slug: cohen-consecutive-cube-twin-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.result28

Cohen Conjecture 28

Problem

Cohen, Journal of Integer Sequences 28 (2025), Article 25.4.7, Section 2.5, Conjecture 28, asserts that there are at least two twin-prime pairs between every pair of consecutive positive cubes and that the count eventually exceeds every positive integer. Its printed threshold table further states N(10) = 11: for every natural n >= 11, there are at least ten such pairs strictly between n^3 and (n+1)^3.

The formal claim28 retains both general assertions and all seventeen printed threshold clauses. The theorem result28 proves its negation.

Motivation

The printed threshold is part of the named conjecture, but the count sequence printed immediately before it gives nine twin-prime pairs for n = 11. Checking that count settles the full conjunction without changing its other clauses.

Gap

Issue #9016 records the source and scoped literature search. The published JIS article and arXiv:2508.08335 state Conjecture 28 with N(10) = 11; the searched repository, open pull requests, and MathDB queries contained no settlement of this threshold. The search does not establish exhaustive publication priority.

Route

Define twinPairCount n as the number of natural numbers p with n^3 < p, p + 2 < (n+1)^3, and both p and p + 2 prime. Kernel evaluation gives twinPairCount 11 = 9. Instantiating the printed threshold clause at n = 11 would require 10 <= 9, a contradiction.

Falsifier

The refutation would fail if the open-interval interpretation disagreed with the paper’s examples, if the filtered count at n = 11 were not nine, or if claim28 omitted or weakened the printed N(10) = 11 clause. The formal definition, checked examples, and theorem retain these three links.

Evidence

• Lean theorem: D5/S0/Certificates/CohenConsecutiveCubePrimePairThresholdRefutation.result28.
• Freeze event: sha256:252f3161cf9445d1de24352155da81e6de9872de419ef43dcc9deaa228547d6b.
• Module statement identity: sha256:d475fe3507eb9b1372aaea14defe909c30cd93c2ff7454a84ec4046863029fb1.
• Result declaration identity: sha256:6a1a9b0c9d12e06f7e1e4b66535d95cad610cc36f8db62234f09ecf5311330d2.
• 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 28 as printed. It does not assert a corrected value of N(10) 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.