New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:=\mathrm{l}\mathrm{c}\mathrm{m}(u_0,u_1,\mathrm{.}\mathrm{.}\mathrm{.},u_n)$ and let $a, l\ge 2$beanyintegers\mathrm{.}Inthispaper,weshowthat,forintegers$\alpha \geq a$and$r\geq \max(a, l-1)$and$n\geq l\alpha r$,wehave$$L_n\geq u_0r^{(l-1)\alpha +a-l}(r+1)^n.$$Particularly,letting$l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers.