Valid Prime Products, Multiplicities, Euclid Witnesses and Sign Units
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batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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1. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=12>1 and candidate factors=[2, 3, 2] 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).
View answer and explanation
(12, [2, 3, 2], Yes, 12, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (12, [2, 3, 2], Yes, 12, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
2. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=18>1 and candidate factors=[2, 3, 3] 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).
View answer and explanation
(18, [2, 3, 3], Yes, 18, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (18, [2, 3, 3], Yes, 18, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
3. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=30>1 and candidate factors=[2, 3, 5] 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).
View answer and explanation
(30, [2, 3, 5], Yes, 30, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (30, [2, 3, 5], Yes, 30, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
4. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=42>1 and candidate factors=[2, 3, 7] 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).
View answer and explanation
(42, [2, 3, 7], Yes, 42, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (42, [2, 3, 7], Yes, 42, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
5. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=66>1 and candidate factors=[2, 3, 11] 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).
View answer and explanation
(66, [2, 3, 11], Yes, 66, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (66, [2, 3, 11], Yes, 66, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
6. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=78>1 and candidate factors=[2, 3, 13] 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).
View answer and explanation
(78, [2, 3, 13], Yes, 78, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (78, [2, 3, 13], Yes, 78, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
7. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=102>1 and candidate factors=[2, 3, 17] 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).
View answer and explanation
(102, [2, 3, 17], Yes, 102, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (102, [2, 3, 17], Yes, 102, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
8. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=114>1 and candidate factors=[2, 3, 19] 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).
View answer and explanation
(114, [2, 3, 19], Yes, 114, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (114, [2, 3, 19], Yes, 114, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
9. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=138>1 and candidate factors=[2, 3, 23] 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).
View answer and explanation
(138, [2, 3, 23], Yes, 138, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (138, [2, 3, 23], Yes, 138, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
10. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=174>1 and candidate factors=[2, 3, 29] 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).
View answer and explanation
(174, [2, 3, 29], Yes, 174, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (174, [2, 3, 29], Yes, 174, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
11. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=186>1 and candidate factors=[2, 3, 31] 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).
View answer and explanation
(186, [2, 3, 31], Yes, 186, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (186, [2, 3, 31], Yes, 186, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
12. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=222>1 and candidate factors=[2, 3, 37] 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).
View answer and explanation
(222, [2, 3, 37], Yes, 222, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (222, [2, 3, 37], Yes, 222, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
13. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=246>1 and candidate factors=[2, 3, 41] 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).
View answer and explanation
(246, [2, 3, 41], Yes, 246, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (246, [2, 3, 41], Yes, 246, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
14. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=258>1 and candidate factors=[2, 3, 43] 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).
View answer and explanation
(258, [2, 3, 43], Yes, 258, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (258, [2, 3, 43], Yes, 258, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
15. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=282>1 and candidate factors=[2, 3, 47] 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).
View answer and explanation
(282, [2, 3, 47], Yes, 282, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (282, [2, 3, 47], Yes, 282, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
16. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=318>1 and candidate factors=[2, 3, 53] 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).
View answer and explanation
(318, [2, 3, 53], Yes, 318, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (318, [2, 3, 53], Yes, 318, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
17. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=354>1 and candidate factors=[2, 3, 59] 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).
View answer and explanation
(354, [2, 3, 59], Yes, 354, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (354, [2, 3, 59], Yes, 354, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
18. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=366>1 and candidate factors=[2, 3, 61] 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).
View answer and explanation
(366, [2, 3, 61], Yes, 366, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (366, [2, 3, 61], Yes, 366, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
19. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=402>1 and candidate factors=[2, 3, 67] 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).
View answer and explanation
(402, [2, 3, 67], Yes, 402, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (402, [2, 3, 67], Yes, 402, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
20. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=426>1 and candidate factors=[2, 3, 71] 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).
View answer and explanation
(426, [2, 3, 71], Yes, 426, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (426, [2, 3, 71], Yes, 426, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
21. Prime Factors and a Complete Product Are Separate Validity Checks
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).
View answer and explanation
(438, [2, 3, 73], Yes, 438, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (438, [2, 3, 73], Yes, 438, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
22. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=474>1 and candidate factors=[2, 3, 79] 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).
View answer and explanation
(474, [2, 3, 79], Yes, 474, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (474, [2, 3, 79], Yes, 474, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
23. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=498>1 and candidate factors=[2, 3, 83] 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).
View answer and explanation
(498, [2, 3, 83], Yes, 498, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (498, [2, 3, 83], Yes, 498, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
24. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=534>1 and candidate factors=[2, 3, 89] 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).
View answer and explanation
(534, [2, 3, 89], Yes, 534, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (534, [2, 3, 89], Yes, 534, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
25. Prime Factors and a Complete Product Are Separate Validity Checks
Use positive integer n=582>1 and candidate factors=[2, 3, 97] 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).
View answer and explanation
(582, [2, 3, 97], Yes, 582, Yes, existence_requires_complete_prime_product)
Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (582, [2, 3, 97], Yes, 582, Yes, existence_requires_complete_prime_product).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate
26. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=12>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(12, [2, 2, 3], [[2, 2], [3, 1]], 12, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (12, [2, 2, 3], [[2, 2], [3, 1]], 12, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
27. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=18>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(18, [2, 3, 3], [[2, 1], [3, 2]], 18, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (18, [2, 3, 3], [[2, 1], [3, 2]], 18, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
28. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=30>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
29. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=42>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
30. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=66>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
31. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=78>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
32. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=102>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
33. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=114>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
34. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=138>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
35. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=174>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
36. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=186>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
37. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=222>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
38. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=246>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
39. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=258>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
40. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=282>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
41. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=318>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
42. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=354>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
43. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=366>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
44. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=402>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
45. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=426>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
46. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=438>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
47. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=474>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
48. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=498>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
49. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=534>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
50. Complete Trial Division Produces the Canonical Prime Multiset
Use positive integer n=582>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).
View answer and explanation
(582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, Yes, complete_trial_division_range)
Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, Yes, complete_trial_division_range).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate
51. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=12>1, candidate factors_left=[2, 3, 2] and factors_right=[2, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 2, 3], [2, 2, 3], [[2, Yes, 0, 2, Yes], [2, Yes, 0, 2, Yes], [3, Yes, 2, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 2, 3], [2, 2, 3], [[2, Yes, 0, 2, Yes], [2, Yes, 0, 2, Yes], [3, Yes, 2, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
52. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=18>1, candidate factors_left=[2, 3, 3] and factors_right=[3, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 3], [2, 3, 3], [[2, Yes, 1, 2, Yes], [3, Yes, 0, 3, Yes], [3, Yes, 0, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 3], [2, 3, 3], [[2, Yes, 1, 2, Yes], [3, Yes, 0, 3, Yes], [3, Yes, 0, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
53. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=30>1, candidate factors_left=[2, 3, 5] and factors_right=[5, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 5], [2, 3, 5], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [5, Yes, 0, 5, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 5], [2, 3, 5], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [5, Yes, 0, 5, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
54. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=42>1, candidate factors_left=[2, 3, 7] and factors_right=[7, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 7], [2, 3, 7], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [7, Yes, 0, 7, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 7], [2, 3, 7], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [7, Yes, 0, 7, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
55. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=66>1, candidate factors_left=[2, 3, 11] and factors_right=[11, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 11], [2, 3, 11], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [11, Yes, 0, 11, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 11], [2, 3, 11], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [11, Yes, 0, 11, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
56. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=78>1, candidate factors_left=[2, 3, 13] and factors_right=[13, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 13], [2, 3, 13], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [13, Yes, 0, 13, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 13], [2, 3, 13], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [13, Yes, 0, 13, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
57. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=102>1, candidate factors_left=[2, 3, 17] and factors_right=[17, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 17], [2, 3, 17], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [17, Yes, 0, 17, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 17], [2, 3, 17], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [17, Yes, 0, 17, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
58. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=114>1, candidate factors_left=[2, 3, 19] and factors_right=[19, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 19], [2, 3, 19], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [19, Yes, 0, 19, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 19], [2, 3, 19], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [19, Yes, 0, 19, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
59. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
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).
