arXiv subjects
Publications and source records attributed to Guoyou Qian.
Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y 0$ is arbitrarily small.
Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $f(x)$ be the product of two linear polynomials with integer coefficients. In this paper, we show that $\log {\rm lcm}_{mn<i\le ln}\{f(i)\}=An+o(n)$, where $A$ is a constant depending only on $l$, $m$ and $f$.
In this paper, we get several new results on permutation polynomials over finite fields. First, by using the linear translator, we construct permutation polynomials of the forms $L(x)+\sum_{j=1}^k γ_jh_j(f_j(x))$ and $x+\sum_{j=1}^kγ_jf_j(x)$. These generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form $L(x)+\sum_{i=1} ^lγ_i {\rm Tr}_{{\bf F}_{q^m}/{\bf F}_{q}}(h_i(x))$, which extends a theorem of Charpin and Kyureghyan obtained in 2009.
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.
Erdös and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $n\geq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1), 1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $m\geq2$ being an integer and $n=1$.
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.
Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $a$ and $b$ be integers with $a\ge 1$ and $a+b\ge 1$. In this paper, we prove that $\log {\rm lcm}_{mn<i\le ln}\{ai+b\} =An+o(n)$, where $A$ is a constant depending on $l, m$ and $a$.
Let $k\ge 1,a\ge 1,b\ge 0$ and $ c\ge 1$ be integers. Let $f$ be a multiplicative function with $f(n)\ne 0$ for all positive integers $n$. We define the arithmetic function $g_{k,f}$ for any positive integer $n$ by $g_{k,f}(n):=\frac{\prod_{i=0}^k f(b+a(n+ic))} {f({\rm lcm}_{0\le i\le k} \{b+a(n+ic)\})}$. We first show that $g_{k,f}$ is periodic and $c {\rm lcm}(1,...,k)$ is its period. Consequently, we provide a detailed local analysis to the periodic function $g_{k,φ}$, and determine the smallest period of $g_{k,φ}$, where $φ$ is the Euler phi function.
Let $f(x)$ be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate: $\log {\rm lcm}(f(1), ..., f(n))\sim An$ as $n\rightarrow\infty $, where $A$ is a constant depending on $f$.
Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1,2,...,k)$ and conjectured that $\frac{{\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved this conjecture and determined the smallest period of $g_{k,1,0}$. For the general integers $a\ge 1$ and $b\ge 0$, it is natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic analysis of the periodic function $g_{k,a,b}$. Finally, we determine the smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from the set of positive integers to general arithmetic progressions.