Note on the Lower Bound of Least Common Multiple

Abstract

Consider a sequence of positive integers in arithmetic progression $u_k=u_0+kr$ with $(u_0,r)=1$. Denote the least common multiple of $u_0,\dots ,u_n$ by $L_n$. We show that if $n⩾r^{\mathrm{2}}+r$, then $L_{\mathrm{n}}⩾u_{\mathrm{0}}{r}^{r+\mathrm{1}}\left(r+\mathrm{1}\right)$, and we obtain optimum result on $n$ in some cases for such estimate. Besides, for quadratic sequences $m^2+\mathrm{c},{\left(m+1\right)}^{\mathrm{2}}+c,\dots ,n^2+c$, we also show that the least common multiple is at least $2^n$ when $m⩽⌈n/2⌉$, which sharpens a recent result of Farhi.