Prime Number Theorem

Primes become less frequent as numbers grow, but they do so in a surprisingly regular way.

The Theorem

$$\pi(x) \sim \frac{x}{\ln x}$$

Where $\pi(x)$ is the number of primes less than or equal to $x$.

What It Means

Around $x = 10^6$, about 1 in every 14 numbers is prime. At $10^{12}$, it is about 1 in 27. The density decreases slowly and predictably.

The distribution of primes is irregular locally but remarkably smooth globally.