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

TL;DR

This paper presents improved lower bounds for the least common multiple of arithmetic progressions with coprime initial terms and common differences, enhancing previous results for most parameter choices.

Contribution

The authors derive new lower bounds for the LCM of arithmetic progressions, extending and improving upon prior bounds for a wide range of parameters.

Findings

New lower bounds for LCM of arithmetic progressions

Improvement over previous bounds for most parameter choices

Applicable for a broad range of initial terms and differences

Abstract

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2.