{"paper":{"title":"Primes of the form $ax+by$","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hui Zhu, Yong-Gao Chen","submitted_at":"2025-06-04T06:58:48Z","abstract_excerpt":"For two coprime positive integers $a,b$, let $T(a,b)=\\{ ax+by : x,y\\in \\mathbb{Z}_{\\ge 0} \\} $ and let $s(a,b)=ab-a-b$. It is well known that all integers which are greater than $s(a,b)$ are in $T(a,b)$. Let $\\pi (a, b)$ be the number of primes in $T(a,b)$ which are less than or equal to $s(a,b)$. It is easy to see that $\\pi (2, 3)=0$ and $\\pi (2, b)=1$ for all odd integers $b\\ge 5$. In this paper, we prove that if $b>a\\ge 3$ with $\\gcd (a, b)=1$, then $\\pi (a, b)>0.005 s(a,b)/\\log s(a,b)$. We conjecture that $\\frac{13}{66}\\pi (s(a,b))\\le \\pi (a, b)\\le \\frac 12\\pi (s(a,b))$ for all $b>a\\ge 3$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.03620","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/2506.03620/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"}