View answer and explanation
(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)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (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).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
60. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=174>1, candidate factors_left=[2, 3, 29] and factors_right=[29, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 29], [2, 3, 29], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [29, Yes, 0, 29, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 29], [2, 3, 29], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [29, Yes, 0, 29, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
61. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=186>1, candidate factors_left=[2, 3, 31] and factors_right=[31, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 31], [2, 3, 31], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [31, Yes, 0, 31, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 31], [2, 3, 31], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [31, Yes, 0, 31, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
62. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=222>1, candidate factors_left=[2, 3, 37] and factors_right=[37, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 37], [2, 3, 37], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [37, Yes, 0, 37, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 37], [2, 3, 37], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [37, Yes, 0, 37, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
63. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=246>1, candidate factors_left=[2, 3, 41] and factors_right=[41, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 41], [2, 3, 41], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [41, Yes, 0, 41, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 41], [2, 3, 41], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [41, Yes, 0, 41, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
64. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=258>1, candidate factors_left=[2, 3, 43] and factors_right=[43, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 43], [2, 3, 43], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [43, Yes, 0, 43, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 43], [2, 3, 43], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [43, Yes, 0, 43, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
65. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=282>1, candidate factors_left=[2, 3, 47] and factors_right=[47, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
66. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=318>1, candidate factors_left=[2, 3, 53] and factors_right=[53, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 53], [2, 3, 53], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [53, Yes, 0, 53, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 53], [2, 3, 53], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [53, Yes, 0, 53, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
67. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=354>1, candidate factors_left=[2, 3, 59] and factors_right=[59, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 59], [2, 3, 59], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [59, Yes, 0, 59, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 59], [2, 3, 59], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [59, Yes, 0, 59, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
68. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=366>1, candidate factors_left=[2, 3, 61] and factors_right=[61, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 61], [2, 3, 61], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [61, Yes, 0, 61, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 61], [2, 3, 61], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [61, Yes, 0, 61, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
69. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
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).
View answer and explanation
(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)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (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).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
70. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=426>1, candidate factors_left=[2, 3, 71] and factors_right=[71, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 71], [2, 3, 71], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [71, Yes, 0, 71, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 71], [2, 3, 71], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [71, Yes, 0, 71, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
71. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=438>1, candidate factors_left=[2, 3, 73] and factors_right=[73, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 73], [2, 3, 73], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [73, Yes, 0, 73, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 73], [2, 3, 73], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [73, Yes, 0, 73, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
72. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=474>1, candidate factors_left=[2, 3, 79] and factors_right=[79, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 79], [2, 3, 79], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [79, Yes, 0, 79, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 79], [2, 3, 79], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [79, Yes, 0, 79, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
73. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=498>1, candidate factors_left=[2, 3, 83] and factors_right=[83, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 83], [2, 3, 83], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [83, Yes, 0, 83, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 83], [2, 3, 83], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [83, Yes, 0, 83, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
74. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=534>1, candidate factors_left=[2, 3, 89] and factors_right=[89, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 89], [2, 3, 89], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [89, Yes, 0, 89, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 89], [2, 3, 89], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [89, Yes, 0, 89, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
75. Only Two Valid Prime Products Can Supply This Uniqueness Certificate
Use positive integer n=582>1, candidate factors_left=[2, 3, 97] and factors_right=[97, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).
View answer and explanation
(Yes, Yes, [2, 3, 97], [2, 3, 97], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [97, Yes, 0, 97, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)
Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 97], [2, 3, 97], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [97, Yes, 0, 97, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate
76. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=0. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(0, No, null, [], zero_has_no_prime_factorization)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (0, No, null, [], zero_has_no_prime_factorization).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
77. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(1, No, 1, [], units_have_empty_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (1, No, 1, [], units_have_empty_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
78. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-1, No, -1, [], units_have_empty_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-1, No, -1, [], units_have_empty_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
79. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-42. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-42, No, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-42, No, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
80. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-66. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-66, No, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-66, No, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
81. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-78. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-78, No, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-78, No, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
82. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-102. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-102, No, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-102, No, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
83. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-114. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-114, No, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-114, No, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
84. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-138. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-138, No, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-138, No, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
85. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-174. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-174, No, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-174, No, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
86. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-186. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-186, No, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-186, No, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
87. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-222. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-222, No, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-222, No, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
88. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-246. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-246, No, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-246, No, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
89. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-258. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-258, No, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-258, No, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
90. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-282. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-282, No, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-282, No, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
91. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-318. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-318, No, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-318, No, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
92. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-354. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-354, No, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-354, No, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
93. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-366. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-366, No, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-366, No, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
94. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-402. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-402, No, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-402, No, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
95. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-426. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-426, No, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-426, No, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
96. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-438. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-438, No, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-438, No, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
97. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-474. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-474, No, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-474, No, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
98. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-498. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-498, No, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-498, No, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
99. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-534. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-534, No, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-534, No, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
100. Separate Zero, Sign Units and the Positive Prime Multiset
Use original bounded integer n=-582. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).
This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.
View answer and explanation
(-582, No, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset)
Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-582, No, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset).
Reviewed step-by-step derivation
Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate
Canonical question JSONL · English JSONL · Validation report