{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:L4U7DN7EKXP6I4MISEMMKGQ6CD","short_pith_number":"pith:L4U7DN7E","schema_version":"1.0","canonical_sha256":"5f29f1b7e455dfe471889118c51a1e10f36f796099ce9f849787acf04e330dea","source":{"kind":"arxiv","id":"2605.18555","version":1},"attestation_state":"computed","paper":{"title":"Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexey Dolotov","submitted_at":"2026-05-18T15:35:56Z","abstract_excerpt":"The Wagstaff numbers $W_p = (2^p + 1)/3$ for odd primes $p$ are the natural $+1$ companions of the Mersenne numbers. Known primality proofs for $W_p$ with $p \\geq 2617$ rely on the elliptic-curve primality proving algorithm of Atkin-Morain; Chebyshev/Lucas-type tests, while available as compositeness criteria, remain conjectural on the sufficiency side. We present fully verified primality proofs of $W_{2617}$ (788 digits), $W_{10501}$ (3161 digits), and $W_{12391}$ (3730 digits), independent of ECPP and relying only on classical $N-1$ machinery. The proofs apply the Brillhart-Lehmer-Selfridge "},"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":"2605.18555","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-18T15:35:56Z","cross_cats_sorted":[],"title_canon_sha256":"06ffef627d8313a088f3528b5268b080d84a4e2be4f2daf06d6edac2bd79f5d8","abstract_canon_sha256":"ae14e5b4c965bad8507a82456fc7f0448b9061eed1710dec05a0c779acb94b2c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:06:07.516585Z","signature_b64":"qXg5woXQQDDB5dkpuWKhxj35x8Db8lEziJ1YqJ30cV0gQj1JWiUIi0EuWdrsu89TnOKwiLT4JLVpDuRUfN7iCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f29f1b7e455dfe471889118c51a1e10f36f796099ce9f849787acf04e330dea","last_reissued_at":"2026-05-20T00:06:07.515723Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:06:07.515723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexey Dolotov","submitted_at":"2026-05-18T15:35:56Z","abstract_excerpt":"The Wagstaff numbers $W_p = (2^p + 1)/3$ for odd primes $p$ are the natural $+1$ companions of the Mersenne numbers. Known primality proofs for $W_p$ with $p \\geq 2617$ rely on the elliptic-curve primality proving algorithm of Atkin-Morain; Chebyshev/Lucas-type tests, while available as compositeness criteria, remain conjectural on the sufficiency side. We present fully verified primality proofs of $W_{2617}$ (788 digits), $W_{10501}$ (3161 digits), and $W_{12391}$ (3730 digits), independent of ECPP and relying only on classical $N-1$ machinery. The proofs apply the Brillhart-Lehmer-Selfridge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.18555","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/2605.18555/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T00:01:59.378191Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"0b07db0b4a8645793de6d410572492d744166d6a706a8d069c1c7fbf678c9e1b"},"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":"2605.18555","created_at":"2026-05-20T00:06:07.515860+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.18555v1","created_at":"2026-05-20T00:06:07.515860+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.18555","created_at":"2026-05-20T00:06:07.515860+00:00"},{"alias_kind":"pith_short_12","alias_value":"L4U7DN7EKXP6","created_at":"2026-05-20T00:06:07.515860+00:00"},{"alias_kind":"pith_short_16","alias_value":"L4U7DN7EKXP6I4MI","created_at":"2026-05-20T00:06:07.515860+00:00"},{"alias_kind":"pith_short_8","alias_value":"L4U7DN7E","created_at":"2026-05-20T00:06:07.515860+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/L4U7DN7EKXP6I4MISEMMKGQ6CD","json":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD.json","graph_json":"https://pith.science/api/pith-number/L4U7DN7EKXP6I4MISEMMKGQ6CD/graph.json","events_json":"https://pith.science/api/pith-number/L4U7DN7EKXP6I4MISEMMKGQ6CD/events.json","paper":"https://pith.science/paper/L4U7DN7E"},"agent_actions":{"view_html":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD","download_json":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD.json","view_paper":"https://pith.science/paper/L4U7DN7E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.18555&json=true","fetch_graph":"https://pith.science/api/pith-number/L4U7DN7EKXP6I4MISEMMKGQ6CD/graph.json","fetch_events":"https://pith.science/api/pith-number/L4U7DN7EKXP6I4MISEMMKGQ6CD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD/action/storage_attestation","attest_author":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD/action/author_attestation","sign_citation":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD/action/citation_signature","submit_replication":"https://pith.science/pith/L4U7DN7EKXP6I4MISEMMKGQ6CD/action/replication_record"}},"created_at":"2026-05-20T00:06:07.515860+00:00","updated_at":"2026-05-20T00:06:07.515860+00:00"}