Note on the lower bound of least common multiple
Summary: 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 \(u_0, \dots, u_n\) by \(L_n\). We show that if \(n \geq r^2 + r\), then \(L_n \geq u_0 r^{r+1}(r + 1)\), and we obtain optimum result on \(n\) in some cases for such estimate. Besides, for quadratic sequences \(m^2 + c, (m + 1)^2 + c, \dots, n^2 + c\), we also show that the least common multiple is at least \(2^n\) when \(m \leq \lceil n/2 \rceil\), which sharpens a recent result of \textit{B. Farhi} [C. R., Math., Acad. Sci. Paris 341, No. 8, 469--474 (2005; Zbl 1117.11005)].
sequence of positive integers in arithmetic progression
scientific article; zbMATH DE number 6161313
scientific article