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Prime Factorization Calculator

Decomposes an integer into its prime factors by trial division, tests primality, and derives the number and sum of its divisors from the exponents.

When to use

You need the prime factors of an integer up to 10^12, want to know whether it is prime, or need its divisor count.

Do not use when: You need the GCD or LCM of several numbers (use gcd-lcm) or factorials and binomial coefficients (use factorial, combinations-permutations).

Formula

n = p₁^k₁ × p₂^k₂ × … (trial division by 2, then odd numbers up to √n); τ(n) = Π (kᵢ + 1); σ(n) = Π (pᵢ^(kᵢ+1) − 1) / (pᵢ − 1)

Inputs

Parameter	Type	Unit	Required	Description
`n`	integer		yes	Integer to factor, from 2 to 1,000,000,000,000. Range: ≥ 2, ≤ 1000000000000

Outputs

Output	Type	Description
`is_prime`	boolean	true when n has no divisors other than 1 and itself.
`factors`	number_list	Prime factors in ascending order with repetition, e.g. [2, 2, 2, 3, 3, 5].
`factorization`	string	Prime-power form, e.g. '2^3 × 3^2 × 5'.
`distinct_primes`	number_list	Each prime factor once.
`divisor_count`	integer	τ(n) = Π (kᵢ + 1) over the exponents kᵢ of the factorization.
`sum_of_divisors`	integer	σ(n) = Π (1 + pᵢ + pᵢ² + … + pᵢ^kᵢ), including 1 and n.

Example

360: {"n":360} → {"is_prime":false,"factors":[2,2,2,3,3,5],"factorization":"2^3 × 3^2 × 5","distinct_primes":[2,3,5],"divisor_count":24,"sum_of_divisors":1170}

97: {"n":97} → {"is_prime":true,"factors":[97],"factorization":"97","divisor_count":2,"sum_of_divisors":98}

GET https://tttkmbb.com/api/v1/calculate/prime-factorization?n=360

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/prime-factorization (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/prime-factorization · Markdown: https://tttkmbb.com/math/prime-factorization.md · JSON definition: https://tttkmbb.com/math/prime-factorization.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="prime-factorization"
OpenAPI operationId: calculate_prime_factorization
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Integer factorization (reference)
Wikipedia – Divisor function (reference)
Wolfram MathWorld – Prime Factorization (reference)

FAQ

Why is 1 not accepted?

1 is neither prime nor composite and has an empty factorization, so the input must be at least 2.

How fast is this for large numbers?

Trial division needs at most about 500,000 divisions for n near 10^12, which takes a few milliseconds; larger inputs are rejected.

Related calculators

GCD and LCM Calculator — Common factors and multiples of several integers.
Modulo Calculator — Remainders used in divisibility tests.
Factorial Calculator — Products of consecutive integers.