Uniform lower bound for the least common multiple of a polynomial sequence

TL;DR

This paper establishes a new lower bound for the least common multiple of polynomial sequences with nonnegative integer coefficients, improving previous bounds and identifying specific exceptions.

Contribution

It provides a uniform lower bound for the LCM of polynomial sequences, extending and refining earlier results by Nair, Farhi, and Oon.

Findings

Lower bound of 2^n for the LCM of polynomial sequences

Identification of specific exceptions for linear and power polynomials

Extension of previous bounds in the literature

Abstract

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $s\ge 2$ being an integer and $n=1$, where $\lceil n/2\rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.