On the Distribution of the Sums of Binomial Coefficients Modulo a Prime

 Let be the binomial coefficient modulo b (b prime), with if l is greater than c, and let be the sum of binomial coefficients modulo b, that is (mod b). We prove the following property: the for which the couples c, l verify and are uniformly distributed in the residue classes modulo b as n tends to infinity. The method, using the Perron-Frobenius theory, applies also to and gives a new proof of the well known result for the non-zero binomial coefficients modulo b. (Received 21 June 1999; in revised form 13 July 2000)     

On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g., for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions. Received November 5, 1998, Accepted December 7, 2000.     

Prime Divisors of Shifted Factorials

For any positive integer n we let P(n) be the largest primefactor of n. We improve and generalize several results of P.Erds and C. Stewart on P(n!+1). In particular, we show thatlim sup_ninfin; P(n!+1)/n2.5, which improves their lower boundof lim sup_ninfin; P(n!+1)/n2. 2000 Mathematics Subject Classification11A05, 11A07, 11J86.     

On an Identity with Binomial Coefficients

Mathematical Notes -     

一类特殊Beatty序列中的素因子问题

研究了数论函数Ω和ω取值于一类特殊的非齐次Beatty序列[αn+β](n=1,2,…)的问题.特别地,证明了渐近公式∑ω([αn+β)=N log log N+O(N(log log N) )以及∑(一1)Ω(αn+β)=O( n/(log N 1/2)),其中α,β∈R,n∈Z+,α>0是型为1的无理数,Ω(k)和ω(k)表示整数k(≠0)的素因子个数(Ω记重数,ω不计重数).     

函数域的双重根式扩域及素除子的密度

设K/F_q是整体函数域,l是与q互素的素数,ξ_1是K的固定代数闭包中的本原l次单位根.对于a,b∈K~*-(K~*)~l,本文主要讨论了根式扩域K(a~(1/2))与K(a~(1/l),(b~(1/l))的性质,利用Kummer理论给出了K(a~(1/l))/K与K(a~(1/l),b~(1/l))/K不是几何扩张的充要条件.当a,b是l-无关时,对于K的素除子P及对应的离散赋值环θ_P,利用这两类扩张的性质,通过分析a,b生成循环群(θ_P/P)~*的充要条件,本文明确给出了满足使得a,b生成循环群(θ_P/P)~*的全体素除子集合M_(a,b)的Dirichlet密度公式.     

Binomial Coefficients and Zero-Sum Ramsey Numbers

A counting argument is developed and divisibility properties of the binomial coefficients are combined to prove, among other results, that[formula]whereK_n, resp.K^k_n, is the complete, resp. completek-uniform, hypergaph andR(K_n, Z_p),R(K^k_n, Z₂) are the corresponding zero-sum Ramsey numbers.     

Integral Domains Having Nonzero Elements with Infinitely Many Prime Divisors

In a factorial domain, every nonzero element has only finitely many prime divisors. We study integral domains having nonzero elements with infinitely many prime divisors.     

二项式系数的更多整除性质

设$a$, $b$和$n$为正整数,且$a>b$,我们证明了下面的整除性质: 对所有正整数$n$, 我们有$$(2bn+1)(2bn+3)(2bn+5){2bn\choose bn}\Big|15(a-b)(3a-b)(5a-b)(5a-3b){2an \choose an}{an\choose bn},$$ 上述整除式推广了杨全会一文中的相关结论.且对所有正整数$n$,我们证明了下面的整除性质:$$(6n+1){4n\choose n}\Big|{12n\choose 6n}{2n\choose n},\ (12n+1){5n\choose n}\Big|{15n\choose 3n}{3n-1\choose n-1},$$ $$(18n+1){12n\choose 9n}{8n\choose 2n}\Big| {24n\choose 18n}{4n\choose 2n}{6n\choose 3n}.$$更多类似的整除性质可以给出.     

Extremal Properties of Sums of Binomial Coefficients

Some extremal problems for the sums of binomial coefficients that arise in research on estimating the computational complexity of discrete optimization algorithms are examined. These extremal problems are solved using the theory of majorization and useful inequalities are introduced for the sums of binomial coefficients.     

二项式展开的抵消系数以及$q$-摸拟

Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A(n,k;A) are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n k=0 A(n,k;A)(x - Ak)n-k = xn. (0.1) Similary, we define B(n,k;A) to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n k=0 B(n,k;A)(x - Ak)k = xn. (0.2) The main result of this paper is an explicit formula for B(n,k;A), which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations (0.1) and (0.2).     

二项式系数倒数级数恒等式

利用已知级数,通过裂项构造出一批新的二项式系数倒数级数,它们的分母分别含有1到4个奇因子与二项式系数的乘积表达式.所给出二项式系数倒数级数的和式是封闭形的.     

包含幂和二项式系数倒数的等式

In this paper,we give several identities of finite sums and some infinite series involving powers and inverse of binomial coefficients,which extends the results of T.Trif.