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Improvements of lower bounds for the least common multiple of finite arithmetic progressions


Authors: Shaofang Hong and Yujuan Yang
Journal: Proc. Amer. Math. Soc. 136 (2008), 4111-4114
MSC (2000): Primary 11A05
DOI: https://doi.org/10.1090/S0002-9939-08-09565-8
Published electronically: July 17, 2008
MathSciNet review: 2431021
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ u_0, r, \alpha $ and $ n$ be positive integers such that $ (u_ 0,r)=1$. Let $ u_k=u_0+kr$ for $ 1\leq k\leq n$. We prove that $ L_n :={\rm lcm}\{u_0, u_1,\cdots, u_n\}\geq u_ 0r^\alpha (r+1)^n$ if $ n>r^\alpha $. This improves the lower bound of $ L_n$ obtained previously by Farhi, Hong and Feng.


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Additional Information

Shaofang Hong
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China
Email: s-f.hong@tom.com, hongsf02@yahoo.com, sfhong@scu.edu.cn

Yujuan Yang
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China
Email: y.j.yang@tom.com

DOI: https://doi.org/10.1090/S0002-9939-08-09565-8
Keywords: Arithmetic progression, least common multiple, lower bound.
Received by editor(s): September 18, 2007
Published electronically: July 17, 2008
Additional Notes: The first author was supported in part by the Program for New Century Excellent Talents in University, Grant No. NCET-06-0785.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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