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.
Categories: math.NT