Prime Factorization Calculator

What does this prime factorization calculator compute?

Every integer n > 1 has a unique factorization into primes, up to order. n = 1 has no prime factors and is neither prime nor composite.

The published formula or method is n = ∏ p_i^{e_i}. The following references support the stated mathematics or method:

• MIT Mathematics for Computer Science — Chapter 8: Number Theory: Divisibility, prime factorization, gcd and the Euclidean algorithm, lcm via prime exponents, division with remainder, modular congruence and multiplicative inverses. CalculatorX’s signed-remainder and zero-input policies are stated separately on the page. Mathematical reference only; not validation or endorsement of this implementation.

Which inputs and limits should you check?

Use a safe JSON integer or a signed decimal digit string with 1 ≤ n ≤ 10^12, at most 13 digits (including leading zeros) and 512 characters per string. Integer strings reject decimal points, scientific notation and separators. JSON numeric inputs follow the general binary64 transport policy; use digit strings to preserve the exact source integer. Unknown fields, null, booleans, nested values and conflicting n/nInput aliases are refused.

Both modes return the complete, ascending list of prime factors and positive exponents. The canonical product is exact; ω counts distinct factors and Ω counts multiplicity. For 1, factors is empty, both counts are zero and the classification is neither prime nor composite. Trial division uses exact BigInt arithmetic up to sqrt(n); a one-unit change in n can change its factorization completely. This bounded tool does not accept cryptographic-size integers.

The result also includes the normalized integer, identity, classification, method, complete textual work and calculation version. Clearing produces an error state and disables result exports; Reset restores 12. Matching aliases are accepted; conflicting ones are errors.

The worked example uses Composite with 12 = 2² × 3. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.prime_factorization v1.0.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Sharing, exports and privacy

Typing computes locally. A copied URL reopens the current engine; an older calculation-version query displays a current-version notice rather than replaying a historical runtime. Native CSV and JSON retain the complete factor list, exponents, counts, work and raw inputs. Sharing URLs and calling REST/MCP send inputs to the server and may enter logs; browser calculation history may persist locally and can be cleared. Site analytics and hosting policies are described in the privacy policy; this page does not promise zero logs or a fixed retention period.