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 $m \leq n$. 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.