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 isanyinteger, 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”.

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