素数幂、双素数幂阶元的共轭类长的个数为4的 有限群的结构

设群G为一个有限群.如果群G中素数幂、双素幂阶元的共轭类长的集合为{1,p~a,m,p~bm},那么群G是可解的,其中ab为正整数,p为素数且与m互素.进一步,给出了群G/Z(G)的结构,这是对文"Chen R F,Zhao X H.A criterion for a group to have nilpotent p-complements[J].Monatsh Math,2016,179(2):221-225"中定理A主要结论的一个推广.     

k-素数和唯一分解

在本文中,作者揭示了唯一k-素因数分解的更深层原因.在第二节中,首先引入S_k中的k-组合条件和费马定理;并证明了下面4论断是等价的:(1) k-组合条件成立,(2)中唯一k-素因数分解成立,(3) Sk中费马定理成立,(4)k=1或2.为了更好地理解k-素数,在第三节中作者考察了一类特殊的k-素数,即3-素数.众所周知唯一3-素因数分解一般是不成立的,那么S₃中的哪些正整数具有唯一3-素因数分解性质呢?在第三节中,作者得到一个S₃中的整数具有唯一3-素因数分解的充要条件.在第三节最后,作者引入π₃(x),它表示小于等于x的3-素数个数.由素数定理,作者得到π₃(x)的一个具体公式以及一些近似公式.     

关于素数等差数列的一组结论

文[1]讨论了三个素数成等差数列的问题,给出了三个素数a,a+d,a+2d成等差数列的三个猜想.猜想1设三个数a,a+d,a+2d构成一个等差数列,其中a和d都是正整数.当d是奇数时,不论a取何值,a,a+d,a+2d不可能都是素数.猜想2设三个数a,a+d,a+2d构成一个     

梅森素数趣谈

素数也叫做质数,其特点是它只能被1和它本身整除.比如2009就不是一个素数,它可以被7整除.许多数学家都在寻找素数的秘密,著名的哥德巴赫猜想就与素数有密切关系;世界上最难的猜想当数黎曼猜想,它也是以素数为中心;欧几里得在两千多年以前就利用反证法证明了有无穷无尽的素数,梅森提出了少量素数可以表示成2~p-1(p为正整数)的形式,但科学家们至今也没有找到这种形式     

3个素数平方和的非线性型的整数部分

假设λ,μ,υ是不全为负的非零实数,λ是无理数,k是正整数,则存在无穷多素数p_1,p_2,p_,p,3使得[λp_1~2+μp_2~2+υp_3~2]=kp.特别地,[λp_1~2+μp_2~2+υp_3~2]表示无穷多素数.     

关于幂数的几个问题

S.W.Gotemo 在1976年于[1]中给出了幂数的概念。正整数r若满足p|n则pAn,此处p为素数,则n叫做一个幂数。 S.W.Golomb考虑了连续幂数的问题。显然4个连续整数不可能为幂数,因为其中之一必为2(2b-1)形状。对两个连续幂数问题,他证明了,若其中之一为完全平方数,则可通过pell方程的构造出来,并且,S.W.Golomb 在[1]中指出,对连续奇幂数仅能给出的一对为25、27。     

素数的判定

一个大于1的整数,如果只能被1和它本身所整除,则这个正整数叫做素数,否则叫做合数。开头的几个素数是2,3,5,7,…。为了进一步找出更多的素数,大约在公元前250年,     

一个素数和一个素数的平方和问题

H表示一个正整数N的集合,使对任意的正整数q,同余方程a+b~2≡N(mod q)在模q的既约剩余系中有解a;b.E(x)表示N≤x,N∈H,但不能表成p_1+p_2~2=N的数的个数,其中p_1,p_2个表示素数,则E(x)<相似文献   

推广的孪生素数问题

王元,Halberstam和Richert及其他一些作者曾证明:存在无穷多个正整数n使F(n)有至多5个素因子.他们的方法都可用来处理猜想(二).并得到类似的结果. 对猜想(一)、(二)的特例即孪生素数猜想和哥德巴赫猜想,陈景润首先得到著名的(1,2)的结果.后来潘承洞、丁夏畦、王元及其他一些作者给出了一些简化的证明. 本文的目的是证明下面两个定理.     

陈永高的两个猜想

证明了陈永高提出了下面的两个猜想是正确的:(1)设n为正整数,p为奇素数,则能够表为2n-p形式的正整数在正奇整数的伞体中有正的F渐近密度; (2)设n为正整数,p为奇素数,则能够表为p-2n形式的正整数在正奇整数集合中有正的下渐近密度.     

One Prime,Two Squares of Primes and Powers of 2

It is proved that every sufficiently large odd integer is the sum of one prime, two squares of primes and 35 powers of 2. This improves a previous result with 35 replaced by 83.     

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.     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.     

Prime numbers and the first digit phenomenon

The set of primes which have lead digit 1 does not have relative natural density in the prime numbers. However, Bombieri has shown that this set does have relative Zeta density equal to log₁₀ 2. This means that a prime chosen at random (w.r.t. the Zeta distribution) will have lead digit 1 with the determined probability. Here the question, Is this a special property of Zeta density or a more universal property of primes? is answered. It is shown that for any generalization of relative natural density (obeying a few basic assumptions) if a value is assigned to the relative density of primes of lead digit 1 then this value is always log₁₀ 2. Another density which does converge on this set is also exhibited. Additionally the relative densities of primes beginning with any specified string of digits are found.     

Associated Primes of the Example of Brenner and Monsky

Recently, Brenner and Monsky found an example of an ideal in a hypersurface ring whose tight closure does not commute with localization, thus answered the localization problem in tight closure theory in the negative. In this article, we use Monsky's calculations to analyze the set of associated primes of the Frobenius powers of this ideal and show that this set is infinite.     

Two results on powers of 2 in Waring-Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.     

Fermat versus Wilson congruences,arithmetic derivatives and zeta values

We look at two analogs each for the well-known congruences of Fermat and Wilson in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, we find interesting relations linking them together, as well as linking them with derivatives and zeta values. The link with the zeta value carries over to the number field case, with the zeta value at 1 being replaced by Euler's constant.     

Prime knowledge about primes

Several proofs demonstrating that there are infinitely manyprimes, different types of primes, tests of primality, pseudoprimes, prime number generators and open questions about primesare discussed in Section 1. Some of these notions are elaboratedupon in Section 2, with discussions of the Riemann zeta functionand how algorithmic complexity enters into tests for primes.Readers may know segments of what follows, but hopefully thiswork will help them place their knowledge into richer landscapes.     

On unequal powers of primes and powers of 2

We consider Linnik’s type of the Waring–Goldbach problem with unequal powers of primes. In particular, it is proved that every sufficiently large even integer can be represented as a sum of one prime, one square of prime, one cube of prime, one fourth power of prime and 18 powers of 2.