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Question 21359: Only Two Valid Prime Products Can Supply This Uniqueness Certificate

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0041-21359 · q-20261014-advance-20261108-r06-unique-prime-factorization-0059

Original question and givens

Use positive integer n=138>1, candidate factors_left=[2, 3, 23] and factors_right=[23, 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=138>1, candidate factors_left=[2, 3, 23] and factors_right=[23, 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

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 dependencies

Rule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

upf LEFT factor index2:ORIGINAL candidate factor=23;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 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 dependencies

Rule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S005

S007 · calculation

Before

upf LEFT complete product at original factor2 := mul(6, 23)

After

upf LEFT complete product at original factor2 = 138

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

upf LEFT:ORIGINAL factors=[2, 3, 23];ALL primality flags=[Yes, Yes, Yes];original n=138

After

empty product convention1;actual complete product=138;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 dependencies

Rule: upf_candidate_prime_and_complete_product_separate · Uses: G001, S007

S009 · implies

Before

upf RIGHT factor index0:ORIGINAL candidate factor=23;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 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 dependencies

Rule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S008

S010 · calculation

Before

upf RIGHT complete product at original factor0 := mul(1, 23)

After

upf RIGHT complete product at original factor0 = 23

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S010

S012 · calculation

Before

upf RIGHT complete product at original factor1 := mul(23, 2)

After

upf RIGHT complete product at original factor1 = 46

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

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 dependencies

Rule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S012

S014 · calculation

Before

upf RIGHT complete product at original factor2 := mul(46, 3)

After

upf RIGHT complete product at original factor2 = 138

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

upf RIGHT:ORIGINAL factors=[23, 2, 3];ALL primality flags=[Yes, Yes, Yes];original n=138

After

empty product convention1;actual complete product=138;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 dependencies

Rule: upf_candidate_prime_and_complete_product_separate · Uses: G001, S014

S016 · implies

Before

SORTED left occurrence index=0,factor=2;ORIGINAL right factors=[23, 2, 3];full right product=138

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 dependencies

Rule: upf_local_first_right_witness_not_occurrence_matching · Uses: G001, S015

S017 · implies

Before

SORTED left occurrence index=1,factor=3;ORIGINAL right factors=[23, 2, 3];full right product=138

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 dependencies

Rule: upf_local_first_right_witness_not_occurrence_matching · Uses: G001, S016

S018 · implies

Before

SORTED left occurrence index=2,factor=23;ORIGINAL right factors=[23, 2, 3];full right product=138

After

[p,p_divides_product,FIRST original right index,right factor,right factor equals p]=[23, Yes, 0, 23, 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 dependencies

Rule: upf_local_first_right_witness_not_occurrence_matching · Uses: G001, S017

S019 · implies

Before

ORIGINAL n=138;left valid=Yes,right valid=Yes;sorted multisets=[[2, 3, 23], [2, 3, 23]];local witness rows=[[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [23, Yes, 0, 23, 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 dependencies

Rule: 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, 23], [2, 3, 23], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [23, Yes, 0, 23, 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: 64a396591545b34660a6cb88a15959b406aea052bc13f4c27d06bd07d66903bd
  • candidate record SHA-256: bd6a927c689e07792b5b779f1711c0290e02250170cf4c609ad34aec8bcea732
  • semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a
  • question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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