The number of primes is infinite

Reference: Proofs from THE BOOK (Third Edition) (Martin Aigner and Gunter M. Ziegler)

Suppose the set of primes, $\mathcal{P}$ is finite and p is the largest prime. We consider the so-called Mersenne number $2^{p}-1$ , and show that any prime factor q of $2^{p}-1$ is bigger than p, which will yield the desired conclusion. Let q be a prime dividing $2^{p}-1$ , so we have $2^{p} \equiv 1 \pmod {q}$ .

Since p is prime, this means that the element 2 has order p in the multiplicative group $Z_{q} { \\ \{0\}}$ of the field $Z_{q}$ . This group has $q-1$ elements. By Lagrange’s theorem, we know that the order of every element divides the size of the group, that is, we have $p|q-1$ , and hence, $p<q$ .