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+--- Thread: Recurring digits (/showthread.php?tid=45)
Recurring digits - Daniel - 08/30/2007
For the Ackermann function with \( n \ge 10 \) and \( k \ge 2 \):
\( 2 \rightarrow n \rightarrow k \equiv 2948736 \; (mod \; 10^7) \)
\( 3 \rightarrow n \rightarrow k \equiv 4195387 \; (mod \; 10^7) \)
\( 4 \rightarrow n \rightarrow k \equiv 1728896 \; (mod \; 10^7) \)
\( 5 \rightarrow n \rightarrow k \equiv 8203125 \; (mod \; 10^7) \)
\( 6 \rightarrow n \rightarrow k \equiv 7238656 \; (mod \; 10^7) \)
\( 7 \rightarrow n \rightarrow k \equiv 5172343 \; (mod \; 10^7) \)
\( 8 \rightarrow n \rightarrow k \equiv 5225856 \; (mod \; 10^7) \)
\( 9 \rightarrow n \rightarrow k \equiv 2745289 \; (mod \; 10^7) \)
\( 11 \rightarrow n \rightarrow k \equiv 2666611 \; (mod \; 10^7) \)
\( 12 \rightarrow n \rightarrow k \equiv 4012416 \; (mod \; 10^7) \)
\( 13 \rightarrow n \rightarrow k \equiv 5045053 \; (mod \; 10^7) \)
\( 14 \rightarrow n \rightarrow k \equiv 7502336 \; (mod \; 10^7) \)
\( 15 \rightarrow n \rightarrow k \equiv 859375 \; (mod \; 10^7) \)
\( 16 \rightarrow n \rightarrow k \equiv 415616 \; (mod \; 10^7) \)
\( 17 \rightarrow n \rightarrow k \equiv 85777 \; (mod \; 10^7) \)
\( 18 \rightarrow n \rightarrow k \equiv 4315776 \; (mod \; 10^7) \)
\( 19 \rightarrow n \rightarrow k \equiv 9963179 \; (mod \; 10^7) \)
RE: Recurring digits - bo198214 - 03/21/2009
Perhaps this article provides a proof:
Blakley, G. R.; Borosh, I. Modular arithmetic of iterated powers. Comput. Math. Appl. 9 (1983), no. 4, 567--581.
Review of Zentralblatt:
http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0535.03018&format=complete