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

Published:2014-05-21
Printed: Sep 2014
• Daniel M. Kane,
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
• Scott Duke Kominers,
Society of Fellows, 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
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Abstract

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. 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: least common multiple, arithmetic progression
 MSC Classifications: 11A05 - Multiplicative structure; Euclidean algorithm; greatest common divisors