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

Abstract

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\mathrm{lcm}}_{\lceil n/2\rceil \le i\le n}\{f(i)\}\ge 2^n$ 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.

6 pages. To appear in Comptes Rendus Mathematique

References in corpus (1)

• Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms