Prime Number Calculator
Check if a number is prime, list all primes up to N using the Sieve of Eratosthenes, or find the prime factorization of any number. Works up to 1 trillion for primality checks.
| Primality Algorithm | Method | Speed | Used In |
|---|---|---|---|
| Trial Division | Test divisors up to √n | Slow for large n | School curricula worldwide |
| Sieve of Eratosthenes | Eliminate composites up to N | Fast for small N (< 10⁶) | Generating lists of primes |
| Miller-Rabin (probabilistic) | Witness-based randomized test | Very fast, may have errors | RSA key generation (OpenSSL) |
| AKS Primality Test | Deterministic polynomial-time | Theoretically optimal | Cryptography research |
| Elliptic Curve Primality | Proves primality for large n | Practical for very large n | NIST / IETF standards |
Frequently Asked Questions
Why is 1 not considered a prime number?
By modern mathematical convention, 1 is excluded from primes because including it would break the Fundamental Theorem of Arithmetic (every integer ≥ 2 has a unique prime factorization). If 1 were prime, 12 could be factored as 2²×3, or 1×2²×3, or 1×1×2²×3, giving infinitely many factorizations. The exclusion of 1 preserves uniqueness. Historically, some mathematicians (including Euclid) did consider 1 prime, but modern number theory standardised the definition around 1900. This is the same convention taught in all countries today.
How large are the primes used in real cryptography?
RSA-2048 (the current standard): uses two random prime numbers each with about 617 decimal digits (1,024 bits each). Their product (the public key modulus) has 2,048 bits. RSA-4096: primes of ~1,234 digits. Factoring a 2,048-bit number with current computers would take longer than the age of the universe. Quantum computers (if sufficiently large) could break RSA using Shor's algorithm — which is why post-quantum cryptography standards (NIST FIPS 203/204, 2024) are based on lattice problems instead of prime factorization. Used by GCHQ (UK), NSA (US), and all major banks and governments worldwide.