Number Theory
Advanced

Prime Number Theorem

Approximates the number of primes below x.

Formula

π(x)∼xln⁡x\pi(x) \sim \frac{x}{\ln x}

Variables

xUpper bound
\pi(x)Prime count

Example

Up to 100 there are 25 primes; the theorem estimates 100/ln100 ≈ 22.

Did You Know?

The prime number theorem reveals that primes thin out but never stop — Euclid proved there are infinitely many.

primesasymptotic