prime number theorem

The number of prime numbers less than x is
about x/log(x). Here "is about" means that the ratio of the
two things tends to 1 as x tends to infinity. This was first
conjectured by Gauss in the early 19th century, and was
proved (independently) by Hadamard and de la Vall'ee Poussin
in 1896. Their proofs relied on complex analysis, but Erdös
and Selberg later found an "elementary" proof.