New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

TL;DR

This paper establishes new lower bounds for the least common multiple of arithmetic progressions, improving previous bounds by leveraging specific parameter conditions and providing tighter estimates.

Contribution

It introduces novel lower bounds for the LCM of arithmetic progressions, extending and strengthening prior results in the field.

Findings

Derived explicit lower bounds for LCM of arithmetic progressions.

Improved previous bounds by Hong and Kominers for certain parameter ranges.

Provided conditions under which the new bounds hold, enhancing theoretical understanding.

Abstract

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and $r\geq \max(a, l-1)$ and $n\geq l\alpha r$, we have $$L_n\geq u_0r^{(l-1)\alpha +a-l}(r+1)^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers.