{"paper":{"title":"Explicit Prime Densities for the Rank of Appearance in Lucas Sequences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Closed-form formulas exist for the Dirichlet density of primes p where a fixed d divides the rank of appearance in any Lucas sequence U.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Joaquim Cera Da Concei\\c{c}\\~ao","submitted_at":"2026-04-21T21:48:43Z","abstract_excerpt":"Let $U$ be a Lucas sequence, $p$ be prime, and $\\rho_U(p)$ be the rank of appearance of $p$ in $U$. We derive closed-form formulas for the Dirichlet density of primes $p$ for which $d\\mid \\rho_U(p)$, where $d\\geq 1$ is a fixed integer. Our results complete the work of Sanna ($2022$) by covering all $U$ and all $d\\geq 1$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive closed-form formulas for the Dirichlet density of primes p for which d∣ρ_U(p), where d≥1 is a fixed integer. Our results complete the work of Sanna (2022) by covering all U and all d≥1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that uniform closed-form expressions exist and can be derived for every Lucas sequence U (including degenerate cases) and every d, relying on the standard algebraic properties of the discriminant and the recurrence without exceptional cases that break the formulas.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Closed-form Dirichlet density formulas are derived for primes p where d divides ρ_U(p) in Lucas sequences U, covering all U and all d ≥ 1.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Closed-form formulas exist for the Dirichlet density of primes p where a fixed d divides the rank of appearance in any Lucas sequence U.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"69e93e3092dd29c7e90ba7e62f4db6db173f6289483cb2329ec9f907c7917ef3"},"source":{"id":"2604.20014","kind":"arxiv","version":2},"verdict":{"id":"ad8d7161-0e59-4486-abf0-13be0dd22bac","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T01:05:26.592003Z","strongest_claim":"We derive closed-form formulas for the Dirichlet density of primes p for which d∣ρ_U(p), where d≥1 is a fixed integer. Our results complete the work of Sanna (2022) by covering all U and all d≥1.","one_line_summary":"Closed-form Dirichlet density formulas are derived for primes p where d divides ρ_U(p) in Lucas sequences U, covering all U and all d ≥ 1.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that uniform closed-form expressions exist and can be derived for every Lucas sequence U (including degenerate cases) and every d, relying on the standard algebraic properties of the discriminant and the recurrence without exceptional cases that break the formulas.","pith_extraction_headline":"Closed-form formulas exist for the Dirichlet density of primes p where a fixed d divides the rank of appearance in any Lucas sequence U."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.20014/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T15:38:39.877364Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T02:24:12.426702Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b3e9de76a78277fa1bf4a59ad97ecdb42921c2221c778542f731f6ea676cdd0c"},"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"}