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Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Published online by Cambridge University Press:  20 November 2018

Daniel M. Kane  and
Scott Duke Kominers
Daniel M. Kane
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA e-mail: dankane@math.stanford.edualadkeenin@gmail.com
Scott Duke Kominers
Affiliation:
Department of Economics, Program for Evolutionary Dynamics and Center for Research on Computation and Society, Harvard University and Harvard Business School, Cambridge, MA 02138-3758, USA e-mail: kominers@fas.harvard.eduskominers@gmail.com
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Abstract

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For relatively prime positive integers ${{u}_{0}}$ and $r$, we consider the least common multiple ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ of the finite arithmetic progression $\left\{ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$. We derive new lower bounds on ${{L}_{n}}$ that improve upon those obtained previously when either ${{u}_{0}}$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n\,+\,1$ for ${{u}_{0}}$ properly chosen, and is also nearly sharp as $n\,\to \,\infty$.

Keywords

11A05least common multiplearithmetic progression
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 3 , 01 September 2014 , pp. 551 - 561
Copyright
Copyright © Canadian Mathematical Society 2014

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