Further improvements of lower bounds for the least common multiples of arithmetic progressions
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- by Shaofang Hong and Scott Duke Kominers PDF
- Proc. Amer. Math. Soc. 138 (2010), 809-813
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 \[ L_n\geq u_0r^{\alpha +a-2}(r+1)^n.\] 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
- 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
- © Copyright 2009 Shaofang Hong and Scott Duke Kominers
- Journal: Proc. Amer. Math. Soc. 138 (2010), 809-813
- MSC (2000): Primary 11A05
- DOI: https://doi.org/10.1090/S0002-9939-09-10083-7
- MathSciNet review: 2566546