Everyone who studies elementary number theory learns two different versions of Fermat’s Little Theorem:
Fermat’s Little Theorem, Version 1: If is prime and is an integer not divisible by , then .
Fermat’s Little Theorem, Version 2: If is prime and is any integer, then .
as well as the following extension of Version 1 of Fermat’s Little Theorem to arbitrary positive integers :
Euler’s Theorem: If is a positive integer and , then , where is Euler’s totient function.
My first goal in this post is to explain a generalization of Version 2 of Fermat’s Little Theorem to arbitrary . I’ll then explain an extension of this result to integer matrices, along with a slick proof of all of these results (and more) via “combinatorial zeta functions”.Continue reading