A318478 Decimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies the congruence 1984^A(k) == A(k) (mod 10^k).

6, 1, 6, 3, 0, 7, 8, 9, 3, 0, 7, 1, 4, 5, 9, 1, 2, 0, 3, 2, 9, 4, 8, 4, 0, 0, 1, 0, 9, 0, 4, 5, 1, 0, 2, 3, 9, 2, 0, 5, 0, 9, 4, 2, 6, 9, 0, 5, 3, 3, 8, 6, 2, 2, 8, 4, 6, 3, 8, 5, 1, 9, 2, 3, 7, 7, 8, 9, 0, 0, 2, 8, 3, 9, 2, 7, 0, 0, 1, 0, 7, 4, 9, 0, 3, 3, 5

Offset: 1

Author

Marco Ripà, Aug 26 2018

Keywords

• nonn
• base

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.

Examples

			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.
		

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

Crossrefs

Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544, A317824, A317903, A317905.