Recurring digits - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- 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