Cohen’s Consecutive-Cube Prime-Pair Thresholds
Abstract
The printed thresholds in Cohen’s Conjectures 28 and 29 fail at n = 11 and n = 12.
Definition 1.1 (Twin-prime pairs between consecutive cubes).
Formalization. D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.twinPairCount (✓ std3).
Citation. Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.
Commentary.
For each natural n, the half-open finite interval starts at n^3 + 1. The filter retains exactly the p for which n^3 < p, p + 2 < (n+1)^3, and both p and p + 2 are prime; its cardinality is twinPairCount(n).
Definition 1.2 (Cousin-prime pairs between consecutive cubes).
Formalization. D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.cousinPairCount (✓ std3).
Citation. Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.
Commentary.
For each natural n, the filter requires n^3 < p and p + 4 < (n+1)^3. It also requires p and p + 4 to be prime and p + 1, p + 2, and p + 3 not to be prime, so the endpoint primes are consecutive.
Definition 1.3 (Conjecture 28).
Formalization. D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.claim28 (✓ std3).
Citation. Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.
Commentary.
The paper states: “Conjecture 28 (number of twin primes between consecutive cubes). For every n ∈ N, the number of pairs of twin primes between n³ and (n + 1)³ is never less than two. More generally, for every k ∈ N, there exists N(k) ∈ N such that for all n ≥ N(k) there are at least k pairs of twin primes between n³ and (n + 1)³. Specifically, N(1) = N(2) = 1, N(3) = 3, N(4) = 5, N(5) = 8, N(6) = N(7) = N(8) = N(9) = 10, N(10) = 11, N(11) = 13, N(12) = N(13) = N(14) = N(15) = N(16) = 15, and N(17) = 20.” Each printed threshold is represented as its own universal conjunct. The existential N ranges over natural numbers; adding 1 ≤ N would strengthen that unused conjunct and does not affect the refutation.
Definition 1.4 (Conjecture 29).
Formalization. D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.claim29 (✓ std3).
Citation. Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.
Commentary.
The paper states: “Conjecture 29 (number of cousin primes between consecutive cubes). For n ∈ N with n > 1, the number of pairs of cousin primes between n³ and (n + 1)³ is never less than two. More generally, for every k ∈ N, there exists N(k) ∈ N such that, for all n ≥ N(k), there are at least k pairs of cousin primes between n³ and (n + 1)³. Specifically, N(1) = N(2) = 2, N(3) = 8, N(4) = N(5) = N(6) = N(7) = 9, N(8) = N(9) = N(10) = 12.” Each printed threshold is represented as its own universal conjunct. The existential N ranges over natural numbers; adding 1 ≤ N would strengthen that unused conjunct and does not affect the refutation.
Theorem 1.5 (Conjecture 28 is false).
Proof. Machine-checked in Lean as D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result28 (✓ std3). ∎
Resolves. Problems/cohen-consecutive-cube-twin-prime-threshold-refutation (refuted) by D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result28.
Source. Repository-derived.
Acknowledgement. Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.
Commentary.
At n = 11 the open interval (1331,1728) contains exactly nine twin-prime pairs: (1427,1429), (1451,1453), (1481,1483), (1487,1489), (1607,1609), (1619,1621), (1667,1669), (1697,1699), and (1721,1723). Thus the printed N(10) = 11 threshold is false.
Theorem 1.6 (Conjecture 29 is false).
Proof. Machine-checked in Lean as D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result29 (✓ std3). ∎
Resolves. Problems/cohen-consecutive-cube-cousin-prime-threshold-refutation (refuted) by D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result29.
Source. Repository-derived.
Acknowledgement. Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.
Commentary.
At n = 12 the open interval (1728,2197) contains exactly seven cousin-prime pairs: (1783,1787), (1867,1871), (1873,1877), (1993,1997), (1999,2003), (2083,2087), and (2137,2141). Thus the printed N(8) = N(9) = N(10) = 12 thresholds are false.
References
- Truth anchor:
D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.claim28 - Truth anchor:
D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.claim29 - Truth anchor:
D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.cousinPairCount - Truth anchor:
D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result28 - Truth anchor:
D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result29 - Truth anchor:
D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.twinPairCount