New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Abstract

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:=\mathrm{lcm}(u_0,u_1,...,u_n)$ and let $a,l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \ge a$ and $r\ge \max (a,l-1)$ and $n\ge l\alpha r$, we have $$L_n\ge u_0{r}^{(l-1)\alpha +a-l}(r+1{)}^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers.

4 pages. To appear in Chinese Annals of Mathematics (Series B)