Search

Let $\mathbb{F}_q$ be the finite field of $q$ elements, where $q=p^{m}$ with $p$ being a prime number and $m$ being a positive integer. Let $\mathcal{C}_{(q, n, \delta, h)}$ be a class of BCH codes of length $n$ and designed $\delta$. A linear code $\mathcal{C}$ is said to be maximum distance separable (MDS) if the minimum distance $d=n-k+1$. If $d=n-k$, then $\mathcal{C}$ is called an almost MDS (AMDS) code. Moreover, if both of $\mathcal{C}$ and its dual code $\mathcal{C}^{\bot}$ are AMDS, then $\mathcal{C}$ is called a near MDS (NMDS) code. In [A class of almost MDS codes, {\it Finite Fields Appl.} {\bf 79} (2022), \#101996], Geng, Yang, Zhang and Zhou proved that the BCH code $\mathcal{C}_{(q, q+1,3,4)}$ is an almost MDS code, where $q=3^m$ and $m$ is an odd integer, and they also showed that its parameters is $[q+1, q-3, 4]$. Furthermore, they proposed a conjecture stating that the dual code $\mathcal{C}^{\bot}_{(q, q+1, 3, 4)}$ is also an AMDS code with parameters $[q+1, 4, q-3]$. In this paper, we first present a characterization for the dual code of an almost MDS code to be an almost MDS code. Then we use this result to show that the Geng-Yang-Zhang-Zhou conjecture is true. Our result together with the Geng-Yang-Zhang-Zhou theorem implies that the BCH code $\mathcal{C}_{(q, q+1,3,4)}$ is a near MDS code., Comment: 25 pages. One reference was added and several typos were corrected in this version

Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial. When ${\rm char}K=0$, Igusa showed the local zeta function is a rational function. However, when ${\rm char}K>0$, the rationality of the local zeta function is unknown in general. In this paper, we study the local zeta functions for the so-called hybrid polynomials in three variables with coefficients in a non-archimedean local field of positive characteristic. These hybrid polynomials were first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We establish the rationality theorem for these local zeta functions and list explicitly all the candidate poles. Our result generalizes the work of Le$\acute{o}$n-Cardenal, Ibadula and Segers and that of Yin and Hong., Comment: 33 pages. arXiv admin note: text overlap with arXiv:1611.02111

Let $n$ and $k$ be positive integers. The Stirling number of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime number and denote by $v_p(n)$ the $p$-adic valuation of $n$. In recent years, Lengyel, Komatsu, Young, Leonetti and Sanna made some progress on $v_p(s(n,k))$. Let $a\in\{1,2\}$. In this paper, by using the properties of $m$-th Stirling number of the first kind developed previously and providing a detailed 3-adic analysis, we arrive at an explicit formula on $v_3(s(a3^n, k))$ with $1\le k\le a3^n$. This gives an evidence to a conjecture of Hong and Qiu proposed in 2020. As a corollary, we show that $v_3(s(a3^n,a3^n-k))=2n-1+v_3(k)-v_3(k-1)$ if $k$ is odd and $3\le k\le a3^{n-1}+1$. This supports a conjecture of Lengyel raised in 2015., Comment: 33pages. arXiv admin note: text overlap with arXiv:1812.04539

Let $n\ge 1, e\ge 1, k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(u, n)=1$. A unit $u$ is called an exceptional unit in the ring $\mathbb{Z}_n$ if $\gcd(1-u,n)=1$. We denote by $\mathcal{N}_{k,c,e}(n)$ the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^e+...+x_k^e\equiv c \pmod n$ with all $x_i$ being exceptional units in the ring $\mathbb{Z}_n$. In 2017, Mollahajiaghaei presented a formula for the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^2+...+x_k^2\equiv c\pmod n$ with all $x_i$ being the units in the ring $\mathbb{Z}_n$. Meanwhile, Yang and Zhao gave an exact formula for $\mathcal{N}_{k,c,1}(n)$. In this paper, by using Hensel's lemma, exponential sums and quadratic Gauss sums, we derive an explicit formula for the number $\mathcal{N}_{k,c,2}(n)$. Our result extends Mollahajiaghaei's theorem and that of Yang and Zhao., Comment: 9 pages

Let $p$ be a prime, $k$ a positive integer and let $\mathbb{F}_q$ be the finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $\mathbb F_q$ and $a\in\mathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of $f(x_1)+\cdots+f(x_s)=a$. In this paper, we show that $$\sum_{s=1}^{\infty}N_{s}(f,0)x^s=\frac{x}{1-qx} -\frac{x { M_f^{\prime}}(x)}{qM_f(x)},$$ where $$M_f(x):=\prod_{m\in\mathbb F_q^{\ast}\atop{S_{f, m}\ne 0}}\Big(x-\frac{1}{S_{f,m}}\Big)$$ with $S_{f, m}:=\sum_{x\in \mathbb F_q}\zeta_p^{{\rm Tr}(mf(x))}$, $\zeta_p$ being the $p$-th primitive unit root and ${\rm Tr}$ being the trace map from $\mathbb F_q$ to $\mathbb F_p$. This extends Richman's theorem which treats the case of $f(x)$ being a monomial. Moreover, we show that the generating series $\sum_{s=1}^{\infty}N_{s}(f,a)x^s$ is a rational function in $x$ and also present its explicit expression in terms of the first $2d+1$ initial values $N_{1}(f,a), ..., N_{2d+1}(f,a)$, where $d$ is a positive integer no more than $q-1$. From this result, the theorems of Chowla-Cowles-Cowles and of Myerson can be derived., Comment: 13 pages

Let $\mathbb{F}_q$ be the finite field of $q=p^m\equiv 1\pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y \in\mathbb{F}_q$ with $y\in\mathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x_1^4+...+x_n^4=c$ and $x_1^4+...+x_{n-1}^4+yx_n^4=0$, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function $\sum_{n=1}^{\infty}N_n(0)x^n$ is a rational function in $x$ and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions $\sum_{n=1}^{\infty}N_n(c)x^n$ and $\sum_{n=1}^{\infty}M_{n+1}(y)x^n$ are rational functions in $x$. We also obtain the explicit expressions of these generating functions. Our result extends Myerson's theorem gotten in 1979., Comment: 22 pages

Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $n\ge 1$ and $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In this paper, we use the principle of cross-classification to derive an explicit formula for the number ${\mathcal N}_{k,f,c}(n)$ of solutions $(x_1,...,x_k)$ of the congruence $x_1+...+x_k\equiv c\pmod n$ with all $x_i$ being $f$-exunits in the ring $\mathbb{Z}_n$. This extends a recent result of Anand {\it et al.} [On a question of $f$-exunits in $\mathbb{Z}/{n\mathbb{Z}}$, {\it Arch. Math. (Basel)} {\bf 116} (2021), 403-409]. We derive a more explicit formula for ${\mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic., Comment: 8 pages. Final version

Let ${\mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, z\in\mathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+\cdots+x_s^3=z$ and $x_1^3+\cdots+x_{s-1}^3+yx_s^3=0$, respectively. Gauss proved that if $q=p, p\equiv1\pmod3$ and $y$ is non-cubic, then $T_3(y)=p^2+\frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined by $4p=c^2+27d^2,~c\equiv 1 \pmod 3$ except for the sign of $d$. In 1978, Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being a non-cubic element of ${\mathbb F}_p$. But the sign problem is kept open for the remaining case of $2$ being cubic in ${\mathbb F}_p$. In this paper, we solve this sign problem by determining the sign of $d$ when $2$ is cubic in ${\mathbb F}_p$. Furthermore, we show that the generating functions $\sum_{s=1}^{\infty} N_{s}(z) x^{s}$ and $\sum_{s=1}^{\infty} T_{s}(y)x^{s}$ are rational functions for any $z, y\in\mathbb F_q^*:=\mathbb F_q\setminus \{0\}$ with $y$ being non-cubic over ${\mathbb F}_q$ and also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles., Comment: 13 pages

Let $n\ge 1$ be an integer and $e_n(x)$ denote the truncated exponential Taylor polynomial, i.e. $e_{n}(x)=\sum_{i=0}^n\frac{x^i}{i!}$. A well-known theorem of Schur states that the Galois group of $e_n(x)$ over $\Q$ is the alternating group $A_n$ if $n$ is divisible by 4 or the symmetric group $S_n$ otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials $\E_n(x):=e_n(x)+e_{n-1}(x)$. We show that $\frac{x^n}{n!}+\sum_{i=0}^{n-1}c_i\frac{x^i}{i!}$ with all $c_i \ (0\le i\le n-1)$ being integers is irreducible over $\Q$ if either $c_0=\pm 1$, or $n$ is not a positive power of $2$ but $|c_0|$ is a positive power of 2. This extends another theorem of Schur. We show also that $\E_n(x)$ is irreducible if $n\not\in\{2,4\}$. Furthermore, we show that ${\rm Gal}_{\Q}(\E_n)$ contains $A_{n}$ except for $n=4$, in which case, ${\rm Gal}_{\Q}(\E_4)=S_3$. Finally, we show that the Galois group ${\rm Gal}_{\Q}(\E_n)$ is $S_n$ if $n\equiv 3 \pmod 4$, or if $n$ is even and $v_p(n!)$ is odd for a prime divisor of $n-1$, or if $n\equiv 1\pmod 4$ and $n-2$ equals the product of an odd prime number $p$ which is coprime to $\sum_{i=1}^{p-1}2^{p-1-i}i!$ and a positive integer coprime to $p$., Comment: 14 pages

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the $p$-adic valuations of $s(n,k)$. In this paper, by using Washington's congruence on the generalized harmonic number and the $n$-th Bernoulli number $B_n$ and the properties of $m$-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound of $v_p(s(ap, k))$ with $a$ and $k$ being integers such that $1\le a\le p-1$ and $1\le k\le ap$. This infers that for any regular prime $p\ge 7$ and for arbitrary integers $a$ and $k$ with $5\le a\le p-1$ and $a-2\le k\le ap-1$, one has $v_p(H(ap-1,k))<-\frac{\log{(ap-1)}}{2\log p}$ with $H(ap-1, k)$ being the $k$-th elementary symmetric function of $1, \frac{1}{2}, ..., \frac{1}{ap-1}$. This gives a partial support to a conjecture of Leonetti and Sanna raised in 2017. We also present results on $v_p(s(ap^n,ap^n-k))$ from which one can derive that under certain condition, for any prime $p\ge 5$, any odd number $k\ge 3$ and any sufficiently large integer $n$, if $(a,p)=1$, then $v_p(s(ap^{n+1},ap^{n+1}-))=v_p(s(ap^n,ap^n-k))+2$. It confirms partially Lengyel's conjecture proposed in 2015., Comment: 23 pages

By $(\mathbb{Z}^+)^{\infty}$ we denote the set of all the infinite sequences $\mathcal{S}=\{s_i\}_{i=1}^{\infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let $\mathcal{S}_n:=\{s_1, ..., s_n\}$ and $H_f(\mathcal{S}_n):=\sum_{k=1}^{n}\frac{1}{f(k)^{s_{k}}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $\mathcal{S}$ of positive integers, $H_f(\mathcal{S}_n)$ is never an integer if $n\ge 2$. Now let deg$f(x)\ge 2$. Clearly, $00$, there are positive integers $n_1$ and $n_2$ and infinite sequences $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$ of positive integers such that $1-\varepsilon

Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_2(s(2^n,k))$ holds for all integers $k$ between 1 and $2^n$. As a corollary, we show that $v_2(s(2^n,2^n-k))=2n-2-v_2(k-1)$ if $k$ is odd and $2\le k\le 2^{n-1}+1$. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if $k\le 2^n$, then $v_2(s(2^n,k)) \le v_2(s(2^n,1))$ and $v_2(H(2^n,k))\leq -n$, where $H(n,k)$ stands for the $k$-th elementary symmetric functions of $1,1/2,...,1/n$. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017., Comment: 23 pages

Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm \Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots p_{r}$ and the $p_i$ are primes, not necessarily distinct. Here $\Phi_p(x):=(x^p-1)/(x-1)$ is the $p-$th cyclotomic polynomial. In \cite{1}, they also proved the conjecture for $N$ odd or a power of 2. In this paper we introduce a so-called $E-$transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture., Comment: This is my first paper written in 2004 as I was a 2nd year Master candidate. When it was finally published in 2009, I have been graduated with a doctorate degree for two years!!!

Let $n$ be a positive integer. In 1915, Theisinger proved that if $n\ge 2$, then the $n$-th harmonic sum $\sum_{k=1}^n\frac{1}{k}$ is not an integer. Let $a$ and $b$ be positive integers. In 1923, Nagell extended Theisinger's theorem by showing that the reciprocal sum $\sum_{k=1}^{n}\frac{1}{a+(k-1)b}$ is not an integer if $n\ge 2$. In 1946, Erd\H{o}s and Niven proved a theorem of a similar nature that states that there is only a finite number of integers $n$ for which one or more of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we present a generalization of Nagell's theorem. In fact, we show that for arbitrary $n$ positive integers $s_1, ..., s_n$ (not necessarily distinct and not necessarily monotonic), the following reciprocal power sum $$\sum\limits_{k=1}^{n}\frac{1}{(a+(k-1)b)^{s_{k}}}$$ is never an integer if $n\ge 2$. The proof of our result is analytic and $p$-adic in character., Comment: 12 pages. To appear in Acta Math. Hungar

Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial. The local zeta function $Z_f(s, \chi)$ was first introduced by Weil, then studied in detail by Igusa. When ${\rm char}(K)=0$, Igusa proved that $Z_f(s, \chi)$ is a rational function of $q^{-s}$ by using the resolution of singularities. Later on, Denef gave another proof of this remarkable result. However, if ${\rm char}(K)>0$, the question of rationality of $Z_f(s, \chi)$ is still kept open. Actually, there are only a few known results so far. In this paper, we investigate the local zeta functions of two-variable polynomial $g(x, y)$, where $g(x, y)=0$ is the superelliptic curve with coefficients in a non-archimedean local field of positive characteristic. By using the notable Igusa's stationary phase formula and with the help of some results due to Denef and Z${\rm \acute{u}}$${\rm\tilde{n}}$iga-Galindo, and developing a detailed analysis, we prove the rationality of these local zeta functions and also describe explicitly all their candidate poles., Comment: 18 pages

