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Zbl 1157.11001
Hong, Shaofang; Yang, Yujuan
Improvements of lower bounds for the least common multiple of finite arithmetic progressions.
(English)
[J] Proc. Am. Math. Soc. 136, No. 12, 4111-4114 (2008). ISSN 0002-9939; ISSN 1088-6826/e

Let $u_0,$ $r$ and $n$ be positive integers with $(u_0,r)=1$. Let $u_k=u_0+kr$ for $1\leq k\leq n$ and $L_n=\text{ lcm}(u_0,u_1,\dots ,u_n)$. In this paper, the authors prove (by elementatry means) that $L_n \geq u_0r^{\alpha}(r+1)^n$ if $n>r^{\alpha}$. \par This result in case $\alpha=0$ was conjectured by {\it B. Farhi} [C. R., Math., Acad. Sci. Paris 341, No. 8, 469--474 (2005; Zbl 1117.11005)] and proved by {\it S. Hong} and {\it W. Feng} [C. R., Math., Acad. Sci. Paris 343, No. 11--12, 695--698 (2006; Zbl 1156.11004)]. In the same paper they also proved the above result in case $\alpha=1$.
[Pieter Moree (Bonn)]
MSC 2000:
*11A05 Multiplicative structure of the integers
11B25 Arithmetic progressions

Keywords: arithmetic progression; least common multiple; lower bound

Citations: Zbl 1156.11004; Zbl 1117.11005

Cited in: Zbl 1196.11007

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