Prime Number Theorem

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.