Prime Natural Numbers and Complete Factor Checks
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261018-r04-prime-number-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=0. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(0, all-positive-naturals, No, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, all-positive-naturals, No, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
2. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=1. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(1, [1], No, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [1], No, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
3. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=2. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(2, [1, 2], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [1, 2], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
4. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=3. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(3, [1, 3], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [1, 3], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
5. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=4. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(4, [1, 2, 4], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2, 4], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
6. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=5. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(5, [1, 5], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [1, 5], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
7. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=6. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(6, [1, 2, 3, 6], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [1, 2, 3, 6], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
8. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=7. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(7, [1, 7], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [1, 7], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
9. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=8. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(8, [1, 2, 4, 8], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [1, 2, 4, 8], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
10. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=9. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(9, [1, 3, 9], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [1, 3, 9], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
11. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=10. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(10, [1, 2, 5, 10], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [1, 2, 5, 10], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
12. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=11. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(11, [1, 11], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, [1, 11], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
13. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=12. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(12, [1, 2, 3, 4, 6, 12], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, [1, 2, 3, 4, 6, 12], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
14. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=13. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(13, [1, 13], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, [1, 13], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
15. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=14. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(14, [1, 2, 7, 14], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, [1, 2, 7, 14], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
16. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=15. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(15, [1, 3, 5, 15], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, [1, 3, 5, 15], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
17. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=16. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(16, [1, 2, 4, 8, 16], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, [1, 2, 4, 8, 16], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
18. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=17. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(17, [1, 17], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, [1, 17], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
19. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=18. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(18, [1, 2, 3, 6, 9, 18], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, [1, 2, 3, 6, 9, 18], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
20. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=19. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(19, [1, 19], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, [1, 19], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
21. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=20. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(20, [1, 2, 4, 5, 10, 20], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, [1, 2, 4, 5, 10, 20], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
22. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=21. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(21, [1, 3, 7, 21], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, [1, 3, 7, 21], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
23. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=22. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(22, [1, 2, 11, 22], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, [1, 2, 11, 22], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
24. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=23. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(23, [1, 23], Yes, Yes)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, [1, 23], Yes, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
25. Complete Positive Divisors and Prime Qualification
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=24. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
View answer and explanation
(24, [1, 2, 3, 4, 6, 8, 12, 24], Yes, No)
Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, [1, 2, 3, 4, 6, 8, 12, 24], Yes, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-divisor-classification · candidate
26. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=0. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(0, not-applicable, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, not-applicable, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
27. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=1. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(1, not-applicable, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, not-applicable, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
28. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=2. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(2, 2, Yes)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 2, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
29. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=33. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(33, 3, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (33, 3, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
30. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=34. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(34, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (34, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
31. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=35. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(35, 5, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (35, 5, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
32. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=36. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(36, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (36, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
33. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=37. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(37, 37, Yes)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (37, 37, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
34. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=38. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(38, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (38, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
35. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=39. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(39, 3, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (39, 3, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
36. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=40. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(40, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (40, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
37. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=41. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(41, 41, Yes)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (41, 41, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
38. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=42. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(42, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (42, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
39. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=43. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(43, 43, Yes)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (43, 43, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
40. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=44. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(44, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (44, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
41. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=45. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(45, 3, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (45, 3, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
42. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=46. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(46, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (46, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
43. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=47. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(47, 47, Yes)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (47, 47, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
44. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=48. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(48, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (48, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
45. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=49. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(49, 7, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (49, 7, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
46. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=50. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(50, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (50, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
47. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=51. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(51, 3, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (51, 3, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
48. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=52. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(52, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (52, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
49. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=53. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(53, 53, Yes)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (53, 53, Yes).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
50. Least Divisor at Least Two and Primality
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=54. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.
