{"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"}