{"paper":{"title":"On a conjecture of Ram\\'{\\i}rez Alfons\\'{\\i}n and Ska{\\l}ba II","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lilu Zhao, Wenguang Zhai, Yuchen Ding","submitted_at":"2023-09-18T14:16:40Z","abstract_excerpt":"Let $1<c<d$ be two relatively prime integers and $g_{c,d}=cd-c-d$. We confirm, by employing the Hardy--Littlewood method, a 2020 conjecture of Ram\\'{\\i}rez Alfons\\'{\\i}n and Ska{\\l}ba which states that $$#\\left\\{p\\le g_{c,d}:p\\in \\mathcal{P}, ~p=cx+dy,~x,y\\in \\mathbb{Z}_{\\geqslant0}\\right\\}\\sim \\frac{1}{2}\\pi\\left(g_{c,d}\\right) \\quad (\\text{as}~c\\rightarrow\\infty),$$ where $\\mathcal{P}$ is the set of primes, $\\mathbb{Z}_{\\geqslant0}$ is the set of nonnegative integers and $\\pi(t)$ denotes the number of primes not exceeding $t$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.09796","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/2309.09796/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"}