梅森素数研究的若干基本理论及其意义

梅森素数的研究历史源远流长,意义非凡.介绍相关的定义、理论及算法,归纳此项工作的意义,并讨论一些有待解决的相关数论问题.     

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     

梅森与梅森素数

数论问题中有许多关于素数的问题,在吸引人们去探索的同时又在磨砺着人类的智慧.许多素数问题的妙趣之处在于人们可以轻而易举地理解问题的表述,但是想要真正将问题解决,却需要坚强的意志、高超的技巧和艰苦的计算.如至今尚未完全解决的哥德巴赫猜想,历经几代数学家的苦苦求索直到1994年才得到求证的费尔马猜想(现在应该叫做费尔马大定理了),还有一个似乎不是那么著名的“梅森猜想”.提到“梅森猜想”,就要先从梅森其人谈起.梅森全名马林.梅森(Marin Mersenne,1588—1648),是法国圣弗朗西斯(St.Francis of Paola)所建的托钵僧团体中的修道…     

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.     

第47个梅森素数被发现

挪威计算机专家奥德·斯特林德莫通过参加一个名为“因特网梅森素数大搜索”（GIMPS）的国际合作项目，最近发现了第47个梅森素数，该素数为“2的42643801次方减1”．它有12837064位数，如果用普通字号将这个巨数连续写下来，它的长度超过50公里!     

稀世珍奇的梅森素数

2009年4月,挪威计算机专家斯特林德莫通过参加一个名为“因特网梅森素数大搜索”（GIMPS）的国际合作项目,发现了第47个梅森素数,该素数为2^42643801-1（即“2的42643801次方减1”）．     

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...     

神奇的伪素数

在浩瀚元垠数的天庭 ,有一种神奇的伪素数 ,它像一块磁石 ,紧紧吸引着数论专家的心灵 .1　费马小定理引发出的奇异法国业余数学家费马 (Ｆｅｒｍａｔ) 1 6 4 0年 6月给本国神甫数学家梅森 (Ｍ .Ｍｅｒｓｅｎｎｅ ,1 5 88- 1 6 4 8)的一封信提出一个命题 :若ｎ是素数 ,则 2 ｎ- 2可被 2ｎ除尽 .同年 1 0月 1 8日给本国数学家德贝西(Ｂ .Ｆ .ｄｅＢｅｓｓｉｅ,1 6 0 5 - 1 6 75 )的信中又说 ,他已证明了一个更广的命题 :若ｐ是一个素数 ,且ａ不能被ｐ整除 ,则ａｐ- 1 - 1能被ｐ整除 (等价的说法是ａｐ-ａ能被素数ｐ整除 ) .后人称此为费马小定…     

关于等差数列在三素数中的分布

基于著名的Hardy-Littlewood猜想,证明了长为3的素数等差数列出现频率最高的公差是素数连乘.另外,举例说明了这种素数等差数列并不一定是三素数最喜欢的分布形式.     

短区间的误差项的双线性形式

本文的目的在于改进 Iwaniec-Laborde 的关于短区间的误差项的双线性形式的估计.这类结果是用筛法和三角和方法研究短区间中殆素数 P_2的分布的关键.     

The exceptional set for the distribution of primes between consecutive powers

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. This paper is concerned with the distribution of prime numbers between two consecutive powers of integers, as a natural generalization of the afore-mentioned conjecture.      

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.     

Estimates for Wieferich numbers

We define Wieferich numbers to be those odd integers w≥3 that satisfy the congruence 2 ^φ(w)≡1 (mod  w ²). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numbers; analogous results for the set of Wieferich primes remain out of reach. Finally, we consider several modifications of the above definition and demonstrate that our methods apply to such sets of integers as well. During the preparation of this paper, W.B. was supported in part by NSF grant DMS-0070628, F.L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I.S. was supported in part by ARC grant DP0211459.     

Well-behaved Beurling primes and integers

In this paper, we study generalised prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately li(x) and ρx, respectively (where ρ is a positive constant), with error terms of order O(x^θ₁) and O(x^θ₂) for some θ₁,θ₂<1. We show that it is impossible to have both θ₁ and θ₂ less than .     

搜寻广义Fermat素数

设ｂ为偶数，本文讨论了广义Ｆｅｒｍａｔ数Ｆ（ｂ，ｍ）＝ｂ＾２＋１为素数的必要条件和充分条件，提出了搜寻广义Ｆｅｒｍａｔ素数的一种效率很高的算法并在微机上实现，得出了ｂ≤２５６，ｍ≤１０的全部广义Ｆａｒｍａｔ素数，其中最大的是４６＾２１２＋１。     

On the sum of the first <Emphasis Type="Italic">n</Emphasis> primes being a square

Let p _n be the nth prime. In this note, we show that the set of n such that is a square is of asymptotic density zero. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 301–306, July–September, 2007.     

Solutions of the congruence

To supplement existing data, solutions of are tabulated for primes with and . For , five new solutions 2^{32}$"> are presented. One of these, for , also satisfies the ``reverse' congruence . An effective procedure for searching for such ``double solutions' is described and applied to the range , . Previous to this, congruences are generally considered for any and fixed prime to see where the smallest prime solution occurs.

     

On the Attached Prime Ideals of Local Cohomology Modules

In this article, we shall prove some new properties about attached prime ideals over local cohomology modules. Also we generalize some of the results of [2 Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press .[Crossref] , [Google Scholar]].