{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:MMVIRB2TNJFXPW7O3V73564NCY","short_pith_number":"pith:MMVIRB2T","schema_version":"1.0","canonical_sha256":"632a8887536a4b77dbeedd7fbefb8d16183cc71e4b94f743a4517b239456532d","source":{"kind":"arxiv","id":"2506.03620","version":1},"attestation_state":"computed","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 $b>a\\ge 3$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.03620","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-04T06:58:48Z","cross_cats_sorted":[],"title_canon_sha256":"2820eaea1e9172ebe4ca38c486f30c091b309f1ea5a7e457c30494b6c00f12fe","abstract_canon_sha256":"ab45d56b81d5f851c62f8b0d7491f24759c13e8059e88948403afd49d7ddbbb4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:15:48.926463Z","signature_b64":"5HL5MZ2wQm2SXOOKIeV+CKFbwHOOifMQNafNYIQpMgzcBj+iCX4tLROY2/Xzl1nVGBt+ByA/PsO9QJ182SoJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"632a8887536a4b77dbeedd7fbefb8d16183cc71e4b94f743a4517b239456532d","last_reissued_at":"2026-07-05T11:15:48.925856Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:15:48.925856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.03620","created_at":"2026-07-05T11:15:48.925930+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.03620v1","created_at":"2026-07-05T11:15:48.925930+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.03620","created_at":"2026-07-05T11:15:48.925930+00:00"},{"alias_kind":"pith_short_12","alias_value":"MMVIRB2TNJFX","created_at":"2026-07-05T11:15:48.925930+00:00"},{"alias_kind":"pith_short_16","alias_value":"MMVIRB2TNJFXPW7O","created_at":"2026-07-05T11:15:48.925930+00:00"},{"alias_kind":"pith_short_8","alias_value":"MMVIRB2T","created_at":"2026-07-05T11:15:48.925930+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY","json":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY.json","graph_json":"https://pith.science/api/pith-number/MMVIRB2TNJFXPW7O3V73564NCY/graph.json","events_json":"https://pith.science/api/pith-number/MMVIRB2TNJFXPW7O3V73564NCY/events.json","paper":"https://pith.science/paper/MMVIRB2T"},"agent_actions":{"view_html":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY","download_json":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY.json","view_paper":"https://pith.science/paper/MMVIRB2T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.03620&json=true","fetch_graph":"https://pith.science/api/pith-number/MMVIRB2TNJFXPW7O3V73564NCY/graph.json","fetch_events":"https://pith.science/api/pith-number/MMVIRB2TNJFXPW7O3V73564NCY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY/action/storage_attestation","attest_author":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY/action/author_attestation","sign_citation":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY/action/citation_signature","submit_replication":"https://pith.science/pith/MMVIRB2TNJFXPW7O3V73564NCY/action/replication_record"}},"created_at":"2026-07-05T11:15:48.925930+00:00","updated_at":"2026-07-05T11:15:48.925930+00:00"}