{"paper":{"title":"Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matt Visser (Victoria University of Wellington)","submitted_at":"2019-03-31T22:33:31Z","abstract_excerpt":"The Firoozbakht, Nicholoson, and Farhadian conjectures can be phrased in terms of increasingly powerful conjectured bounds on the prime gaps $g_n := p_{n+1}-p_n$. \\[ g_n \\leq p_n \\left(p_n^{1/n} -1 \\right)\\qquad\\qquad\\qquad (n \\geq 1; \\; Firoozbakht). \\] \\[ g_n \\leq p_n \\left((n\\ln n)^{1/n} -1 \\right)\\qquad\\qquad (n>4; \\; Nicholson). \\] \\[ g_n \\leq p_n \\left( \\left(p_n {\\ln n\\over\\ln p_n}\\right)^{1/n} -1 \\right)\\qquad (n>4; \\; Farhadian). \\] While a general proof of any of these conjectures is far out of reach I shall show that all three of these conjectures are unconditionally and explicitly "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.00499","kind":"arxiv","version":2},"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/1904.00499/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"}