Fermat's little theorem states that if $~p~$ is a prime number then for any integer $~a~$ the number $~a^p - a~$ is divisible by $~p~$.
What if one fixes the exponent $~n~$ and tries to find all $~m~$ such that for any integer $~a~$ the number $~a^n - a~$ is divisible by $~m~$? Is it true that there is at least one such $~m~$? Is the set of all such $~m~$ finite? Do you have any ideas how we can find them all?
I'm not sure that this questions have any reasonable mathematical importance - rather it's just my own curiosity. Thanks in advance for any ideas.