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In this paper, we show that, for integers $\\alpha \\geq a$ and $r\\geq \\max(a, l-1)$ and $n\\geq l\\alpha r$, we have $$L_n\\geq u_0r^{(l-1)\\alpha +a-l}(r+1)^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers."},"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":"1211.4468","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-19T15:45:03Z","cross_cats_sorted":[],"title_canon_sha256":"5b6da642178712597cd48bdf98fd592f397f3d22ca4883940ed1d8f06220c557","abstract_canon_sha256":"3105328ce24a61f28d7b09b93d8785991c0bd6079e09cd6b65c7ef9117911b91"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:14.430869Z","signature_b64":"ifDOoXxY2QnQMoKzwmJzu9BB2il5Th62sLCOe3ceFGC8Z8vAExcf4pldg4pDCVdajgEumSV/pUrKs9QX+hBQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9479278d9e4420ef7298fc82c0a416d62df0e11b6eb9f2d59f5d0021aba2617d","last_reissued_at":"2026-05-18T03:08:14.430391Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:14.430391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Lower Bounds for the Least Common Multiples of Arithmetic Progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Qianrong Tan, Rongjun Wu, Shaofang Hong","submitted_at":"2012-11-19T15:45:03Z","abstract_excerpt":"For relatively prime positive integers $u_0$ and $r$ and for $0\\le k\\le n$, define $u_k:=u_0+kr$. Let $L_n:={\\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\\ge 2$ be any integers. In this paper, we show that, for integers $\\alpha \\geq a$ and $r\\geq \\max(a, l-1)$ and $n\\geq l\\alpha r$, we have $$L_n\\geq u_0r^{(l-1)\\alpha +a-l}(r+1)^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4468","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":"1211.4468","created_at":"2026-05-18T03:08:14.430466+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.4468v1","created_at":"2026-05-18T03:08:14.430466+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4468","created_at":"2026-05-18T03:08:14.430466+00:00"},{"alias_kind":"pith_short_12","alias_value":"SR4SPDM6IQQO","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_16","alias_value":"SR4SPDM6IQQO64UY","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_8","alias_value":"SR4SPDM6","created_at":"2026-05-18T12:27:20.899486+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/SR4SPDM6IQQO64UY7SBMBJAW2Y","json":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y.json","graph_json":"https://pith.science/api/pith-number/SR4SPDM6IQQO64UY7SBMBJAW2Y/graph.json","events_json":"https://pith.science/api/pith-number/SR4SPDM6IQQO64UY7SBMBJAW2Y/events.json","paper":"https://pith.science/paper/SR4SPDM6"},"agent_actions":{"view_html":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y","download_json":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y.json","view_paper":"https://pith.science/paper/SR4SPDM6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.4468&json=true","fetch_graph":"https://pith.science/api/pith-number/SR4SPDM6IQQO64UY7SBMBJAW2Y/graph.json","fetch_events":"https://pith.science/api/pith-number/SR4SPDM6IQQO64UY7SBMBJAW2Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y/action/storage_attestation","attest_author":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y/action/author_attestation","sign_citation":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y/action/citation_signature","submit_replication":"https://pith.science/pith/SR4SPDM6IQQO64UY7SBMBJAW2Y/action/replication_record"}},"created_at":"2026-05-18T03:08:14.430466+00:00","updated_at":"2026-05-18T03:08:14.430466+00:00"}