On Farey Fractions with Small Prime Factors

We investigate the Farey fractions, i.e., the set of irreducible fractions m /n, 0 < m < n x. We derive an asymptotic equality for the number of Farey fractions having no large prime factors.     

关于整数素因数间的一个新的对偶公式

主要目的是利用Liouville反转公式来给出整数素因数间的一个新的对偶公式,从而推广了K.Alladi的基于Mbius反转公式的对偶引理.     

Another Note on the Greatest Prime Factors of Fermat Numbers

For every positive integer k > 1, let P(k) be the largest prime divisor of k. In this note, we show that if F_m = 2^2m + 1 is the mth Fermat number, then P(F_m) 2^m+2(4m + 9) + 1 for all m 4. We also give a lower bound of a similar type for P(F_a,m), where F_a,m = a^2m + 1 whenever a is even and m a¹⁸.AMS Subject Classification (1991) 11A51 11J86     

Sums of the Numbers of Prime Factors of Positive Integers

Denote by ω(n) and Ω(n) the number of distinct prime factors of n and the total number of prime factors of n,respectively.For any positive integer ι,we prove that ∑↑2≤n≤x1/ω（n)=ι↑∑↑κ=0(ι↑∑↑i=κ（-1)^i-κCi^κF^(i-κ)(1)κ！x/（loglogx）^i 1 O(x/(loglogx)^ι 2) ∑↑2≤n≤xΩ（n)/ω（n)=x ι↑∑↑κ=0ι↑∑↑i=κ∑↑p1/p^κ 2-p^κ 1（-1)^i-κCi^κF^(i-κ）（1）κ!)x/(loglogx)^i 1 O(x/(loglogx)ι 2） where F(z)=1/г(z)pⅡ（1 z/p-1)(1-1/p)^z,and the constant O despends on ι.This improves previous result of R.L.Duncan and Chao Huizhong.     

On the Number of Restricted Prime Factors of an Integer

Let \({q\geqq2}\) be an integer and denote S(n) the sum of the digits in base q of the positive integer n. Our main result is to estimate the sum \({\Sigma_{n\leqq x}\tilde{\omega}(n)}\) where \({\tilde{\omega}(n)}\) is the number of distinct prime factors p of n such that \({S(p) \equiv a \,{\rm mod} \,b \,(a \in \mathbb{Z}, b\geqq 2)}\) .     

五个具有固定素因子个数的整数之平方和

设N≡5(mod24)为充分大的正整数,若GRH(广义Riemann假设)成立, 则方程N=n12+n22+…+n52有解,此处ni,i=1,2,…,5具有固定素因子个数,并且方程解数具有渐近式．     

Height of Prime and Weakly Prime Submodules

We will introduce and study new definitions for the height of prime submodules and weakly prime submodules. Also new module versions of catenary are discussed.     

关于表自然数因子个数的函数

设ｇ（Ｎ）是满足ｇ（０）＝０的任一实值数论函数．当是ｎ的标准分解式时，定义和ｆ（１）＝０．本文给出了和的渐近公式，此处Ω（ｎ）表ｎ的全部素因子的个数．     

幂格的素理想和素对偶理想

在文 [1 ]中 ,引入了幂格的概念 ,并讨论了其相关性质 .本文在此基础上 ,讨论格与其幂格的理想 ,对偶理想的关系 ,以及格与其幂络的素理想 ,素对偶理想的关系 .     

On the Distribution of the Degrees of Prime Element Factors in Additive Arithmetical Semigroups

 Results of Arnold Knopfmacher about the distribution of the degrees of irreducible factors in the canonical decomposition of monic polynomials in ? _q [X] are generalized to additive arithmetical semigroups (G,∂) satisfying a weak condition called axiom ?     

Prime Orders All of Whose Prime Suborders Are Selfdual

Let P be an order on a set V. A subset A of V is autonomous in P if every element of V not in A is either less than or greater than or incomparable to all elements of A. The empty set, the singletons from V and the set V are autonomous sets and are called trivial. Call an order prime if all its autonomous sets are trivial. We give the complete list of all finite prime orders all of whose prime suborders are selfdual.     

半群的完全素和素模糊理想

通过由模糊点生成的模糊理想给出了半单半群的刻画。同时也刻画了两类半群：一类是所有模糊理想是素理想。另一类是所有模糊理想为安全素理想。     

模糊幂格的素理想与素对偶理想

讨论格与其模糊幂格的理想,对偶理想的关系,以及格与其模糊幂格的素理想,素对偶的关系.     

Quantale中的素理想及弱素理想

本文把Quantale中的序结构与代数运算＆结合在一起给出了Quantale中素理想和弱素理想的概念。讨论了它们之间的关系,得到了Quantale中理想是(弱)素理想的充要条件。证明了与序半群中的一些经典结论相一致的命题。     

半环上的模糊素理想和模糊素根

引入了半环的模糊素理想的概念,给出了它的几种刻画方式,得到了它的一些性质;进一步定义了模糊素理想的模糊素根,获得了一些漂亮结果。     

On the Distribution of the Degrees of Prime Element Factors in Additive Arithmetical Semigroups

 Results of Arnold Knopfmacher about the distribution of the degrees of irreducible factors in the canonical decomposition of monic polynomials in ? _q [X] are generalized to additive arithmetical semigroups (G,∂) satisfying a weak condition called axiom ? (Received 25 August 1998; in revised form 7 June 1999)