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We prove that ${\\rm lcm}_{\\lceil n/2\\rceil \\le i\\le n} \\{f(i)\\}\\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $s\\ge 2$ being an integer and $n=1$, where $\\lceil n/2\\rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013."},"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":"1308.6458","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-08-29T13:23:07Z","cross_cats_sorted":[],"title_canon_sha256":"e5d8b60748a6192dfcc08b824f247549ec865e6f89af1086d7a409e34851be0d","abstract_canon_sha256":"33da840cea659cb06562b99aa9eb74124340fa24874a6236bfd43a13bd4ba3d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:32.384482Z","signature_b64":"yQt7iIkzy/89Yglt9E4h8dvaghqwDys+s+bEKMSmmeuBKpn+CojolgNMkp4kGrBaWIzSketEocZpuKaVLrvOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4d4480246e749b8666f8ef48b8609f86830e3462e08e51d68c9fd054db1c1be","last_reissued_at":"2026-05-18T03:06:32.383854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:32.383854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniform lower bound for the least common multiple of a polynomial sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chunlin Wang, Guoyou Qian, Shaofang Hong, Yuanyuan Luo","submitted_at":"2013-08-29T13:23:07Z","abstract_excerpt":"Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\\rm lcm}_{\\lceil n/2\\rceil \\le i\\le n} \\{f(i)\\}\\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $s\\ge 2$ being an integer and $n=1$, where $\\lceil n/2\\rceil$ denotes the smallest integer which is not less than $n/2$. 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