Proof of Peter Schorn: Assume there exist only m prime numbers; let n = m + 1.
For two numbers i and j with 1 ≤ i < j ≤ n, then
gcd[ (n!)i + 1, (n!)j + 1 ] = 1.
Indeed, j = i + d, with 1 ≤ d < n, so
gcd[ (n!)i + 1, (n!)j + 1 ] = gcd[ (n!)i + 1, (n!)d ] = 1.
Therefore the n integers (n!)i + 1 (for i = 1, 2, …, n) are pairwise relatively prime.
If pi is a prime dividing (n!)i + 1, then p1, p2,
…, pn
are distinct primes with n = m + 1, which is a contradiction
[Paulo Ribenboim, The New Book of Prime Number Records, p. 5, Springer-Verlag 1996, 3rd edition].