Note on the Lower Bound of Least Common Multiple

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), ... ,u(n) by L-n. We show that if n >= r(2) + r, then L-n >= u(0)r(r+1) (r + 1), and we obtain optimum result on.. in some cases for such estimate. Besides, for quadratic sequences m(2) + c, (m + 1)(2) + c, ... ,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.