Remarks

• A useful consequence is that X^pb - X divides X^pa - X if and only if b divides a.
• Another consequence is that indeed Mersenne primes, that is, primes of the form 2ⁿ - 1 only occur for n prime. For, if n = ab, then, by the lemma, 2^a - 1 is a divisor of 2ⁿ - 1.