View answer and explanation
(54, 2, No)
Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (54, 2, No).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-smallest-factor · candidate
51. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=20, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(20, 2, Yes, 10, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 2, Yes, 10, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
52. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=22, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(22, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
53. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=24, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(24, 5, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 5, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
54. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=26, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(26, 2, Yes, 13, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (26, 2, Yes, 13, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
55. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=28, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(28, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (28, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
56. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=30, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(30, 5, Yes, 6, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (30, 5, Yes, 6, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
57. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=32, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(32, 2, Yes, 16, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (32, 2, Yes, 16, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
58. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=34, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(34, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (34, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
59. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=36, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(36, 5, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (36, 5, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
60. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=38, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(38, 2, Yes, 19, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (38, 2, Yes, 19, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
61. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=40, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(40, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (40, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
62. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=42, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(42, 5, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (42, 5, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
63. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=44, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(44, 2, Yes, 22, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (44, 2, Yes, 22, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
64. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=46, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(46, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (46, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
65. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=48, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(48, 5, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (48, 5, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
66. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=50, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(50, 2, Yes, 25, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (50, 2, Yes, 25, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
67. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=52, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(52, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (52, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
68. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=54, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(54, 5, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (54, 5, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
69. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=56, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(56, 2, Yes, 28, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (56, 2, Yes, 28, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
70. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=58, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(58, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (58, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
71. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=60, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(60, 5, Yes, 12, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (60, 5, Yes, 12, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
72. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=62, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(62, 2, Yes, 31, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (62, 2, Yes, 31, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
73. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=64, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(64, 3, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (64, 3, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
74. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=66, candidate d=5. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(66, 5, No, not-applicable, not-a-nontrivial-factor-witness)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (66, 5, No, not-applicable, not-a-nontrivial-factor-witness).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
75. Qualify a Nontrivial Composite-Factor Witness
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=68, candidate d=2. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
View answer and explanation
(68, 2, Yes, 34, composite-witness-valid)
Require all nontrivial-factor conditions before computing the quotient; a failed candidate alone does not certify primality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (68, 2, Yes, 34, composite-witness-valid).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-factor-witness · candidate
76. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=0, hi=20, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(0, 20, [2, 3, 5, 7, 11, 13, 17, 19], 8)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 20, [2, 3, 5, 7, 11, 13, 17, 19], 8).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
77. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=5, hi=25, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(5, 25, [5, 7, 11, 13, 17, 19, 23], 7)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 25, [5, 7, 11, 13, 17, 19, 23], 7).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
78. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=10, hi=30, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(10, 30, [11, 13, 17, 19, 23, 29], 6)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 30, [11, 13, 17, 19, 23, 29], 6).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
79. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=15, hi=35, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(15, 35, [17, 19, 23, 29, 31], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 35, [17, 19, 23, 29, 31], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
80. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=20, hi=40, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(20, 40, [23, 29, 31, 37], 4)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 40, [23, 29, 31, 37], 4).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
81. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=25, hi=45, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(25, 45, [29, 31, 37, 41, 43], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (25, 45, [29, 31, 37, 41, 43], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
82. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=30, hi=50, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(30, 50, [31, 37, 41, 43, 47], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (30, 50, [31, 37, 41, 43, 47], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
83. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=35, hi=55, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(35, 55, [37, 41, 43, 47, 53], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (35, 55, [37, 41, 43, 47, 53], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
84. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=40, hi=60, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(40, 60, [41, 43, 47, 53, 59], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (40, 60, [41, 43, 47, 53, 59], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
85. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=45, hi=65, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(45, 65, [47, 53, 59, 61], 4)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (45, 65, [47, 53, 59, 61], 4).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
86. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=50, hi=70, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(50, 70, [53, 59, 61, 67], 4)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (50, 70, [53, 59, 61, 67], 4).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
87. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=55, hi=75, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(55, 75, [59, 61, 67, 71, 73], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (55, 75, [59, 61, 67, 71, 73], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
88. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=60, hi=80, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(60, 80, [61, 67, 71, 73, 79], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (60, 80, [61, 67, 71, 73, 79], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
89. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=65, hi=85, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(65, 85, [67, 71, 73, 79, 83], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (65, 85, [67, 71, 73, 79, 83], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
90. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=70, hi=90, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(70, 90, [71, 73, 79, 83, 89], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (70, 90, [71, 73, 79, 83, 89], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
91. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=75, hi=95, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(75, 95, [79, 83, 89], 3)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (75, 95, [79, 83, 89], 3).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
92. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=80, hi=100, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(80, 100, [83, 89, 97], 3)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (80, 100, [83, 89, 97], 3).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
93. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=85, hi=105, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(85, 105, [89, 97, 101, 103], 4)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (85, 105, [89, 97, 101, 103], 4).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
94. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=90, hi=110, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(90, 110, [97, 101, 103, 107, 109], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (90, 110, [97, 101, 103, 107, 109], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
95. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=95, hi=115, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(95, 115, [97, 101, 103, 107, 109, 113], 6)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (95, 115, [97, 101, 103, 107, 109, 113], 6).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
96. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=100, hi=120, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(100, 120, [101, 103, 107, 109, 113], 5)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (100, 120, [101, 103, 107, 109, 113], 5).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
97. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=105, hi=125, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(105, 125, [107, 109, 113], 3)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (105, 125, [107, 109, 113], 3).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
98. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=110, hi=130, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(110, 130, [113, 127], 2)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (110, 130, [113, 127], 2).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
99. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=115, hi=135, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(115, 135, [127, 131], 2)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (115, 135, [127, 131], 2).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
100. Every Prime in an Inclusive Bounded Interval
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=120, hi=140, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).
View answer and explanation
(120, 140, [127, 131, 137, 139], 4)
Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (120, 140, [127, 131, 137, 139], 4).
Reviewed step-by-step derivation
Prime natural number · qf-ext18-prime-interval-inventory · candidate
Canonical question JSONL · English JSONL · Validation report