Prime Number Checker

What is a prime number?

We call a number prime when it's a natural number greater than 1 that admits only two divisors: 1 and itself. The sequence opens with 2, 3, 5, 7, 11, 13, 17, 19, 23 and keeps going from there. By definition, 1 stays out of this list.

Prime numbers: definition and primality testing

A prime number is a natural number greater than 1 with exactly two divisors: 1 and itself. By convention, 1 is not prime (the modern definition was settled in the early 20th century; before that, it was debated). The naive primality test is trial division: to check whether n is prime, test divisibility by every integer from 2 up to √n — no factor can exceed √n without pairing with a smaller one. Small primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. The Prime Number Theorem states that primes thin out with density roughly 1/ln(n). The Sieve of Eratosthenes (≈260 BC) remains the most efficient way to list all primes up to a bound. Mersenne primes have the form Mₚ = 2^p − 1; the distributed project GIMPS found the latest known prime in 2024 — M82589933, with more than 24 million digits.

Applications: cryptography and probabilistic tests

Primes power RSA: security relies on the difficulty of factoring n = p·q when p and q are primes of ~1024 bits each. They are used in hash function design and in probabilistic primality tests like Miller-Rabin and Solovay-Strassen, shipped in libraries such as OpenSSL, Pari/GP and SageMath.

FAQ

Is 1 prime? No. Modern mathematics excludes 1 so that the Fundamental Theorem of Arithmetic (unique prime factorization) holds.

Is 2 the only even prime? Yes. Every other even number is divisible by 2 and therefore composite.

How many primes are there? Infinitely many — Euclid proved this around 300 BC by contradiction.

What is the largest known prime? M82589933 = 2^82589933 − 1, a Mersenne prime with 24,862,048 digits, discovered by GIMPS in 2024.