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Zhi-Wei Sun

Publications and source records attributed to Zhi-Wei Sun.

232 records · Page 13Linked to original sources
arXiv2006

Extensions of Wilson's lemma and the Ax-Katz theorem

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to present a useful lemma on polynomials modulo prime powers, and applied this lemma to reprove the Ax-Katz theorem on solutions of congruences modulo p and deduce various results on codewords in p-ary linear codes with weights. In light of the recent generalizations of Fleck's congruence given by D. Wan, and D. M. Davis and Z. W. Sun, we obtain new extensions of Wilson's lemma and the Ax-Katz theorem.

math.NT↗
arXiv2006

Fleck quotients and Bernoulli numbers

Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\sum_{k=r(mod p)}\binom{n}{k}(-1)^k\in\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ modulo p in terms of Bernoulli numbers, where m>0 is an integer with $m\not=n$ and $m=n (mod p(p-1))$. Consequently, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $0 0$ and $l\ge 0$ are integers with $2\le n-l\le p$ then $$\frac{1}{p^{n-l}}\sum_{l<k\le n} \binom{p^a n-d}{p^a k-d}(-1)^{pk}\binom{k-1}{l} =\frac{(-1)^{l-1}n!}{l!(n-l)}B_{p-n+l} (mod p)$$ for all d=1,...,max{p^{a-2},1}.

math.NT↗
arXiv2006

On q-Euler numbers, q-Salie numbers and q-Carlitz numbers

Let $(a;q)_n=\prod_{0\le k<n}(1-aq^k)$ for n=0,1,2,.... Define q-Euler numbers $E_n(q)$, q-Salié numbers $S_n(q)$ and q-Carlitz numbers $C_n(q)$ as follows: $$\sum_{n=0}^{\infty}E_n(q)\frac{x^n}{(q,q)_n} =1/\sum_{n=0}^{\infty}\frac{q^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$ $$\sum_{n=0}^{\infty}S_n(q)\frac{x^n}{(q;q)_n} =\sum_{n=0}^{\infty}\frac{q^{n(n-1)}x^{2n}}{(q;q)_{2n}} /\sum_{n=0}^{\infty}\frac{(-1)^nq^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$ $$\sum_{n=0}^{\infty}C_n(q)\frac{x^n}{(q;q)_n} =\sum_{n=0}^{\infty}\frac{q^{n(n-1)}x^{2n+1}}{(q;q)_{2n+1}} /\sum_{n=0}^{\infty}\frac{(-1)^nq^{n(2n+1)}x^{2n+1}}{(q;q)_{2n+1}}.$$ We show that $$E_{2n}(q)-E_{2n+2^{s}t}(q)=[2^s]_{q^t} (mod (1+q)[2^s]_{q^t})$$ for any nonnegative integers n,s,t with t odd, where $[k]_q=(1-q^k)/(1-q)$; this is a q-analogue of Stern's congruence $E_{2n+2^s}=E_{2n}+2^s (mod 2^{s+1})$. We also prove that $(-q;q)_n=\prod_{0<k\le n}(1+q^k)$ divides $S_{2n}(q)$ and the numerator of $C_{2n}(q)$; this extends Carlitz's result that $2^n$ divides the Salié number $S_{2n}$ and the numerator of the Carlitz number $C_{2n}$. Our result on q-Salié numbers implies a conjecture of Guo and Zeng.

math.CO↗
arXiv2006

A connection between covers of the integers and unit fractions

For integers a and n>0, let a(n) denote the residue class {x\in Z: x=a (mod n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1} does not. We show that if n_k is a period of the covering function w_A(x)=|{1\le s\le k: x\in a_s(n_s)}| then for any r=0,...,n_k-1 there are at least m integers in the form $\sum_{s\in I}1/n_s-r/n_k$ with I contained in {1,...,k-1}.

