Unique prime factorization of integers

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

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 — Every listed factor must be prime.
• Natural-number divisibility — Prime divisibility supports the theorem.
• Mathematical induction — Existence has an induction obligation.
• Euclidean algorithm — Bezout reasoning supports Euclid lemma.
• Greatest common divisor and coprimality — 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.