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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 $b>a\\ge 3$ ","authors_text":"Hui Zhu, Yong-Gao Chen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-04T06:58:48Z","title":"Primes of the form $ax+by$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.03620","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e721352f514d01ed2f17ea0a238b59dc960c0fe6fadb376b68917b87a0786dfa","target":"record","created_at":"2026-07-05T11:15:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ab45d56b81d5f851c62f8b0d7491f24759c13e8059e88948403afd49d7ddbbb4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-04T06:58:48Z","title_canon_sha256":"2820eaea1e9172ebe4ca38c486f30c091b309f1ea5a7e457c30494b6c00f12fe"},"schema_version":"1.0","source":{"id":"2506.03620","kind":"arxiv","version":1}},"canonical_sha256":"632a8887536a4b77dbeedd7fbefb8d16183cc71e4b94f743a4517b239456532d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"632a8887536a4b77dbeedd7fbefb8d16183cc71e4b94f743a4517b239456532d","first_computed_at":"2026-07-05T11:15:48.925856Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:15:48.925856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5HL5MZ2wQm2SXOOKIeV+CKFbwHOOifMQNafNYIQpMgzcBj+iCX4tLROY2/Xzl1nVGBt+ByA/PsO9QJ182SoJBg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:15:48.926463Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.03620","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e721352f514d01ed2f17ea0a238b59dc960c0fe6fadb376b68917b87a0786dfa","sha256:1ba816ee472cf5b3c3499bee52bf7f4bd387fbbf35199800439e6e0e14d48c1f"],"state_sha256":"787b7dc8186784d52c46df16411514e9b490938110f9a6f49798771cf76feead"}