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R0017-09111 · q-20261013-advance-20261018-r04-prime-number-0011
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=10. 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=10. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).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
positive factor pair candidate1:natural dividend=10;positive divisor=1
After
10=10·1+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, S001
Before
candidate d=1;quotient=10;remainder=0
After
factor pair=[1, 10]
Conditions: none
Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors.
Rule: complete_positive_factor_pair_inventory · Uses: G001, S002
Before
positive factor pair candidate2:natural dividend=10;positive divisor=2
After
10=5·2+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, S003
Before
candidate d=2;quotient=5;remainder=0
After
factor pair=[2, 5]
Conditions: none
Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors.
Rule: complete_positive_factor_pair_inventory · Uses: G001, S004
Before
positive factor pair candidate3:natural dividend=10;positive divisor=3
After
10=3·3+1
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, S005
Before
candidate d=3;quotient=3;remainder=1
After
factor pair=none
Conditions: none
Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors.
Rule: complete_positive_factor_pair_inventory · Uses: G001, S006
Before
all positive divisors=[1, 2, 5, 10];n=10
After
n>=2=Yes;exactly1 and n=No;prime=No
Conditions: none
Primality requires n≥2 and the complete positive-divisor set exactly{1,n}.At n=1 there is only one divisor1,not two copies in[1,1].
Rule: prime_positive_divisors_with_lower_bound · Uses: G001, S007
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.(10, [1, 2, 5, 10], Yes, 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: 7d34f9eec385ca03d86ffbc850d555eae12240e8807ada29a0a6f01cd0dc798bcandidate record SHA-256: 471ab2e35436bf40c751ff8ebc06a9d089fdd62adaa8eb1987f3fa3a6f0607fasemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED