談談“篩法”

在这篇短文里,我们向读者介绍一个在数论中常用的方法,即所谓“节法”.为了避免引用较高深的数学工具,我们除了谈谈最古典的埃拉多斯染尼氏(Eratosthenes)节法及其些微应用外,只略为涉及一点这一方法在近代的发展.其实在这里所写的一些结果,有关的数论书籍中都有记载,作者只是加以整理与归纳,以便于读者更易于了解这一方法及提供一点作者认为较有趣的数论知识.     

2500年研究探寻相亲数

设σ(n)为n的所有正因子(包括1和n本身在内)之和.正整数对(m,n)被称之为相亲数(或双亲数,因为这种数总是成双成对出现的)如果他们满足 σ(m)=σ(n) = m + n.如果n=n, σ(m)=2m,则m被称之为完全数(或单亲数,因为这种数总是单独出现的).更一般的,如果κ个(κ＞2)正整数(m1,m2,…mmk)满足下列条件σ(m1)=m1+m2,σ(m2)=m2+m3,σ(mk)=mκ+m1.则这κ个正整数被称之为多亲数.第一对相亲数(220,284)是在2500年前的古希腊数学家毕达哥拉斯发现的.不过迄今为止,人们对相亲数的情况、尤其对相亲数的分布情况仍然知之甚少.与相亲数有关的难题、尤其是悬而未决千百年的难题还很多就是在今夭,我们仍然不知道是不是有无穷多对相亲数,我们甚至连一个生成相亲数的充分必要条件(定义除外)都没有.在这篇文章中,我们试图给出人类在2500年的漫长历史长河中研究、探寻相亲数的大致情况与重要结果,并着重介绍从古至今生成相亲数的各种数值方法与代数方法.完全数的研究探寻史几乎与相亲数的研究探寻史是一样长的.比如2350年前的古希腊数学家欧几理德就在其数学名著<几何原本>中列出了前四个完全数,不过迄今为止,人们总共也只找到39个完全数,并且这些完全数还都是偶完全数.至于有没有奇完全数的存在,则是一个悬而未决两千多年的著名数学难题.最早的两串多亲数(一串为5个.另一串为28个),则是由法国数学家Poulet于1918年发现的.多亲数的研究探寻史虽然比相亲数的研究探寻史要短得多,但目前人们对它们的注意力与日俱增.由于相亲数与完全数及多亲数密切相关、紧密相连(我们可以将其统一称之为亲和数,因为它们都与相关数的因子和有关),因此在本文中,我们除了要讨论介绍相亲数外,也将顺便介绍完全数与多亲数的研究与探寻简史、以及人们在研究探寻这些数时所获得的一些重要结果.附注截止2004年3月25日作者校勘清样时,人们已经发现了共40个完全数和6262871对相亲数.     

浅论矩阵初等变换法在数论中的应用

矩阵论是一个应用十分广泛的数学学科 .本文将以矩陈的初等变换法为理论工具 ,谈谈它在数论中的两个应用 .本文约定 :小写拉丁字母表示整数 ,大写拉丁字母表示整数矩阵 ,对矩阵实施初等变换的过程中所用到 (得到 )的数均为整数 .1　一个命题命题 1　设 (ａ1 ,ａ2 ,… ,ａｎ) =ｄ ,则存在可逆方阵Ａ =[ａｉｊ]ｎ×ｎ,使得ａ1 ａ2 …ａｎ Ａ =[ｄ 0… 0 ](ｎ≥ 2 ) .证明 (数学归纳法 )(1 )当ｎ =2时 ,不妨设ａ1 >ａ2 >0 (否则可以施以倍法变换或换位变换 ,使得ａ1 >ａ2 >0 ) ,由辗转相除法知 :ａ1 =ｑ1 ａ2 +ｒ1 ,0 <ｒ1 <ａ2ａ2 =ｑ2 ｒ1 +…     

广义Moebius变换和算术Fourier变换

离散Fourier变换(DFT)在数字信号处理等许多领域中占有重要地位．近年来，出现一种优于FFT的算术Fourier变换来计算DFT．在广义Moebius变换的基础上，本文采用了一种改进的AFT来计算DFT，这种方法可以直接提取DFT的系数，且用数论的方法阐明了这一过程，并展开了进一步的讨论．这也代表了数论方法应用在计算数学领域的一个新的发展方向．     

有趣的连分数

连分数是初等数论中一个古老的课题 ,而且是极具艺术魅力的一部分 ,它曾经吸引了不少数学大师为之着迷 ,今日仍是一个十分有用的数学工具 .它既可以用于整数理论的探索 ,也可以用于整数、函数的近似求解 ,还可以用于线性递推数列性质的研究以及不定方程 (如 Pell方程 x2 -dy2=± 1,d是大于 1的非平方数 )的求解等等 .在计算机领域 ,连分数为复杂的数学计算近似求解提供了有效的理论工具 ,有力地推动了应用数学的发展 .在中学数学竞赛中 ,应用连分数解题也是常用的解题技巧 .本文将简要地介绍连分数的一些基本概念 ,并对 d与 di的连分数展开…     

数论在信息安全中的应用举例

数论在信息安全中的应用是新课标选修系列3中“信息安全与密码”模块的基本内容之一,新课标明确要求“了解通讯安全的有关概念(如明文、密文、密钥)和通讯安全中的基本问题(如保密、数字签名、密钥管理、分配和共享);理解公钥体制(单向函数概念),以及加密和数字签名的方法(基于大数分解的RSA方案)”.全文将对RSA方案的相关原理和算法进行探讨,并提供一个模拟RSA方案加密、解密过程的实验方案.1密码学基本概念简介1.1密码体制定义1一个密码体制是满足下列条件的五元组(M,C,K,E,D);(1)M表示所有可能的明文组成的有限集,即明文空间.(2)C…     

数学公式中的精品——话说公式e^in＋1=O

在数学天地里,有数不尽的公式,其中不少是非常重要的甚至是美妙绝伦的,它们构成了数学世界一幅幅绚丽多姿的风景,令众多数学爱好心醉神迷．比如名数学家陈省身教授在一篇章中谈道：“当代有名的数论大家A．塞尔伯格(Selberg)曾经说,他喜欢数学的一个动因,是以下的公式：     

评华罗庚与王元合著的《数论在近似分析中的应用》

华罗庚教授与王元教授合著的《数论在近似分析中的应用》一书,作为纯粹数学与应用数学专著,已于1978年底由科学出版社出版。这本书(为简便起见,以下简称《数论应用》)是近二十年来用数论方法对于多维积分近似计算研究成果的一个系统全面的总结。 众所周知,1859年Riemann引进复变数函数的函数,1896年Hadamard用解析方法证实了素数定理,1921年Hardy—Littlewood创建“圆法”,以及特别是1937年N.M.BoB创造了“三角和方法”解决了大奇数Goldbach问题以来,分析方法渗透到     

非标准分析及其应用

本文简要介绍非标准分析基础及其在其他数学分支中的应用,特别是在组合数论中的应用.所介绍的基础包含数理逻辑常识、非标准模型构造以及非标准分析常用原理和性质.所介绍的应用包含随机微分方程强解的存在性、关于局部群的Hilbert第五问题、精确大数定律及其在经济学中的应用、组合数论中的和集现象、关于密度的Pl¨unnecke类型不等式和Freiman逆问题.     

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...     

Variations on seven points: An introduction to the scope and methods of coding theory and finite geometries

We use a simple example (the projective plane on seven points) to give an introductory survey on the problems and methods in finite geometries — an area of mathematics related to geometry, combinatorial theory, algebra, group theory and number theory as well as to applied mathematics (e.g., coding theory, information theory, statistical design of experiments, tomography, cryptography, etc.). As this list already indicates, finite geometries is — both from the point of view of pure mathematics and from that of applications related to computer science and communication engineering — one of the most interesting and active fields of mathematics. It is the aim of this paper to introduce the nonspecialist to some of these aspects.To Professor Günter Pickert on the occasion of his 65th birthday     

Packing,covering and tiling in two-dimensional spaces

Packing, covering and tiling is a fascinating subject in pure mathematics. It mainly deals with arrangement patterns and efficiencies of geometric objects. This subject has a long and rich history, even back to Kepler, Newton, Lagrange and Gauss. Inspired by its applications and with the help of computing methods, in recent years it has become a very active research area in mathematics once again. Most of the fundamental problems in this subject can be characterized as simple sounding but challenging. This subject has important applications in many other areas such as Number Theory, Logic, Complex Analysis, Optimization, Coding Theory, Crystallography, Material Science, Industry, and even Biology. In spite of the long history, many of its key problems are still open, even in the plane. The purpose of this paper is to present a comprehensive review for packing, covering and tiling in the two-dimensional spaces. We will focus on the key problems, the fundamental results, the creative ideas, some important applications, and some significant connections with other areas.     

