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R0017-09182 · q-20261013-advance-20261018-r04-prime-number-0032
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=36. 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=36. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.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.
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.
Rule: prime_complete_evidence_scope · Uses: G001
Before
phase=given n;n=36
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.
Rule: prime_lower_bound_guard · Uses: G001, S001
Before
given n trial divisor2:natural dividend=36;positive divisor=2
After
36=18·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S002
Before
phase=given n;tried ascending divisors=[2];sqrt floor=6
After
least factor at least2=2;prime=No
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.
Rule: complete_small_factor_bound · Uses: G001, S003
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.(36, 2, No)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: da7596f970946078c382126e11b628ded56a06f56521601573f8eeefc9fc7438candidate record SHA-256: bc1633c7aaf2fb6a81b48353c13f5746a8a03563dd533b3c5d52a65d3ff03c80semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED