All positive integers N, when raised to the 7th power are equivalent to N mod m, where m is any of the following: {2, 3, 6, 7, 14, 21, 42}. So (N^7 - N) is 0 mod m for that same set.
Drilling down slightly farther, this set is the union of the set of primes {2,3,7} and the product of any combination of those primes taken 1, 2, or 3 at a time.
But I have not proved why the observation is true ...
Observation, not necessarily a proof
