# In the case of power residue being one, the relation between its index and modulus by an order of its base

Fermat found his little theorem by examining Mersenne prime. This thesis discloses the numerical structure of Fermat’s observation on Mersenne prime and submits its expanded theorems, which diﬀer from his little theorem. The structure is deduced from the more essential theorems which describe the relation between the index and the modulus of power residue being one.

# フェルマーの小定理の証明

フェルマーの小定理を簡潔に証明する。フェルマーの小定理が成立する理由、数の仕組みを理解できるとともに、平易な形でそれを明らかにするよう試みた。【証明】ｐによる余りは1,2,〜,ｐ-2,ｐ-1のいずれかしかない。ａにａを掛けて余りをとる操作Ｔを繰り返し行うと、ｐ-1以下ですでに現われた余りが再び、、、つづく