Note on the Lower Bound of Least Common Multiple

Abstract EN

Consider a sequence of positive integers in arithmetic progression uk=u0+kr with (u0,r)=1.

Denote the least common multiple of u0,…,un by Ln.

We show that if n⩾r2+r, then Ln⩾u0rr+1r+1, and we obtain optimum result on n in some cases for such estimate.

Besides, for quadratic sequences m2+c,m+12+c,…,n2+c, we also show that the least common multiple is at least 2n when m⩽n/2, which sharpens a recent result of Farhi.