ABSTRACT
There are infinitely many primes. This was easily proved using only the fact that every integer greater than 1 has a prime factor. Eventually, in 1949, Selberg and Paul Erdos discovered "elementary" proofs of the Prime Number Theorem. These proofs are elementary in the sense that they avoid any real or complex analysis, but they are in many ways more technical and harder to understand than the "non-elementary" proofs. The theorem consists of two inequalities. Although they look similar, they tell different things. The first inequality tells that there are lots of primes. The second inequality gives an upper bound on the number of primes up to x.
