Prime Number Checker — Is It Prime?

What is a prime number?

A prime number is a positive integer greater than 1 whose only divisors are 1 and itself. The first few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. There are infinitely many primes (Euclid's theorem) and they form the building blocks of all integers via the fundamental theorem of arithmetic — every integer greater than 1 is either prime or can be uniquely factorised as a product of primes.

Why primes matter

• Cryptography — RSA, Diffie-Hellman and many other crypto algorithms depend on the difficulty of factorising large composites into their primes.
• Hash functions — many hash table sizes are chosen as primes to reduce collision rates.
• Coding theory — primes appear in error-correcting codes and number-theoretic transforms.
• Mathematics — primes are central to number theory and analytic number theory.
• Random number generators — many PRNGs use prime moduli.

How this checker works

For numbers up to a few billion, the tool uses optimised trial division up to the square root of N. For numbers from a few billion up to JavaScript's safe-integer limit (2^53 − 1), it uses a deterministic Miller-Rabin primality test with witnesses chosen to be correct for all 64-bit integers. The result is exact, not probabilistic.

If the number is composite, the prime factorisation is also computed and displayed.

Tips

• 2 is the only even prime — every other prime is odd.
• 3, 5, 7 are the only "single-digit" primes beyond 2.
• Primes get sparser as numbers grow — among the first 100 integers, 25 are prime; among the first million, 78,498 are.
• The "primes ending in 1, 3, 7, 9" rule covers all primes greater than 5 (since 2 and 5 are the only primes that don't fit).

FAQs

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