Question 9189: Least Divisor at Least Two and Primality

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0017-09189 · q-20261013-advance-20261018-r04-prime-number-0039

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. n=43. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.

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. n=43. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.

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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: prime_complete_evidence_scope · Uses: G001

S002 · implies

Before

phase=given n;n=43

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependencies

Rule: prime_lower_bound_guard · Uses: G001, S001

S003 · implies

Before

given n trial divisor2:natural dividend=43;positive divisor=2

After

43=21·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S002

S004 · implies

Before

given n trial divisor3:natural dividend=43;positive divisor=3

After

43=14·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S003

S005 · implies

Before

given n trial divisor4:natural dividend=43;positive divisor=4

After

43=10·4+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S004

S006 · implies

Before

given n trial divisor5:natural dividend=43;positive divisor=5

After

43=8·5+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S005

S007 · implies

Before

given n trial divisor6:natural dividend=43;positive divisor=6

After

43=7·6+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S006

S008 · implies

Before

phase=given n;tried ascending divisors=[2, 3, 4, 5, 6];sqrt floor=6

After

least factor at least2=43;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependencies

Rule: complete_small_factor_bound · Uses: G001, S007

Reverse checks

S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

(43, 43, Yes)

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: 2c4153d4482a38609451bbe0e8051aed8d7233be94b69b4d096680d4b228de81
candidate record SHA-256: 45283391c358531348ddc3a8e12223ccf77de02e97728a71b81f75ef38880186
semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
Lean: NOT_RUN · human review: NOT_PERFORMED

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