{"paper":{"title":"The least common multiple of consecutive arithmetic progression terms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Guoyou Qian, Shaofang Hong","submitted_at":"2009-03-03T13:31:38Z","abstract_excerpt":"Let $k\\ge 0,a\\ge 1$ and $b\\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\\rm lcm}(1,2,...,k)$ and conjectured that $\\frac{{\\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.0530","kind":"arxiv","version":3},"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"}