{"paper":{"title":"Consecutive primes and Beatty sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Victor Z. Guo, William D. Banks","submitted_at":"2016-12-05T18:45:06Z","abstract_excerpt":"Fix irrational numbers $\\alpha,\\hat\\alpha>1$ of finite type and real numbers $\\beta,\\hat\\beta\\ge 0$, and let $B$ and $\\hat B$ be the Beatty sequences $$ B:=(\\lfloor\\alpha m+\\beta\\rfloor)_{m\\ge 1}\\quad\\text{and}\\quad\\hat B:=(\\lfloor\\hat\\alpha m+\\hat\\beta\\rfloor)_{m\\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\\sharp)$ of consecutive primes for which $p\\in B$ and $p^\\sharp\\in\\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \\big|\\{p\\le x:p\\in B\\text{ and }p^\\sharp\\in\\hat B\\}\\big|=(\\alpha\\hat\\alpha)^{-1}\\pi(x)+O\\big(x(\\log "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01468","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":""},"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"}