Prime Factorization Calculator

Writing a number as a product of prime numbers — numbers divisible only by 1 and themselves. Every integer above 1 has exactly one prime factorization (ignoring order); for example, 72 = 2³ × 3².

It uses trial division with a 6k ± 1 wheel: after checking 2 and 3, it only tests divisors of the form 6k − 1 or 6k + 1 (5, 7, 11, 13, …), skipping every multiple of 2 or 3. That cuts the work by two-thirds and factors anything up to a trillion in a fraction of a second.

If trial division finds no divisor up to the square root, the number must be prime — any composite number has a factor no bigger than its square root. When that happens, the factorization is just the number itself.

Counting each prime factor as many times as it appears. 72 = 2³ × 3² has 5 prime factors with multiplicity (2, 2, 2, 3, 3) but only 2 distinct prime factors.

1 is neither prime nor composite — it has no prime factors at all, so there’s nothing to show. 0 is divisible by every prime infinitely, so its factorization is undefined. The calculator starts at 2, the smallest prime.