# Unique prime factorization of integers

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order.

mko-unique-prime-factorization · theorem · v0.10.0



## Mathematical statement

Separate existence from uniqueness. Induction/well-founded reasoning gives existence, with complete bounded trial-division instances as computational companions. Uniqueness keeps the Euclid-lemma witness p|ab implies p|a or p|b, not just a few matching outputs. General factorizations may contain several distinct primes. Separate a negative sign, units±1 with the empty prime product and zero, which has no such factorization.





## Representative formula (exact TeX)

n=\prod_{i=1}^{r}p_i^{e_i}





## Assumptions


- The main theorem concerns positive integers n>1.
- Bounded computational companions search completely through the square root of the remaining factor and check exact primality/product.





## Symbol scope


- p_i — prime factor
- e_i — prime exponent
- p — ordered prime family
- e — positive exponent family
- i — bound factor index
- r — number of distinct prime factors
- n — positive integer to factor





## Intuition, definition, and misconceptions

Existence and uniqueness are separate.

Uniqueness compares prime multisets and ignores order, not multiplicity.

Units, signs and zero need separate branches.

Notation candidate: for the main domain n>1,r≥1,the primes p_1<...<p_r are distinct and every exponent is positive. The product includes all prime factors,not only a prime-power special case. Existing unit/empty-product and zero exclusions remain; the displayed representation is not itself an existence/uniqueness proof.

Finding one product does not prove uniqueness, and zero/units cannot use the n>1 theorem unchanged.





## Adopted derivation guide

[Existence, Euclid's lemma and unique factors](https://ocme.evemisslab.com/en/derivations/derivation-mko-unique-prime-factorization-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Prime natural number](https://ocme.evemisslab.com/en/mko/mko-prime-number/) — Every listed factor must be prime.
- [Natural-number divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/) — Prime divisibility supports the theorem.
- [Mathematical induction](https://ocme.evemisslab.com/en/mko/mko-mathematical-induction/) — Existence has an induction obligation.
- [Euclidean algorithm](https://ocme.evemisslab.com/en/mko/mko-euclidean-algorithm/) — Bezout reasoning supports Euclid lemma.
- [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/mko/mko-gcd-coprime/) — Prime divisibility and coprimality are linked.





## Verification boundary

Only a complete bounded trial-division range gives the stated instance certificate. The shared prime-power display is a special case, not the general multi-prime formula; final product/symbol alignment remains under review.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add p-adic valuations and the Mobius function.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/source13-20261018-unique-prime-factorization.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-unique-prime-factorization/index.md)