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 Book Id: WPLBN0002828532 Subjects: Non Fiction, Education, Number Theory ► Abstract Description DetailsThis book contains 23 papers, most of which were written by participants to the fifth International Conference on Number Theory and Smarandache Notions held in Shangluo University, China, in March, 2009. In this Conference, several professors gave a talk on Smarandache Notions and many participants lectured on them both extensively and intensively. All these papers are original and have been refereed. The themes of these papers range from the mean value or hybrid mean value of Smarandache type functions, the mean value of some famous number theroretic functions acting on the Smarandache sequences, to the convergence property of some infinite series involving the Smarandache type sequences. ... Excerpt Details3. Remarks Sandor [2] has considered the problem of finding the S-perfect and completely S-perfect numbers, but his proof is not complete. He has proved that the only S-perfect of the form n = p q is n = 6 and there is no S-perfect number of the form n = 2kq where k ¸ 2 is an integer and q is an odd prime. On the other hand, Theorem 2.1 gives all the S-perfect numbers. Again, Sandor only proved that, the only completely S-perfect number of the form n = p2q is n = 28, and all completely S-perfect numbers are given by Theorem 2.2. Theorem 2.1 and Theorem 2.2 ¯nd respectively the S-perfect and completely S-perfect numbers when S(1) = 1. The situation is quite different if one adopts the convention that S(0) = 1. In the latter case, as has been proved by Gronas [3], all completely S-perfect numbers are n = p(prime), 9, 16, 24. All that is known about the S-perfect numbers is that, among the ¯rst 106 numbers, n = 12 is the only S-perfect number (see Ashbacher [4]). In exactly the same way, the Z-perfect and completely Z-perfect numbers may be defined. Thus, given an integer n, 1. ... Table of Contents DetailsJ. Wang : An equation related to the Smarandache power function 1 X. Lu and J. Hu : On the F.Smarandache 3n-digital sequence 5 B. Cheng : An equation involving the Smarandache double factorial function and Euler function 8 A. A. K. Majumdar : S-perfect and completely S-perfect numbers 12 B. Zhao and S. Wang: Cyclic dualizing elements in Girard quantales 21 P. Chun and Y. Zhao : On an equation involving the Smarandache function and the Dirichlet divisor function 27 F. Li and Y. Wang : An equation involving the Euler function and the Smarandache m-th power residues function 31 H. Gunarto and A. Majumdar : On numerical values of Z(n) 34 K. Nagarajan, etc.: M-graphoidal path covers of a graph 58 H. Liu : On the cubic Gauss sums and its fourth power mean 68 A. A. K. Majumdar : On the dual functions Z¤(n) and S¤(n) 74 S. S. Gupta : Smarandache sequence of Ulam numbers 78 L. Li : A new Smarandache multiplicative function and its arithmetical properties 83 Y. Yang and X. Kang : A predator-prey epidemic model with infected predator 86 R. Fu and H. Yang : An equation involving the Lucas numbers 90 W. Yang : Two rings in IS-a...
 Date: 2013, Volume: 5 Book Id: WPLBN0002828166 Subjects: Non Fiction, Education, Number Theory ► Abstract Description Details前言 数论这门学科最初是从研究整数开始的, 所以叫做整数论. 后来整数 论又进一步发展, 就叫做数论了. 确切的说, 数论就是一门研究整数性质 的学科. 它是最古老的数学分支. 按照研究方法来说, 数论可以分成初等 数论, 解析数论, 代数数论, 超越数论, 计算数论, 组合数论等. Foreword Number theory, this discipline was originally started from the study integer, so called Number Theory. Later integer on further development of number theory called it. Rather, number theory is an integer nature of disciplines and it is the oldest branch of mathematics concerned by the study methods, can be divided into elementary number theory, number theory, analytic number theory, algebraic number theory, transcendental number theory, computational number theory, combinatorics number theory and so on.... Table of Contents Details第一章Smarandache函数. . . . . . . . . . . . 1 1.1 引言. . . . . . . . . . . . . . . . . . . . . . . 1 1.2 关于F.Smarandache可乘数函数的一类均值. . . . . . 1 1.3 Smarandache函数值的分布. . . . . . . . . . . . . 5 1.3.1 几个引理. . . . . . . . . . . . . . . . . . . 6 1.3.2 证明. . . . . . . . . . . . . . . . . . . . . 7 1.4 Smarandache函数df (n) 的均值. . . . . . . . . . . . 9 1.5 关于F.Smarandache LCM 函数以及它的主值. . . . . . 12 1.6 Smarandache Pierced 链. . . . . . . . . . . . . . . 16 1.7 Smarandache 函数的几个相关结论. . . . . . . . . . 18 1.7.1 关于Smarandache 函数的一个等式. . . . . . . . 18 1.7.2 关于文章\一个新的算术函数的主值"的一些注释. . 20 1.7.3 Smarandache 函数的一个推广. . . . . . . . . . 23 1.7.4 关于F.Smarandache函数及其k次补数. . . . . . . 27 1.7.5 关于F.Smarandache函数的奇偶性. . . . . . . . 32 第二章伪Smarandache 函数. . . . . . . . . . . . 36 2.1 伪Smarandache 函数的定义及性质. . . . . . . . . . 36 2.2 关于伪Smarandache函数的几个定理. . . . . . . . . 38 2.3 关于伪Smarandache 函数的几个方程. . . . . . . . . 40 2.3.1 一个与Smarandache函数有关的函数方程及其正整 数解. . . . . . . . . . . . . . . . . . . . . 41 2.3.2 一个包含伪Smarandache函数及其对偶函数的方程. 42 2.3.3 一个包含伪Smarandache 函数及Smarandache 可乘 函数的方程. . . . . . . . . . . . . . . . . . 45 2.4 伪Smarandache函数的...
