{"paper":{"title":"A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Josu\\'e Alexander Ibarra","submitted_at":"2026-07-09T05:17:23Z","abstract_excerpt":"An integer $n \\ge 1$ is cyclic if $\\gcd(n,\\varphi(n))=1$ (equivalently, if every group of order $n$ is cyclic), and Sophie Germain cyclic if both $n$ and $2n+1$ are cyclic. Let $C_\\sigma(N)$ count the Sophie Germain cyclic integers in $[1,N]$. Cohen conjectured that $C_\\sigma$ is subadditive, $C_\\sigma(m+n) \\le C_\\sigma(m)+C_\\sigma(n)$ for all $1 \\le m \\le n$ (his Conjecture 66), having checked $m,n \\le 10^6$ without finding a counterexample. We give one: at $m=31$, $n=3928$, $C_\\sigma(3959)=697 > 696 = C_\\sigma(31)+C_\\sigma(3928)$. The argument is short, and is verified by the Lean 4 kernel."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09793","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.09793/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}