Determining deep holes is an important topic in decoding Reed-Solomon codes. Let $l\ge 1$ be an integer and $a_1,\ldots,a_l$ be arbitrarily given $l$ distinct elements of the finite field ${\bf F}_q$ of $q$ elements with the odd prime number $p$ as its characteristic. Let $D={\bf F}_q\backslash\{a_1,\ldots,a_l\}$ and $k$ be an integer such that $2\le k\le q-l-1$. In this paper, we study the deep holes of generalized projective Reed-Solomon code ${\rm GPRS}_q(D, k)$ of length $q-l+1$ and dimension $k$ over ${\bf F}_q$. For any $f(x)\in {\bf F}_q[x]$, we let $f(D)=(f(y_1),\ldots,f(y_{q-l}))$ if $D=\{y_1, ..., y_{q-l}\}$ and $c_{k-1}(f(x))$ be the coefficient of $x^{k-1}$ of $f(x)$. By using D\"ur's theorem on the relation between the covering radius and minimum distance of ${\rm GPRS}_q(D, k)$, we show that if $u(x)\in {\bf F}_q[x]$ with $\deg (u(x))=k$, then the received codeword $(u(D), c_{k-1}(u(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if and only if the sum $\sum\limits_{y\in I}y$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$. We show also that if $j$ is an integer with $1\leq j\leq l$ and $u_j(x):= \lambda_j(x-a_j)^{q-2}+\nu_j x^{k-1}+f_{\leq k-2}^{(j)}(x)$ with $\lambda_j\in {\bf F}_q^*$, $\nu_j\in {\bf F}_q$ and $f_{\leq{k-2}}^{(j)}(x)\in{\bf F}_q[x]$ being a polynomial of degree at most $k-2$, then $(u_j(D), c_{k-1}(u_j(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if and only if the sum $\binom{q-2}{k-1}(-a_j)^{q-1-k}\prod\limits_{y\in I}(a_j-y)+e$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$, where $e$ is the identity of the group ${\bf F}_q^*$. This implies that $(u_j(D), c_{k-1}(u_j(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if $p|k$., Comment: 16 pages

Let $n$ and $k$ be integers such that $1\le k\le n$ and $f(x)$ be a nonzero polynomial of integer coefficients such that $f(m)\ne 0$ for any positive integer $m$. For any $k$-tuple $\vec{s}=(s_1, ..., s_k)$ of positive integers, we define $$H_{k,f}(\vec{s}, n):=\sum\limits_{1\leq i_{1}<\cdots

In this paper, we study the Igusa's local zeta functions of a class of hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. Such class of hybrid polynomial was first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We prove the rationality of these local zeta functions and describe explicitly their poles. The proof is based on Igusa's stationary phase formula., Comment: 18 pages

For the hypergeometric series with parameters from the rational fields, there is an effective criterion due to Christol to decide whether the hypergeometric series is N-integral or not. Christol criterion is a basic and vital tool in the recent striking work of Delaygue, Rivoal and Roques on the N-integrality of the hypergeometric mirror maps with rational parameters. In this paper, we develop a systematic theory on the N-integrality of the hypergeometric series with parameters from quadratic fields. We first present a detailed $p$-adic analysis to set up a criterion of the $p$-adic integrality of the hypergeometric series with parameters from rational fields. Consequently, we present two equivalent statements for the hypergeometric series with parameters from algebraic number fields to be N-integral. Finally, by using these results, introducing a new function that extends the Christol's function and developing a further $p$-adic analysis, we establish a criterion for the N-integrality of the hypergeometric series with parameters from the quadratic fields. In the process, there are two important ingredients. One is the uniform distribution result of roots of a quadratic congruence which is due to Duke, Friedlander and Iwaniec together with Toth. Another one is an upper bound on the number of solutions of polynomial congruences given by Stewart in 1991., Comment: 62 pages

In this paper, we obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,3}(1,x)$ of the fourth kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,3}(1,a)$., Comment: 16 pages

Let $\mathbb{F}_q$ denote the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ represent the nonzero elements of $\mathbb{F}_{q}$. In this paper, by using the Smith normal form we give an explicit formula for the number of rational points of the algebraic variety defined by the following system of equations over $\mathbb{F}_{q}$: \begin{align*} {\left\{\begin{array}{rl} &\sum_{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} ...x_{n_1}^{e^{(1)}_{i,n_1}} +\sum_{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}} ...x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0,\\ &\sum_{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}} ...x_{n_3}^{e^{(2)}_{j,n_3}} +\sum_{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}} ...x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, \end{array}\right.} \end{align*} where the integers $1\leq r_1