math.NT↗
arXiv2006

Polynomial extension of Fleck's congruence

Let $p$ be a prime, and let $f(x)$ be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the $p$-adic order of the sum $$\sum_{k=r(mod p^β)}\binom{n}{k}(-1)^k f([(k-r)/p^α]),$$ where $α\geβ\ge 0$, $n\ge p^{α-1}$ and $r\in Z$. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as $$\sum_{k=r(mod p^α)}\binom{n}{k} a^k\equiv 0 (mod p^{[(n-p^{α-1})/ϕ(p^α)]})$$ provided that $α>1$ and $a\equiv-1(mod p)$.

math.NT↗
arXiv2005

On odd covering systems with distinct moduli

A famous unsolved conjecture of P. Erdos and J. L. Selfridge states that there does not exist a covering system {a_s(mod n_s)}_{s=1}^k with the moduli n_1,...,n_k odd, distinct and greater than one. In this paper we show that if such a covering system {a_s(mod n_s)}_{s=1}^k exists with n_1,...,n_k all square-free, then the least common multiple of n_1,...,n_k has at least 22 prime divisors.

math.NT↗
arXiv2005

On various restricted sumsets

For finite subsets A_1,...,A_n of a field, their sumset is given by {a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various restricted sumsets of A_1,...,A_n with restrictions of the following forms: a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j (mod m_{ij}). Furthermore, we gain an insight into relations among recent results on this area obtained in quite different ways.

math.CO↗
arXiv2005

Binomial Coefficients and Quadratic Fields

Let E be a real quadratic field with discriminant d and let p be an odd prime not dividing d. For ρ=1 or -1, we determine $\prod_{0<c<d, (d/c)=ρ} binomial coeff.{p-1}{\lfloor pc/d\rfloor}$ modulo p^2 in terms of Lucas numbers, the fundamental unit and the class number of E, where (d/c) is the Kronecker symbol.

math.NT↗
arXiv2004

On the Herzog-Schönheim conjecture for uniform covers of groups

Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Schönheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be distinct. In this paper we show that if $\Cal A$ covers all the elements of G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal to G, then $M=\max_{1\le j\le k}|\{1\le i\le k:n_i=n_j\}|$ is not less than the smallest prime divisor of $n_1... n_k$, moreover $\min_{1\ls i\ls k}\log n_i=O(M\log^2 M)$ where the O-constant is absolute.

math.GR↗
arXiv2004

On m-covers and m-systems

Let A={a_s(mod n_s)}_{s=0}^k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in more than m=[sum_{s=1}^k 1/n_s] members of A, then for any a=0,1,2,... there are at least binom{m}{[a/n_0]} subsets I of {1,...,k} with sum_{s in I}1/n_s=a/n_0. We also characterize when any integer lies in at most m members of A, where m is a fixed positive integer.

math.NT↗
arXiv2004

On Euler numbers modulo powers of two

To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M. A. Stern.

math.NT↗
arXiv2004

Groups and Combinatorial Number Theory

In this talk we introduce several topics in combinatorial number theory which are related to groups; the topics include combinatorial aspects of covers of groups by cosets, and also restricted sumsets and zero-sum problems on abelian groups. A survey of known results and open problems on the topics is given in a popular way.

math.GR↗
arXiv2004

On the range of a covering function

Let {a_s(mod n_s)}_{s=1}^k (k>1) be a finite system of residue classes with the moduli n_1,...,n_k distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1\le s\le k: x=a_s (mod n_s)}| is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.

math.NT↗
arXiv2004

New identities involving Bernoulli and Euler polynomials. II

We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003), 709--718].

math.NT↗
arXiv2004

Arithmetic Properties of Periodic Maps

Let $ψ_1,...,ψ_k$ be periodic maps from $\Bbb Z$ to a field of characteristic p (where p is zero or a prime). Assume that positive integers $n_1,...,n_k$ not divisible by p are their periods respectively. We show that $ψ_1+...+ψ_k$ is constant if $ψ_1(x)+...+ψ_k(x)$ equals a constant for |S| consecutive integers x where S={r/n_s: r=0,...,n_s-1; s=1,...,k}. We also present some new results on finite systems of arithmetic sequences.

math.NT↗