{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:YXJL7PYTOLSJBQTD2GVAWLCWK3","short_pith_number":"pith:YXJL7PYT","schema_version":"1.0","canonical_sha256":"c5d2bfbf1372e490c263d1aa0b2c5656f7c91bde64984fa0bf329ac17fa4dbdc","source":{"kind":"arxiv","id":"1010.3322","version":1},"attestation_state":"computed","paper":{"title":"Sums and Products with Smooth Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"David Covert, William D. Banks","submitted_at":"2010-10-16T05:38:39Z","abstract_excerpt":"We estimate the sizes of the sumset A + A and the productset A $\\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y."},"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":"1010.3322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-10-16T05:38:39Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"beed7de99193d6e33375897eef0422682d79aa535600480f0dd8aeee1455c334","abstract_canon_sha256":"77c0072cce4d2bc464b43d1cc0825471a6ee06b1c0897fc96649ae5528c88e30"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:09.466567Z","signature_b64":"tnuka01XT9P45tUCxkb6nIKt7L05CMK3R7/WvgQH9W5u8vKTzjkgVLLwK5w/WmiTyUE4z6rAKV5Npwg3tpnrBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5d2bfbf1372e490c263d1aa0b2c5656f7c91bde64984fa0bf329ac17fa4dbdc","last_reissued_at":"2026-05-18T04:39:09.465922Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:09.465922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums and Products with Smooth Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"David Covert, William D. Banks","submitted_at":"2010-10-16T05:38:39Z","abstract_excerpt":"We estimate the sizes of the sumset A + A and the productset A $\\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.3322","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":""},"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":"1010.3322","created_at":"2026-05-18T04:39:09.466022+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.3322v1","created_at":"2026-05-18T04:39:09.466022+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.3322","created_at":"2026-05-18T04:39:09.466022+00:00"},{"alias_kind":"pith_short_12","alias_value":"YXJL7PYTOLSJ","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"YXJL7PYTOLSJBQTD","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"YXJL7PYT","created_at":"2026-05-18T12:26:17.028572+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/YXJL7PYTOLSJBQTD2GVAWLCWK3","json":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3.json","graph_json":"https://pith.science/api/pith-number/YXJL7PYTOLSJBQTD2GVAWLCWK3/graph.json","events_json":"https://pith.science/api/pith-number/YXJL7PYTOLSJBQTD2GVAWLCWK3/events.json","paper":"https://pith.science/paper/YXJL7PYT"},"agent_actions":{"view_html":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3","download_json":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3.json","view_paper":"https://pith.science/paper/YXJL7PYT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.3322&json=true","fetch_graph":"https://pith.science/api/pith-number/YXJL7PYTOLSJBQTD2GVAWLCWK3/graph.json","fetch_events":"https://pith.science/api/pith-number/YXJL7PYTOLSJBQTD2GVAWLCWK3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3/action/storage_attestation","attest_author":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3/action/author_attestation","sign_citation":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3/action/citation_signature","submit_replication":"https://pith.science/pith/YXJL7PYTOLSJBQTD2GVAWLCWK3/action/replication_record"}},"created_at":"2026-05-18T04:39:09.466022+00:00","updated_at":"2026-05-18T04:39:09.466022+00:00"}