  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 $m \leq n$. 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 $n- \frac{1}{2} n^{2/3} \leq m \leq n$, then we have $L_{c,m,n}
\geq \lambda(c) \cdot n e^{3 (n - m)}$, 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.
