=188, appears in M. Ripà's book "La strana coda della serie n^n^...^n", where the author took his birth year (1984), as a random base in order to prove some general properties about tetration, and calculating 1984^^n(mod 10^187) as a test for his paper-and-pencil procedure.
REFERENCES
M. Gardner, Mathematical Games, Scientific American 237, 18 - 28 (1977).
M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 78-79. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
LINKS
J. Jimenez Urroz and J. Luis A. Yebra, On the equation a^x == x (mod b^n), J. Int. Seq. 12 (2009) #09.8.8.
Robert P. Munafo, Large Numbers
Wikipedia, Graham's number
Wikipedia, Tetration
EXAMPLE
1984^^1984 (mod 10^8) == 98703616.
Thus, 1984^^1984 = ...61630789307145912032948400109045102(...)7490335.
Consider the sequence 1984^^n: 1984, 1984^1984, 1984^(1984^1984), ... From 1984^^3 onwards, all terms end with the digits 16. This follows from Euler's generalization of Fermat's little theorem.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Marco Ripà, Aug 26 2018
STATUS
approved