There Are Infinitely Many Primes

then by Proposition 1.1.20 there is a factorization

with each prime and . If for some , then . Because of (1.2.1), we also have , so , which is a contradiction. Thus the prime is not in the list , and we have constructed our new prime.

For example,

Multiplying together the first primes and adding doesn't produce a prime, but it produces an integer that is merely divisible by a new prime.

William 2007-06-01