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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Further improvements of lower bounds for the least common multiples of arithmetic progressions


Authors: Shaofang Hong and Scott Duke Kominers
Journal: Proc. Amer. Math. Soc. 138 (2010), 809-813
MSC (2000): Primary 11A05
Published electronically: September 4, 2009
MathSciNet review: 2566546
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Abstract: For relatively prime positive integers $ u_0$ and $ r$, we consider the arithmetic progression $ \{u_k:=u_0+kr\}_{k=0}^n$.

Define $ L_n:=\operatorname{lcm}\{u_0, u_1, ..., u_n\}$ and let $ a\ge 2$ be any integer. In this paper, we show that for integers $ \alpha , r\geq a$ and $ n\geq 2\alpha r$, we have

$\displaystyle L_n\geq u_0r^{\alpha +a-2}(r+1)^n.$

In particular, letting $ a=2$ yields an improvement to the best previous lower bound on $ L_n$ (obtained by Hong and Yang) for all but three choices of $ \alpha , r\geq 2$.


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Additional Information

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

Scott Duke Kominers
Affiliation: Department of Mathematics and Department of Economics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Baker Library 420C, Harvard Business School, Soldiers Field, Boston, Massachusetts 02163
Email: kominers@fas.harvard.edu, skominers@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10083-7
Keywords: Least common multiple, arithmetic progression
Received by editor(s): June 12, 2009
Published electronically: September 4, 2009
Additional Notes: The first author was partly supported by the National Science Foundation of China and by the Program for New Century Excellent Talents in University, Grant No. NCET-06-0785. The second author was partly supported by a U.S. National Science Foundation Graduate Research Fellowship and is the corresponding author.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2009 Shaofang Hong and Scott Duke Kominers