{"paper":{"title":"The distribution of powers of primes related to the Frobenius problem","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Enxun Huang, Tengyou Zhu","submitted_at":"2024-12-25T13:18:44Z","abstract_excerpt":"Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\\mathbb{P}$ is the set of primes. For any given integer $k \\geq 1$, we prove that $$\\#\\left\\{p^k\\le g_{c,d}:p\\in \\mathbb{P}, ~p^k=cx+dy,~x,y\\in \\mathbb{Z}_{\\geqslant0} \\right\\}\\sim \\frac{k}{k+1}\\frac{g^{1/k}}{\\log g} \\quad (\\text{as}~c\\rightarrow\\infty),$$ which gives an extension of a recent result of Ding, Zhai and Zhao."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.18898","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/2412.18898/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"}