Original question and givens
Use positive integer n=438>1 and candidate factors=[2, 3, 73] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).
G001 — Use positive integer n=438>1 and candidate factors=[2, 3, 73] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=upf-factorization-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, factors, all_factors_prime, factor_product, valid_factorization, 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 candidate 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 candidate complete product at original factor0 := mul(1, 2)
After
upf candidate 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 candidate 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 candidate complete product at original factor1 := mul(2, 3)
After
upf candidate 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 candidate factor index2:ORIGINAL candidate factor=73;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4, 5, 6, 7, 8]
After
ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 1], [5, 3], [6, 1], [7, 3], [8, 1]];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 candidate complete product at original factor2 := mul(6, 73)
After
upf candidate complete product at original factor2 = 438
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 candidate:ORIGINAL factors=[2, 3, 73];ALL primality flags=[Yes, Yes, Yes];original n=438
After
empty product convention1;actual complete product=438;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
Result
(438, [2, 3, 73], Yes, 438, Yes, existence_requires_complete_prime_product)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.