Reference: Proofs from THE BOOK (Third Edition) (Martin Aigner and Gunter M. Ziegler)
Suppose the set of primes, is finite and p is the largest prime. We consider the so-called Mersenne number , and show that any prime factor q of is bigger than p, which will yield the desired conclusion. Let q be a prime dividing , so we have .
Since p is prime, this means that the element 2 has order p in the multiplicative group of the field . This group has elements. By Lagrange’s theorem, we know that the order of every element divides the size of the group, that is, we have , and hence, .