The **Prime Number Theorem** (or the PNT) is a theorem that concerns the distribution of primes and, subsequently, the gaps between primes. Its first proof date is not known.

## Statement of Theorem

The theorem, formally stated, says that:

$ \lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1 $

where $ \pi(x) $ is the number of primes up to and including $ x $. This means that for any number $ x $, the number of primes up to and including $ x $ approaches $ x $ divided by the the log to base e (or the natural log) of $ x $ as $ x $ approaches infinity and $ {x/\ln(x)} $ becomes a better approximation of $ \pi(x) $ as $ x $ grows larger. This also means that:

$ P_x \approx x\ln(x) $, where $ P_x $ is the $ x $th prime number.