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

This paper is devoted to studying the numbers $L_{c,m,n} := $\mathrm{l}\mathrm{c}\mathrm{m}${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 $$\frac{{\displaystyle \prod _{k=m}^n(k^2+c)}}{c\cdot (n-m)!{\displaystyle \prod _{k=1}^{n-m}(k^2+4c)}},$$ and we derive (as consequences) some nontrivial lower bounds for $L_{c,m,n}$. We prove for example that if $n-\frac{1}{2}{n}^{2\mathrm{/}3}\le m\le n$, then we have $L_{c,m,n} \geq \lambda© \cdot n e^{3 (n - m)}$,where$\lambda© := \frac{e^{- \frac{2 \pi^2}{3} c - \frac{5}{12}}}{(2 \pi)^{3/2} c}$. 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.