Prime Factorization Calculator

Prime factorization breaks a positive integer into a product of prime numbers. Every integer greater than 1 has exactly one such factorization (up to order), a result known as the Fundamental Theorem of Arithmetic. For example, 360 = 2^3 * 3^2 * 5.

1 is neither prime nor composite. By convention it is treated as the empty product (no factors at all), which keeps the Fundamental Theorem clean: every integer greater than 1 factors uniquely into primes, and 1 sits outside that rule. This calculator returns no result for 1, 0, or negative inputs.

It uses trial division with BigInt arithmetic. After stripping the small primes 2, 3, 5, 7, and 11, it tests odd candidates up to the square root of the remaining value. For inputs up to 1 trillion (10^12), that is at most about 1 million iterations, which finishes in well under a second in your browser.

If n = p1^e1 * p2^e2 * ... then the divisor count (tau function) is (e1+1)(e2+1)... and the sum of divisors (sigma function) is the product of (p^(e+1) - 1) / (p - 1) over each prime. For 12 = 2^2 * 3, that gives 6 divisors (1, 2, 3, 4, 6, 12) summing to 28.

The maximum input is 1,000,000,000,000 (10^12). That cap keeps trial division responsive on every device. For much larger semiprimes (the kind used in cryptography), trial division is hopelessly slow and specialized algorithms like the General Number Field Sieve are needed instead.