Nontrivial effective lower bounds for the least common multiple of some quadratic sequences

Abstract

This paper is devoted to studying the numbers $L_{c,m,n}:=\mathrm{lcm}\{m^2+c,(m+1{)}^2+c,\dots ,n^2+c\}$, where $c,m,n$ are positive integers such that $m\le n$. Precisely, we prove that $L_{c,m,n}$ is a multiple of the rational number \[\frac{\displaystyle\prod_{k=m}^{n}\left(k^2+c\right)}{c \cdot (n-m)!\displaystyle\prod_{k=1}^{n-m}\left(k^2+4c\right)} ,\] and we derive (as consequences) some nontrivial lower bounds for $L_{c,m,n}$. We prove for example that if $n-\frac{1}{2}{n}^{2/3}\le m\le n$, then we have $L_{c,m,n}\ge \lambda (c)\cdot n{e}^{3(n-m)}$, where $\lambda (c):=\frac{e^{-\frac{2{\pi }^2}{3}c-\frac{5}{12}}}{(2\pi {)}^{3/2}c}$. Further, it must be noted that our approach (focusing on commutative algebra) is new and different from those using previously by Farhi, Oon and Hong.

15 pages