"大互联网梅森素数寻求(GIMPS)"研究计划进展

梅森素数是一种特殊的素数,它的研究与寻求一直是数论研究的代表性问题之一.寻求梅森素数之路艰辛曲折,其计算复杂性对现代计算能力极具挑战.计算机网络技术的发展,特别是能使虚拟组织共享计算资源的全球分布计算技术,使得寻求速度大大加快.本文综述寻求梅森素数的最新进展及历史进程,并介绍寻求梅森数所用的分布计算技术.     

整除在初等数论中的应用

利用格点图内因子的分布规律,推导出素数判断函数,孪生素数判断函数,歌德巴赫素数判断函数;推导出可计算不大于某正整数的素数个数,不大于某正整数的孪生素数个数和大偶数包含的歌德巴赫素数个数精确和近似的计算公式.     

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     

Vandiver猜想与非正则素数分布的一个猜想

本文介绍Vandiver猜想与相关研究结果;我们证明A₂=A₄=…=A₃₂=0,这里A是Q(ζ_p)的理想类群的p-Sylow子群;我们提出一个关于非正则素数分布的猜想,给出数值验算.     

关于调和数的一个结论

设n是大于1的正常数,并且设n=pα11p2α2…ptαt,其中pi为素数,i=1,2,…,t,ω(n)表示n的不同素因子的个数,即ω(n)=t.若n的所有因子的倒数和为整数,即0≤∑ij≤αjj=1,2,…,t1p1i1pi22…ptit为整数,称n是调和数.证明了和调和数相关的一个结论.     

An approximate expression related with RSA fixed points

Let T=T(n,e,a)be the number of fixed points of RSA(n,e)that are co-prime with n=pq,and A,B be sets of prime numbers in (1,x)and(1,y) respectively.An estimation on the mean-value M(A,B,e,a)=1/(#A)(#B)∑p∈A,q∈B,(p.q)=1 logT(pq,e,a)is given.     

关于等差数列型素数集的分解

设C为非负整数集.若存在非负整数集A和B,使得C=A+B,其中|A|,|B|≥2,则称C为可分解的,否则称C为不可分解的或本原的.讨论特殊的等差数列型素数集合,证明了4k+1型素数集不可分解,以及4k+3型素数集不可分解.     

漫长的寻觅梅森素数的历程

漫长的寻觅梅森素数的历程徐品方（四川西昌师专）数论中有一些猜想，是世界最坚硬的磐石，它能轻而易举地挫去人的智慧的锋芒，耗尽人的才华和心血，甚至幸福．荷兰数学家丹齐格（Ｖ．Ｄ．Ｄａｎｚｉｇ，１９００—１９０９）说：“数论是数学中所有部门最难的一门．不错...     

Finite p-groups whose nonnormal subgroups have orders at most p 3

We classify finite p-groups all of whose nonnormal subgroups have orders at most p3, p odd prime. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.     

b≤200O,m≤1 O的广义Fermat素数

设b为偶数,本文基于作者的原有结果,进一步改进了算法,对于b≤2000,m≤10给出了所有广义Fermat素数F(b,m)＝b2m+1,其中最大的是16321024+1,有3290位.     

A class of noninjective circle transformations without periodic points

We construct a class of noninjective aperiodic circle transformations. As a by-product, we establish some new properties of Mersenne numbers.     

Mersenne's solution for Galileo's problem of the rotating earth

More than twenty years before Huygens and Newton developed formulas for centrifugal acceleration, Mersenne contrived a statisfactory solution for Galileo's problem of the extrusion of bodies from the earth as a result of its daily rotation. Mersenne was able to overcome an error in Galileo's approach without the use either of an explicit notion of infinitesimals or of any clear concept of force. His solution depends on comparing the lengths of two lines, a technique that several historians have claimed to be inadequate for this problem.     

Re-seeding invalidates tests of random number generators

Kim et al. [C. Kim, G.H. Choe, D.H. Kim, Test of randomness by the gambler’s ruin algorithm, Applied Mathematics and Computation 199 (2008) 195-210] recently presented a test of random number generators based on the gambler’s ruin problem and concluded that several generators, including the widely used Mersenne Twister, have hidden defects. We show here that the test by Kim et al. suffers from a subtle, but consequential error: re-seeding the pseudorandom number generator with a fixed seed for each starting point of the gambler’s ruin process induces a random walk of the test statistic as a function of the starting point. The data presented by Kim et al. are thus individual realizations of a random walk and not suited to judge the quality of pseudorandom number generators. When generating or analyzing the gambler’s ruin data properly, we do not find any evidence for weaknesses of the Mersenne Twister and other widely used random number generators.     

Secondary representation of some modules over a commutative ring

Let R be a commutative ring and let M be an R-module with the property that its zero submodule has a primary decomposition. Let E be an injective R-module with W.Ass _R (E) = Ass _R (E) (here W.Ass _R (E) denotes the set of weakly associated primes of E). Then we will show that Hom _R (M,E) has a secondary representation and we will specify the set of its attached prime ideals. This revised version was published online in June 2006 with corrections to the Cover Date.     

Idempotents and the multiplicative group of some totally bounded rings

In this paper, we extend some results of D.Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power 2^ℵ0 commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.     

A congruence relation of the Catalan-Mersenne numbers

The Catalan-Mersenne numbers c_n are double Mersenne numbers defined by c₀ = 2 and \({c_n} = {2^{{c_{n - 1}}}} - 1\) for positive integers n. We prove a certain congruence relation of the Catalan- Mersenne numbers.     

On the representation of large integers as sums of four almost equal squares of primes

In this paper, we are able to establish two localized results on the conjecture that each large integer congruent to 4 modulo 24 can be written as the sum of four squares of primes. The proof is based on the new estimates for exponential sums over primes in short intervals and a technique to get the asymptotic formula on the enlarged major arcs in the circle method. This work is supported by the National Natural Science Foundation of China (Grant Nos. 10701048 and 10771127).     

Primes in arithmetic progressions with friable indices

We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.     

On arithmetic numbers

An integer n is said to be arithmetic if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, we give an interesting characterization of Mersenne numbers.     

A note on fractal strings and spacetime

A short discussion is presented on the role of Mersenne primes and even perfect numbers in fractal strings, Cantorian-fractal spacetime, spin structures on Riemann surfaces and the classification of consistent string theories. Since in principle the number of Mersenne primes should be infinite this entails that the family of consistent quantum string theories in higher dimensions than 26 could be infinite as well and, consequently, this could be the physical reason why the number of quantum-spacetime dimensions is infinite. The Tsallis entropy of fractal strings is constructed followed by the exact expression for the fractal string mass spectrum.