Prime Numbers List Calculator

A prime is a whole number greater than 1 whose only positive divisors are 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The number 1 is neither prime nor composite by modern convention, which keeps prime factorization unique for every integer greater than 1.

The sieve lists the integers from 2 to N. For each unmarked prime p, it marks composite multiples starting at p^2 because smaller multiples were already handled by earlier primes. Once p exceeds sqrt(N), every remaining unmarked number is prime. Its time complexity is O(N log log N).

Twin primes are pairs of primes that differ by exactly 2, such as (3, 5), (5, 7), (11, 13), (17, 19), and (29, 31). The twin prime conjecture says infinitely many such pairs exist. It remains unproven, so avoid treating a computational search record as evidence that the conjecture has been settled.

On average, the gap between consecutive primes near n grows like ln(n), so primes become less dense. The prime number theorem says the count of primes up to n is approximately n / ln(n). Individual gaps vary widely. Arbitrarily long runs of composite numbers are known, while infinitely many twin-prime gaps remain conjectural.

The largest known prime is M136279841 = 2^136,279,841 - 1. GIMPS reported its discovery in October 2024. It has 41,024,320 decimal digits and is the 52nd known Mersenne prime. This browser calculator lists primes only within its much smaller input limit.