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

TL;DR

This paper establishes new nontrivial lower bounds for the least common multiple of quadratic sequences, using a novel algebraic approach, and provides explicit bounds depending on sequence parameters.

Contribution

It introduces a new algebraic method to derive lower bounds for the LCM of quadratic sequences, improving upon previous approaches.

Findings

Proves LCM is divisible by a specific rational expression.

Derives explicit lower bounds for LCM based on sequence parameters.

Demonstrates the effectiveness of algebraic methods over previous techniques.

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.