Prime Number Checker

Check if a number is prime free online using real trial division up to the square root. Also lists prime factors for composite numbers, with genuine algorithmic verification.

Enter a Number

Browser only -- real trial division algorithm.

How to Use This Tool

Enter any positive integerThe number you want to check

See if it's primeGenuine algorithmic verification, not a lookup

Check prime factors if compositeFull factorization shown

Prime Number Checker is a free online tool that works entirely in your browser — no signup, no data uploaded, no limits.

A Genuinely Complete Test, Not Just a Shortcut

Checking divisibility only up to the square root of a number isn't a shortcut that sometimes misses things -- it's mathematically complete. If a number has any factor larger than its square root, that factor must pair with a corresponding factor smaller than the square root (since their product equals the original number), so finding nothing up to √n proves there's nothing beyond it either.

Prime Factorization for Composite Numbers

When a number isn't prime, this calculator shows its complete prime factorization -- the specific prime numbers that multiply together to produce it -- giving genuinely useful mathematical context beyond a simple yes/no answer.

Why 1 Is Neither Prime Nor Composite

By formal mathematical definition, prime numbers have exactly two distinct positive divisors (1 and themselves). Since 1 only has one divisor (itself), it fails this definition and is correctly classified as neither prime nor composite by this checker.

Frequently Asked Questions

Trial division up to the square root of the number -- if no integer from 2 up to √n divides evenly into n, then n is prime. Checking only up to the square root (rather than all the way to n) is a real mathematical optimization, since any factor pair must have one factor at or below the square root.

If a number n has a factor larger than √n, it must also have a corresponding factor smaller than √n (since their product equals n) -- so finding no factors up to √n guarantees there are none beyond it either, making this a complete, not just a shortcut, primality test.

The calculator also lists the complete prime factorization -- breaking the number down into the prime numbers that multiply together to produce it, which is genuinely useful context beyond just a yes/no primality answer.

Trial division remains efficient for numbers within typical practical ranges, though for extremely large numbers (cryptographic-scale), more sophisticated probabilistic primality tests become necessary -- this tool's straightforward approach suits everyday and educational use well.

No -- by mathematical definition, 1 is neither prime nor composite, a definition this checker correctly applies rather than treating 1 as trivially prime.