{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:SJRQON3SJPQMXLFVFT3ARPM7YO","short_pith_number":"pith:SJRQON3S","schema_version":"1.0","canonical_sha256":"92630737724be0cbacb52cf608bd9fc39dbd53dce2a70470afdbe4af049e17d1","source":{"kind":"arxiv","id":"2607.09793","version":1},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.09793","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-09T05:17:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6e74a5890f630d34f8ddf3a8940e5b96168d9c821a8adc4f47e6c5b0ece0e01b","abstract_canon_sha256":"b8d6691f0ac877e5b392ff4e6838a00112f9e4cdd5a68c77e123e06b88399e17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T00:18:36.137413Z","signature_b64":"qkhCfiu6xcxZOckd02kQpGhBNu1fmnTjM1h4Sjmk1nZ9mY6h47RBagkPv6M2XzPJaymBbfBs7yHqJXszTIS0Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92630737724be0cbacb52cf608bd9fc39dbd53dce2a70470afdbe4af049e17d1","last_reissued_at":"2026-07-14T00:18:36.136548Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T00:18:36.136548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.09793","created_at":"2026-07-14T00:18:36.137011+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.09793v1","created_at":"2026-07-14T00:18:36.137011+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.09793","created_at":"2026-07-14T00:18:36.137011+00:00"},{"alias_kind":"pith_short_12","alias_value":"SJRQON3SJPQM","created_at":"2026-07-14T00:18:36.137011+00:00"},{"alias_kind":"pith_short_16","alias_value":"SJRQON3SJPQMXLFV","created_at":"2026-07-14T00:18:36.137011+00:00"},{"alias_kind":"pith_short_8","alias_value":"SJRQON3S","created_at":"2026-07-14T00:18:36.137011+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO","json":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO.json","graph_json":"https://pith.science/api/pith-number/SJRQON3SJPQMXLFVFT3ARPM7YO/graph.json","events_json":"https://pith.science/api/pith-number/SJRQON3SJPQMXLFVFT3ARPM7YO/events.json","paper":"https://pith.science/paper/SJRQON3S"},"agent_actions":{"view_html":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO","download_json":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO.json","view_paper":"https://pith.science/paper/SJRQON3S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.09793&json=true","fetch_graph":"https://pith.science/api/pith-number/SJRQON3SJPQMXLFVFT3ARPM7YO/graph.json","fetch_events":"https://pith.science/api/pith-number/SJRQON3SJPQMXLFVFT3ARPM7YO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO/action/storage_attestation","attest_author":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO/action/author_attestation","sign_citation":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO/action/citation_signature","submit_replication":"https://pith.science/pith/SJRQON3SJPQMXLFVFT3ARPM7YO/action/replication_record"}},"created_at":"2026-07-14T00:18:36.137011+00:00","updated_at":"2026-07-14T00:18:36.137011+00:00"}