Novel methods on comparing grey numbers

Uncertain decision-making is an important branch of decision-making theory. It is crucial to describe uncertain information, which determine the decision-making is effective or not. This paper first presents a brief survey of the existing methods on denoting uncertain information, such as fuzzy mathematics, stochastic and interval methods, analyzes the merits and demerits of these methods. Then the paper proposes a novel method grey systems theory to describe uncertain information and gives the novel definition of grey number on the basis of probability distribution. Subsequently a novel probability method on comparing grey numbers, especially discrete grey numbers and interval grey numbers, is studied. When an interval grey number satisfied to continuous uniform distribution, it will be degenerated into an interval number. Finally three numerical examples are investigated to demonstrate the effectiveness of the present method.     

Inverse Problems in Darboux’ Theory of Integrability

The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the case of prescribed integrating factors. A?number of relevant examples and applications is included.     

Discrete mathematics in the high school curriculum

In this paper we present some topics from the field of discrete mathematics which might be suitable for the high school curriculum. These topics yield both easy to understand challenging problems and important applications of discrete mathematics. We choose elements from number theory and various aspects of coding theory. Many examples and problems are included.     

广义Lebesgue-Ramanujan-Nagell方程研究的新进展

广义Lebesgue-Ramanujan-Nagell方程是数论中一类重要的Diophantine方程.本文介绍了此类方程的近期结果和尚未解决的问题.     

Linearization of holomorphic operator functions II

This paper is a continuation of [2]. It contains the proof of the main linearization theorem announced in [2], some modifications of the construction and some unsolved problems for the case of several variables. Without further explanation we shall use the notations of [2], and we shall continue the numbering of formulae, sections, theorems and lemmas as begun in [2].     

信息技术与大学数学课程整合的方式与理论探讨

信息技术与大学数学课程整合正成为当前我国信息技术教育乃至整个教育信息化进程中的一个热点问题,探讨其整合的方式及理论基础是非常必要的.本文首先结合大学数学案例探讨了整合的三种方式:动态的课堂演示型、单机的数学实验型和全交互的网络教学型,其次探讨了整合所依据的四种理论:传播理论、建构主义理论、教学设计理论和系统理论,最后提出了自己对信息技术与大学数学课程整合的一些思考.     

The many aspects of counting lattice points in polytopes

A wide variety of topics in pure and applied mathematics involve the problem of counting the number of lattice points inside a convex bounded polyhedron, for short called a polytope. Applications range from the very pure (number theory, toric Hilbert functions, Kostant’s partition function in representation theory) to the most applied (cryptography, integer programming, contingency tables). This paper is a survey of this problem and its applications. We review the basic structure theorems about this type of counting problem. Perhaps the most famous special case is the theory of Ehrhart polynomials, introduced in the 1960s by Eugène Ehrhart. These polynomials count the number of lattice points in the different integral dilations of an integral convex polytope. We discuss recent algorithmic solutions to this problem and conclude with a look at what happens when trying to count lattice points in more complicated regions of space.     

INVITED PAPER: Some mathematical aspects of security in the automated business environment

In the automated business world, personal contact and paper-basedtransactions are being replaced by digital electronic communications.This transition raises many new security issues, and has seenthe emphasis of cryptography broaden from secrecy to includemessage integrity, user verification, digital signature, andaccess control. In this paper, we consider some of the waysin which mathematics is playing a crucial role in securing theautomated business environment. We begin with a brief introduction to cryptography and the principlesupon which the design of cipher systems is based. We then considerstream ciphers and block ciphers in turn, discussing their respectiveuses, design requirements, and crypt-analysis. Here statisticalmethods are of prime importance, as well as algebraic methodsfor generating and combining bit strings and blocks. The section on public key systems includes a discussion of severalmathematical techniques that provide one way functions. Theseare functions that are easy to perform but difficult to invert,often based on some difficult mathematical problem, and areat the heart of public key systems. Two public key systems (RSAand El Gamal) are described, and the status of algorithms forsolving the difficult problems upon which they are based factoringand discrete logarithms is reviewed. Finally we discuss applications of digital-signature and user-verificationtechniques. These are becoming of increasing importance in today'sautomated business world.