Prime Counting Calculator

About this calculator

The prime-counting function π(x) returns how many prime numbers are less than or equal to x, and this calculator finds it exactly using the Sieve of Eratosthenes — an ancient algorithm that marks off multiples of each prime starting from 2, leaving only primes unmarked, then simply counts what remains up to x.

π(x) sits at the center of analytic number theory: the prime number theorem, proven independently by Hadamard and de la Vallée Poussin in 1896, shows that π(x) is approximately x divided by the natural logarithm of x for large x, and refining that approximation (including through Riemann’s work connecting it to the zeta function) remains one of the deepest topics in mathematics. More practically, students studying number theory and computer science use exact prime counts to verify sieve implementations and to reason about prime density in cryptographic key-generation contexts.

Enter a number x and this calculator sieves all integers up to that point and returns exactly how many of them are prime.