# Prime natural number

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

A prime natural number is at least 2 and has no positive divisors other than 1 and itself.

mko-prime-number · definition · v0.10.0



## Mathematical statement

A natural number p is prime exactly when 2≤p and every natural divisor d of p satisfies d=1 or d=p. Neither 0 nor 1 is prime.





## Representative formula (exact TeX)

p\in N





## Assumptions


- The numbers p and d are natural, including 0; the prime condition additionally requires 2≤p.
- Use natural-number divisibility, not approximate square roots or floating-point division, to check factors.





## Symbol scope


- p — candidate prime
- N — natural-number domain





## Intuition, definition, and misconceptions

For a finite candidate p, enumerate every positive divisor, or check every candidate from 2 through floor(sqrt(p)): a composite number has a nontrivial factor no greater than its square root.

The divisor behavior of 0 and 1 does not satisfy the prime definition. The number 2 is the smallest prime and the only even prime.

One nontrivial factor provides a composite witness. Failure to find a factor in an incomplete candidate list alone is not a primality proof.

Mistakes include listing 1 as prime or claiming a large number prime after checking only a few small factors.





## Adopted derivation guide

[Prime natural numbers: proving 97 prime and 91 composite, and excluding 0 and 1](https://ocme.evemisslab.com/en/derivations/derivation-mko-prime-number-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Natural-number divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/) — Primality is defined through the structure of natural-number factors.





## Verification boundary

This slice uses complete divisor or square-root-bound checks inside a safe-integer range; it is not a formal primality proof for arbitrary large integers.

not_formalized

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





## Next obligation

Bind Nat.Prime, Nat.dvd_prime, least factors and decidability; develop a separate certificate route for large integers.





## 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/prime-number.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-prime-number/index.md)