A Dynamical Systems Proof of the Little Theorem of Fermat

Abstract

Counting the periodic orbits of a one parameter family of dynamical systems generated by linear expasions of the circumference we probe the following Euler-Fermat theorem: if a and n are positive integers, with no common prime divisor, the a (n) 1 mod (n), where is the Euler function.

Using an ad hoc one-parameter family of circle maps, we proved in the "Little theorem of fermat" If a is an integer and p is a prime number such that (a, p)=1, then a p-1=mod (p)

Euler invented the function that counts the numbers smaller than n, whic are relatively prime to it and observed that the following generalization of the little theorem is also true.