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Power Tower n^^n, mod 100

Ryohei Miyadera
Organization: Kwansei Gakuin High School
Department: Mathematics Department
Koichi Tukiyama
Daisuke Minematu
Satoshi Hashiba
Revision date


power tower of order n.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 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.

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.

*Mathematics > Number Theory
*Mathematics > Recreational Mathematics

power tower
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power tower of order n .nb (29.1 KB) - Mathematica Notebook
powertowers2.nb (111.3 KB) - Mathematica Notebook