# Question 21369: Only Two Valid Prime Products Can Supply This Uniqueness Certificate [Return to catalogue page 107](https://ocme.evemisslab.com/en/questions/page/107/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations/#q-20261014-advance-20261108-r06-unique-prime-factorization-0069) 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. R0041-21369 · q-20261014-advance-20261108-r06-unique-prime-factorization-0069 ## Original question and givens Use positive integer n=402>1, candidate factors_left=[2, 3, 67] and factors_right=[67, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate). - G001 — Use positive integer n=402>1, candidate factors_left=[2, 3, 67] and factors_right=[67, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate). ## 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-unique-prime-factorization](https://ocme.evemisslab.com/en/mko/mko-unique-prime-factorization/) ## Step-by-step derivation S001 · impliesBefore source operation=upf-uniqueness-audit;original snapshot and mathematics retained After explicit ordered source output fields=[left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate] Conditions: none Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore upf LEFT factor index0:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[] After ALL [divisor,remainder] rows=[];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S001 S003 · calculationBefore upf LEFT complete product at original factor0 := mul(1, 2) After upf LEFT complete product at original factor0 = 2 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore upf LEFT factor index1:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[] After ALL [divisor,remainder] rows=[];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S003 S005 · calculationBefore upf LEFT complete product at original factor1 := mul(2, 3) After upf LEFT complete product at original factor1 = 6 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore upf LEFT factor index2:ORIGINAL candidate factor=67;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4, 5, 6, 7, 8] After ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3], [5, 2], [6, 1], [7, 4], [8, 3]];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S005 S007 · calculationBefore upf LEFT complete product at original factor2 := mul(6, 67) After upf LEFT complete product at original factor2 = 402 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore upf LEFT:ORIGINAL factors=[2, 3, 67];ALL primality flags=[Yes, Yes, Yes];original n=402 After empty product convention1;actual complete product=402;all_prime=Yes;valid_prime_product=Yes Conditions: none Keep original factor order.An empty list vacuously has all-prime entries,but its product1 differs from n>1.A matching composite product is not a prime factorization.The SAFE guard concerns source projection,not theorem truth. Step identifiers and dependenciesRule: upf_candidate_prime_and_complete_product_separate · Uses: G001, S007 S009 · impliesBefore upf RIGHT factor index0:ORIGINAL candidate factor=67;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4, 5, 6, 7, 8] After ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3], [5, 2], [6, 1], [7, 4], [8, 3]];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S008 S010 · calculationBefore upf RIGHT complete product at original factor0 := mul(1, 67) After upf RIGHT complete product at original factor0 = 67 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore upf RIGHT factor index1:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[] After ALL [divisor,remainder] rows=[];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S010 S012 · calculationBefore upf RIGHT complete product at original factor1 := mul(67, 2) After upf RIGHT complete product at original factor1 = 134 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore upf RIGHT factor index2:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[] After ALL [divisor,remainder] rows=[];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S012 S014 · calculationBefore upf RIGHT complete product at original factor2 := mul(134, 3) After upf RIGHT complete product at original factor2 = 402 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore upf RIGHT:ORIGINAL factors=[67, 2, 3];ALL primality flags=[Yes, Yes, Yes];original n=402 After empty product convention1;actual complete product=402;all_prime=Yes;valid_prime_product=Yes Conditions: none Keep original factor order.An empty list vacuously has all-prime entries,but its product1 differs from n>1.A matching composite product is not a prime factorization.The SAFE guard concerns source projection,not theorem truth. Step identifiers and dependenciesRule: upf_candidate_prime_and_complete_product_separate · Uses: G001, S014 S016 · impliesBefore SORTED left occurrence index=0,factor=2;ORIGINAL right factors=[67, 2, 3];full right product=402 After [p,p_divides_product,FIRST original right index,right factor,right factor equals p]=[2, Yes, 1, 2, Yes];same right index MAY be reused Conditions: none Choose the first divisible factor in original right order without consuming an occurrence.Invalid composite candidates give diagnostic rows,not Euclid-lemma claims for composite p.Joint validity still needs two valid prime products and all multiplicities equal. Step identifiers and dependenciesRule: upf_local_first_right_witness_not_occurrence_matching · Uses: G001, S015 S017 · impliesBefore SORTED left occurrence index=1,factor=3;ORIGINAL right factors=[67, 2, 3];full right product=402 After [p,p_divides_product,FIRST original right index,right factor,right factor equals p]=[3, Yes, 2, 3, Yes];same right index MAY be reused Conditions: none Choose the first divisible factor in original right order without consuming an occurrence.Invalid composite candidates give diagnostic rows,not Euclid-lemma claims for composite p.Joint validity still needs two valid prime products and all multiplicities equal. Step identifiers and dependenciesRule: upf_local_first_right_witness_not_occurrence_matching · Uses: G001, S016 S018 · impliesBefore SORTED left occurrence index=2,factor=67;ORIGINAL right factors=[67, 2, 3];full right product=402 After [p,p_divides_product,FIRST original right index,right factor,right factor equals p]=[67, Yes, 0, 67, Yes];same right index MAY be reused Conditions: none Choose the first divisible factor in original right order without consuming an occurrence.Invalid composite candidates give diagnostic rows,not Euclid-lemma claims for composite p.Joint validity still needs two valid prime products and all multiplicities equal. Step identifiers and dependenciesRule: upf_local_first_right_witness_not_occurrence_matching · Uses: G001, S017 S019 · impliesBefore ORIGINAL n=402;left valid=Yes,right valid=Yes;sorted multisets=[[2, 3, 67], [2, 3, 67]];local witness rows=[[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [67, Yes, 0, 67, Yes]] After joint uniqueness certificate=Yes;general proof:prime p divides other product;Euclid lemma gives equal prime factor;match ONE occurrence,cancel nonzero p,then induct on remaining factor count Conditions: none Euclid's lemma follows from multiplying a Bezout identity when gcd(p,a)=1:if p divides ab,it divides b.Iterate over the finite product,match one equal prime,cancel that nonzero factor,and induct to the empty product1;remaining primes≥2 cannot multiply to1.Reused local indices alone do not prove global multiplicity matching. Step identifiers and dependenciesRule: upf_general_uniqueness_Euclid_nonzero_cancel_induction · Uses: G001, S018 ## Reverse checks - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute upf RIGHT complete product at original factor2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute upf RIGHT complete product at original factor1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute upf RIGHT complete product at original factor0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute upf LEFT complete product at original factor2 with exact rational arithmetic, not rounded decimals. - 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 — Recompute upf LEFT complete product at original factor1 with exact rational arithmetic, not rounded decimals. - 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 — Recompute upf LEFT complete product at original factor0 with exact rational arithmetic, not rounded decimals. - 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 (Yes, Yes, [2, 3, 67], [2, 3, 67], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [67, Yes, 0, 67, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor) 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: 36e288a30131666cdec8d5f24b65d5e6a268421e2f3cae137070110088548dd6 - candidate record SHA-256: f6b117d9c55ca0920a16b4cdac24bef682bf4b78a7416922a8b86b3306403028 - semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a - question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0069.json)