# Question 9101: Complete Positive Divisors and Prime Qualification

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0017-09101 · q-20261013-advance-20261018-r04-prime-number-0001



## Original question and givens

Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=0. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).


- G001 — Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=0. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-prime-number](https://ocme.evemisslab.com/en/mko/mko-prime-number/)





## Step-by-step derivation

S001 · impliesBefore

natural-number prime protocol;0 and1 are not prime

After

a failed candidate witness alone is not a primality proof

Conditions: none

Naturals include0.Primality retains both n≥2 and a complete factor bound.A failed single factor candidate only invalidates that witness;it does not certify primality.

Step identifiers and dependenciesRule: prime_complete_evidence_scope · Uses: G001





S002 · impliesBefore

p=0;d is an arbitrary positive natural

After

positive divisors=all positive natural numbers;p>=2=No;prime=No

Conditions: none

Every positive d divides0 via0=d*0,but0 fails the prime lower bound.An infinite positive-divisor collection does not make0 prime.

Step identifiers and dependenciesRule: zero_positive_divisors_not_prime · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(0, all-positive-naturals, No, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 32f566ad680b9b5246ca65b8e7d1a09592443b532a6bfde563d46397b88321b3
- candidate record SHA-256: 055452ba687e5f1acd2426de22c9b152d48cef9cba1f27789cebe0e3fc0de07e
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r04-prime-number-0001.json)