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,
.
More later,
Nalin Pithwa