The least common multiple of consecutive quadratic progression terms

Abstract

Let $k$ be an arbitrary given positive integer and let $f(x)\in \mathbb{Z}[x]$ be a quadratic polynomial with $a$ and $D$ as its leading coefficient and discriminant, respectively. Associated to the least common multiple ${\mathrm{lcm}}_{0\le i\le k}\{f(n+i)\}$ of any $k+1$ consecutive terms in the quadratic progression $\{f(n){\}}_{n\in {\mathbb{N}}^{\ast }}$, we define the function $g_{k,f}(n):=({\prod }_{i=0}^k|f(n+i)|)/{\mathrm{lcm}}_{0\le i\le k}\{f(n+i)\}$ for all integers $n\in {\mathbb{N}}^{\ast }\setminus Z_{k,f}$, where $Z_{k,f}:={\bigcup }_{i=0}^k\{n\in {\mathbb{N}}^{\ast }:f(n+i)=0\}$. In this paper, we first show that $g_{k,f}$ is eventually periodic if and only if $D\ne a^2{i}^2$ for all integers $i$ with $1\le i\le k$. Consequently, we develop a detailed $p$-adic analysis of $g_{k,f}$ and determine its smallest period. Finally, we obtain asymptotic formulas of $\log {\mathrm{lcm}}_{0\le i\le k}\{f(n+i)\}$ for all quadratic polynomials $f$ as $n$ goes to infinity.

32 pages. To appear in Forum Mathematicum