The least common multiple of consecutive quadratic progression terms

TL;DR

This paper investigates the periodicity and asymptotic behavior of the least common multiple of consecutive quadratic polynomial values, providing conditions for periodicity and detailed $p$-adic analysis.

Contribution

It establishes when the function related to the LCM of quadratic progression terms is eventually periodic and determines its smallest period, along with asymptotic formulas.

Findings

g_{k,f} is eventually periodic iff D ≠ a^2 i^2 for all 1 ≤ i ≤ k

The paper determines the smallest period of g_{k,f}

Asymptotic formulas for the logarithm of the LCM are derived

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 ${\rm lcm}_{0\le i\le k}\{f(n+i)\}$ of any $k+1$ consecutive terms in the quadratic progression $\{f(n)\}_{n\in \mathbb{N}^*}$, we define the function $g_{k, f}(n):=(\prod_{i=0}^{k}|f(n+i)|)/{\rm lcm}_{0\le i\le k}\{f(n+i)\}$ for all integers $n\in \mathbb{N}^*\setminus Z_{k, f}$, where $Z_{k,f}:=\bigcup_{i=0}^k\{n\in \mathbb{N}^*: f(n+i)=0\}$. In this paper, we first show that $g_{k,f}$ is eventually periodic if and only if $D\ne a^2i^2$ 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 $\log {\rm lcm}_{0\le i\le k}\{f(n+i)\}$ for all quadratic polynomials $f$ as $n$ goes to infinity.