The least common multiple of consecutive terms in a cubic progression

Zongbing Lin; Shaofang Hong; Zongbing Lin; Shaofang Hong

doi:10.3934/math.2020119

AIMS Mathematics

2020, Volume 5, Issue 3: 1757-1778. doi: 10.3934/math.2020119

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Research article

The least common multiple of consecutive terms in a cubic progression

Zongbing Lin ¹,
Shaofang Hong ^{2
,
,}

1.
School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, P. R. China
2.
Mathematical College, Sichuan University, Chengdu 610064, P.R. China

Received: 30 October 2019 Accepted: 07 February 2020 Published: 14 February 2020
MSC : 11B25, 11N13, 11A05

Let $k$ be a positive integer and $f(x)$ a polynomial with integer coefficients. Associated to the least common multiple ${\rm lcm}_{0\le i\le k}\{f(n+i)\}$, we define the function $\mathcal{G}_{k, f}$ for all positive integers $n\in \mathbb{N}^*\setminus Z_{k, f}$ by $\mathcal{G}_{k, f}(n): = \frac{\prod_{i = 0}^k |f(n+i)|}{{\rm lcm}_{0\le i\le k}\{f(n+i)\}}, $ where $Z_{k, f}: = \bigcup_{i = 0}^k\{n\in \mathbb{N}^*: f(n+i) = 0\}.$ If $f(x) = x$, then Farhi showed in 2007 that $\mathcal{G}_{k, f}$ is periodic with $k!$ as its period. Consequently, Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1, ..., k)$. Later on, Farhi and Kane confirmed a conjecture of Hong and Yang and determined the smallest period of $\mathcal{G}_{k, f}$. For the general linear polynomial $f(x)$, Hong and Qian showed in 2011 that $\mathcal{G}_{k, f}$ is periodic and got a formula for its smallest period. In 2015, Hong and Qian characterized the quadratic polynomial $f(x)$ such that $\mathcal{G}_{k, f}$ is almost periodic and also arrived at an explicit formula for the smallest period of $\mathcal{G}_{k, f}$. If $\deg f(x)\ge 3$, then one naturally asks the following interesting question: Is the arithmetic function $\mathcal{G}_{k, f}$ almost periodic and, if so, what is the smallest period? In this paper, we asnwer this question for the case $f(x) = x^3+2$. First of all, with the help of Hua's identity, we prove that $\mathcal{G}_{k, x^3+2}$ is periodic. Consequently, we use Hensel's lemma, develop a detailed $p$-adic analysis to $\mathcal{G}_{k, x^3+2}$ and particularly investigate arithmetic properties of the congruences $x^3+2\equiv 0 \pmod{p^e}$ and $x^6+108\equiv 0\pmod{p^e}$, and with more efforts, its smallest period is finally determined. Furthermore, an asymptotic formula for ${\rm log \ lcm}_{0 \le i \le k}\{(n+i)^3+2\}$ is given.
- cubic progression,
- least common multiple,
- p-adic valuation,
- almost periodic function,
- the smallest period,
- Hensel’s lemma
Citation: Zongbing Lin, Shaofang Hong. The least common multiple of consecutive terms in a cubic progression[J]. AIMS Mathematics, 2020, 5(3): 1757-1778. doi: 10.3934/math.2020119

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Abstract

Let $k$ be a positive integer and $f(x)$ a polynomial with integer coefficients. Associated to the least common multiple ${\rm lcm}_{0\le i\le k}\{f(n+i)\}$, we define the function $\mathcal{G}_{k, f}$ for all positive integers $n\in \mathbb{N}^*\setminus Z_{k, f}$ by $\mathcal{G}_{k, f}(n): = \frac{\prod_{i = 0}^k |f(n+i)|}{{\rm lcm}_{0\le i\le k}\{f(n+i)\}}, $ where $Z_{k, f}: = \bigcup_{i = 0}^k\{n\in \mathbb{N}^*: f(n+i) = 0\}.$ If $f(x) = x$, then Farhi showed in 2007 that $\mathcal{G}_{k, f}$ is periodic with $k!$ as its period. Consequently, Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1, ..., k)$. Later on, Farhi and Kane confirmed a conjecture of Hong and Yang and determined the smallest period of $\mathcal{G}_{k, f}$. For the general linear polynomial $f(x)$, Hong and Qian showed in 2011 that $\mathcal{G}_{k, f}$ is periodic and got a formula for its smallest period. In 2015, Hong and Qian characterized the quadratic polynomial $f(x)$ such that $\mathcal{G}_{k, f}$ is almost periodic and also arrived at an explicit formula for the smallest period of $\mathcal{G}_{k, f}$. If $\deg f(x)\ge 3$, then one naturally asks the following interesting question: Is the arithmetic function $\mathcal{G}_{k, f}$ almost periodic and, if so, what is the smallest period? In this paper, we asnwer this question for the case $f(x) = x^3+2$. First of all, with the help of Hua's identity, we prove that $\mathcal{G}_{k, x^3+2}$ is periodic. Consequently, we use Hensel's lemma, develop a detailed $p$-adic analysis to $\mathcal{G}_{k, x^3+2}$ and particularly investigate arithmetic properties of the congruences $x^3+2\equiv 0 \pmod{p^e}$ and $x^6+108\equiv 0\pmod{p^e}$, and with more efforts, its smallest period is finally determined. Furthermore, an asymptotic formula for ${\rm log \ lcm}_{0 \le i \le k}\{(n+i)^3+2\}$ is given.

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