
power tower of order n.nb We study n^n^....^n, where we use n ntimes. This is so called the power tower of order n . We found the fact that n^n^....^n under mod 100 depends only on Mod[n,100], and n^n^....^n under mod 10 depends only on Mod[n,20]. By using this fact we can make the fastest algorithm to calculate n^n^....^n under mod100 and mod10. n^n^....^n under mod m (m is an arbitrary integer) has been studied by many authors, but when m = 10 and 100, this problem becomes very simple and interesting. powertowers2.nb We study n^n^ ....^n, where we use n n  times. This is so called the power tower of order n . We found the fact that n^n^ ....^n under mod 10^s depends only on n mod(2*10^s). This study is a generalization of our previous study, and this is an important result, because this shows that very interesting result can be proved for general number 10^s. I have to admit that the proof for the case of mod 10^s is complicated and a little bit difficult to read, so to understand the ideas of our study it might be better to read our previous study.