Let $\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\mathbb{F}_{q}$. Let $m$ and $t$ be positive integers. In this paper, by using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over $\mathbb{F}_{q}$: $$ \sum\limits_{j=0}^{t-1}\sum\limits_{i=1}^{r_{j+1}-r_j} a_{k,r_j+i}x_1^{e^{(k)}_{r_j+i,1}}...x_{n_{j+1}}^{e^{(k)}_{r_j+i,n_{j+1}}}=b_k, \ k=1,...,m. $$ where the integers $t>0$, $r_0=0

Let $m, n, k$ and $c$ be positive integers. Let $\nu_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if $2\le m\le n$ and $c$ is odd, then $\nu_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $\nu_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009., Comment: 20 pages

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

In this paper, we present several necessary conditions for the reversed Dickson polynomial $E_{n}(1, x)$ of the second kind to be a permutation of $\mathbb{F}_{q}$. In particular, we give explicit evaluation of the sum $\sum_{a\in \mathbb{F}_{q}}E_{n}(1,a)$., Comment: 13 pages

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 $L(x)+\sum_{j=1}^k \gamma_jh_j(f_j(x))$ and $x+\sum_{j=1}^k\gamma_jf_j(x)$. These generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form $L(x)+\sum_{i=1} ^l\gamma_i {\rm Tr}_{{\bf F}_{q^m}/{\bf F}_{q}}(h_i(x))$, which extends a theorem of Charpin and Kyureghyan obtained in 2009., Comment: 11 pages. To appear in International Journal of Number Theory

Erd\"{o}s 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 $n=1$., Comment: 4 pages. To appear in Comptes Rendus Mathematique

Let $a$ and $b$ be positive integers. In 1946, Erd\H{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. In this paper, we show that for any integer $k$ with $1\le k\le n$, the $k$-th elementary symmetric function of $1/b, 1/(a+b),..., 1/(an-a+b)$ is not an integer except that either $b=n=k=1$ and $a\ge 1$, or $a=b=1, n=3$ and $k=2$. This refines the Erd\H{o}s-Niven theorem and answers an open problem raised by Chen and Tang in 2012., Comment: 12 pages. To appear in Acta Mathematica Hungarica

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., Comment: 6 pages. To appear in Comptes Rendus Mathematique

In this paper, we construct several new permutation polynomials over finite fields. First, using the linearized polynomials, we construct the permutation polynomial of the form $\sum_{i=1}^k(L_{i}(x)+\gamma_i)h_i(B(x))$ over ${\bf F}_{q^{m}}$, where $L_i(x)$ and $B(x)$ are linearized polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalize a result of Marcos by constructing permutation polynomials of the forms $x h(\lambda_{j}(x))$ and $xh(\mu_{j}(x))$, where $\lambda_{j}(x)$ is the $j$-th elementary symmetric polynomial of $x, x^{q}, ..., x^{q^{m-1}}$ and $\mu_{j}(x)=\textup{Tr}_{{\bf F}_{q^{m}}/{\bf F}_{q}}(x^{j})$. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form $L_1(x)+L_{2}(\gamma)h(f(x))$ over ${\bf F}_{q^{m}}$, which extends a result of Kyureghyan., Comment: 9 pages. To appear in Bulletin of the Australian Mathematical Society

Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let $e\ge 1$ be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ equal to its weight? We introduce some new simple sub-functions and develop new technique to get several recursive formulas. Then we use these recursive formulas to show that the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ is the same as its weight. So we answer the above question affirmatively. Finally, we conjecture that if $l\ge 4$ is an integer, then the nonlinearity of the rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}...x_{le}$ equals its weight., Comment: 10 pages

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and $r\geq \max(a, l-1)$ and $n\geq l\alpha r$, we have $$L_n\geq u_0r^{(l-1)\alpha +a-l}(r+1)^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers., Comment: 4 pages. To appear in Chinese Annals of Mathematics (Series B)

Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $\nu_2(k)$ and $s_2(k)$ be the 2-adic valuation of $k$ and the sum of binary digits of $k$, respectively. Let $S(n,k)$ be the Stirling number of the second kind. It is shown that $\nu_2(S(c2^n,b2^{n+1}+a))\geq s_2(a)-1,$ where $04$ is a power of 2, and $\delta(k)=0$ otherwise. This confirms a conjecture of Lengyel raised in 2009 except when $k$ is a power of 2 minus 1., Comment: 23 pages. To appear in Journal of Number Theory

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., Comment: 32 pages. To appear in Forum Mathematicum

Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word $u$ is a deep hole of the standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. In this paper, we extend this result by giving a new class of deep holes of the generalized Reed-Solomon codes., Comment: 5 pages

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

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 $g_{k,\varphi}$, and determine the smallest period of $g_{k,\varphi}$, where $\varphi$ is the Euler phi function., Comment: 16 pages. To appear in Journal of Number Theory

Erdos and Niven proved that for any positive integers $m$ and $d$, there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/m,1/(m+d), ..., 1/(m+nd)$ are integers. Recently, Chen and Tang proved that if $n\ge 4$, then none of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we show that if $n\ge 2$, then none of the elementary symmetric functions of $1, 1/3, ..., 1/(2n-1)$ is an integer., Comment: 7 pages

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes $[p-1, k]_p$ with $p$ a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes $[q-1, k]_q$ with $q$ a prime power of $p$. Let $q \geq 4$ and $2 \leq k\leq q-2$. We show that the received word $u$ is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. So there are at least $2(q-1)q^k$ deep holes if $k \leq q-3$., Comment: 10 pages. To appear in SCIENCE CHINA Mathematics

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 $f$., Comment: To appear in Acta Mathematica Hungarica

In 1982, Nair proved the identity: ${\rm lcm}({k \choose 1}, 2{k \choose 2}, >..., k{k \choose k}) ={\rm lcm}(1, 2, ..., k), \forall k\in \mathbb{N}.$ Recently, Farhi proved a new identity: ${\rm lcm}({k \choose 0}, {k \choose 1}, >..., {k \choose k}) =\frac{{\rm lcm}(1, 2, ..., k+1)}{k+1}, \forall k\in \mathbb{N}.$ In this note, we show that Nair's and Farhi's identities are equivalent., Comment: 3 pages

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll., Comment: 9 pages. To appear in International Journal of Number Theory

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., Comment: 10 pages. To appear in Proceedings of the Edinburgh Mathematical Society

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2., Comment: 5 pages; v3: improved results and updated references

In this paper, we determine all the squares in the sequence $\{\prod_{k=2}^n(k^2-1)\}_{n=2}^\infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence., Comment: 4 pages. To appear in Asian-European J. Math

In this paper we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod $r$). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod $r$). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results., Comment: 23 pages. To appear in International Journal of Number Theory

Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A positive semi-definite matrix $A=(a_{ij})$ with $a_{ij}\ge 0$ for all $i$ and $j$ is called infinitely divisible if the fractional Hadamard power $A^{\circ r}=(a_{ij}^r)$ is positive semi-definite for every nonnegative real number $r$. Let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive integers. In this paper, we show that if $f$ is a multiplicative function such that $(f*\mu)(d)\ge 0$ whenever $d|x$ for any $x\in S$, then the $n\times n$ matrices $(f(x_i, x_j))$, $(\frac{1}{f[x_i, x_j]})$ and $(\frac{f(x_i, x_j)}{f[x_i, x_j]})$ are infinitely divisible. Finally we extend these results to the Dirichlet convolution case which produces infinitely many examples of infinitely divisible matrices. Our results extend the results obtained previously by Bourque, Ligh, Bhatia, Hong, Lee, Lindqvist and Seip., Comment: 8 pages. to appear in Acta Arith

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$., Comment: 20 pages. To appear in Journal of the Australian Mathematical Society