slug: cohen-consecutive-cube-cousin-prime-threshold-refutation bibkey: cohen2025cyclic doi: 10.48550/arXiv.2508.08335 url: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf triage: theorem motivation_gids:

• D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result29

Cohen Conjecture 29

Problem

Cohen, Journal of Integer Sequences 28 (2025), Article 25.4.7, Section 2.5, Conjecture 29, asserts that there are at least two cousin-prime pairs between consecutive cubes for every natural n > 1 and that the count eventually exceeds every positive integer. Its printed threshold table further states N(8) = N(9) = N(10) = 12: every natural n >= 12 has at least eight, nine, and ten such pairs, respectively.

The formal claim29 retains both general assertions and all ten printed threshold clauses. The theorem result29 proves its negation.

Motivation

The printed thresholds are part of the named conjecture, but the count sequence printed immediately before them gives seven cousin-prime pairs for n = 12. Checking that count settles the full conjunction while preserving the paper’s requirement that the two primes be consecutive.

Gap

Issue #9016 records the source and scoped literature search. The published JIS article and arXiv:2508.08335 state Conjecture 29 with all three thresholds at 12; the searched repository, open pull requests, and MathDB queries contained no settlement of these thresholds. The search does not establish exhaustive publication priority.

Route

Define cousinPairCount n as the number of natural numbers p with n^3 < p, p + 4 < (n+1)^3, prime endpoints, and no prime among p + 1, p + 2, and p + 3. Kernel evaluation gives cousinPairCount 12 = 7. Instantiating the printed N(8) = 12 clause at n = 12 would require 8 <= 7, a contradiction.

Falsifier

The refutation would fail if cousin pairs did not require consecutive primes, if the filtered count at n = 12 were not seven, or if claim29 omitted or weakened the printed N(8) = 12 clause. The formal definition, checked examples, and theorem retain these three links.

Evidence

• Lean theorem: D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result29.
• Freeze event: sha256:252f3161cf9445d1de24352155da81e6de9872de419ef43dcc9deaa228547d6b.
• Module statement identity: sha256:d475fe3507eb9b1372aaea14defe909c30cd93c2ff7454a84ec4046863029fb1.
• Result declaration identity: sha256:db584f0741fdb5011858ba4dfb9af452e41ff850d66da49abad77cd9b6405557.
• proof_shape: bind-only; escape_witness: none; admission_basis: open-problem-resolution (issue #9016).
• The axiom closure is propext, Classical.choice, and Quot.sound.

Triage

theorem. The certified count refutes Conjecture 29 as printed. It does not assert corrected threshold values or settle the eventual-growth conjunct in isolation.

ASSUMED-UNVERIFIED

The scoped literature search does not establish exhaustive publication priority. The JIS errata archive and Google Scholar were not exhaustively checked.