 Date: 2013, Volume: 4 Book Id: WPLBN0002828164 Subjects: Non Fiction, Education, Number Theory ► Abstract Description Details前言 数论这门学科最初是从研究整数开始的, 所以叫做整数论. 后来整数 论又进一步发展, 就叫做数论了. 确切的说, 数论就是一门研究整数性质 的学科. 在我国, 数论也是发展最早的数学分支之一. 许多著名的数学著 作中都有关于数论内容的论述, 比如求最大公约数、勾股数组、某些不 定方程整数解的问题等等... Foreword Number theory, this discipline was originally started from the study integer, so called Number Theory. Later integer on further development of number theory called it. Rather, number theory is an integer nature of Discipline in our country, the development of number theory is one of the oldest branches of mathematics and many well-known mathematical forward work on number theory in both the content of discourse, such as the common denominator, Pythagorean, some do not Equation given integer solution problems, and so….... Table of Contents Details第一章Smarandache函数的问题及其新进展1 1.1 引言. . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Smarandache非构造序列. . . . . . . . . . . . . . 1 1.3 Smarandache数字和. . . . . . . . . . . . . . . . 2 1.4 Smarandache数字乘积. . . . . . . . . . . . . . . 2 1.5 Smarandache Pierced链. . . . . . . . . . . . . . . 3 1.6 Smarandache因子乘积. . . . . . . . . . . . . . . 4 1.7 Smarandache真因子乘积. . . . . . . . . . . . . . 5 1.8 Smarandache平方补数. . . . . . . . . . . . . . . 6 1.9 Smarandache立方补数. . . . . . . . . . . . . . . 7 1.10 Smarandache广义剩余序列. . . . . . . . . . . . . 7 1.11 Smarandache素数列. . . . . . . . . . . . . . . . 8 1.12 Smarandache平方列. . . . . . . . . . . . . . . . 13 1.13 Smarandache素数可加补数. . . . . . . . . . . . . 15 1.14 Smarandache函数S(n) . . . . . . . . . . . . . . . 19 1.15 Smarandache双阶乘函数. . . . . . . . . . . . . . 31 1.16 Smarandache商函数. . . . . . . . . . . . . . . . 42 1.17 Smarandache p次幂原函数. . . . . . . . . . . . . . 43 1.18 第一类伪Smarandache素数. . . . . . . . . . . . . 43 1.19 第一类伪Smarandache平方数. . . . . . . . . . . . 44 1.20 Goldbach-Smarandache序列. . . . . . . . . . . . . 46 1.21 Vinogradov-Smarandache序列. . . . . . . . . . . . 46 ...
 Date: 2013, Volume: 2 Book Id: WPLBN0002828158 Subjects: Non Fiction, Education, Number Theory ► Abstract Description DetailsThis book includes part of the research results about the Smarandache problems written by Chinese scholars at present, and its main purpose is to introduce various results about the Smarandache problems, such as Smarandache function and its asymptotic properties, series convergence, solutions about special equations. At the same time, we put forward to some new interesting problems either in order to research further. We hope this booklet will guide and inspire readers to these fields.... Excerpt Details前言 数论这门学科最初是从研究整数开始的, 所以叫整数数论. 后来整数 数论又进一步发展, 就叫做数论了. 确切地说, 数论就是一门研究整数性 质的学科. 数论和几何学一样, 是古老的数学分支. 数论在数学中的地位是特殊的, 高斯曾经说过:“数学是科学的皇后, 数论是数学中的皇冠”. 虽然数论中的许多问题在很早就开始了研究, 并得到了丰硕的成果, 但是至今仍有许多被数学家称之为“皇冠上的明 珠”的悬而未解的问题等待人们去解决. 正因如此, 数论才能不断地充 实和发展, 才能既古老又年轻, 才能始终活跃在数学领域的前沿. Foreword Number theory, this discipline was originally started from the study integer, so called integer number theory. Later integer further development of number theory, number theory called up. Rather, number theory is an integer of study qualitative disciplines. Number theory and geometry, is an ancient branch of mathematics. Number theory in mathematics position is special, Gauss once said: "Mathematics is the queen of sciences, number theory is the mathematics of the crown. "Although many of the problems in number theory began very early in the research, And has been fruitful, but there are still many of the mathematicians call "crown Ming Pearl "of unsolved problems waiting to be solved for this reason, number theory can continue to charge Real and development in order to both old and young, can always active in the forefront of the field of mathematics.... Table of Contents Details目录 第一章Smarandache 函数1 1.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 S(n) 函数和d(n) 函数的混合均值. . . . . . . . . . . . 4 1.3 关于F.Smarandache 函数S(mn) 的渐近性质. . . . . . . . 6 1.4 复合函数S(Z(n)) 的均值. . . . . . . . . . . . . . . . 7 1.5 是否为整数的问题. . . . . . . . . . . . . . . 10 1.6 关于函数S(n) 的一个方程. . . . . . . . . . . . . . . 13 1.7 关于函数S(nk) 的一个方程. . . . . . . . . . . . . . . 15 1.8 关于Smarandache 函数值的分布. . . . . . . . . . . . . 17 1.9 S(ak(n)) 函数的值分布. . . . . . . . . . . . . . . . . 21 1.10 两个包含Smarandache 函数的方程. . . . . . . . . . . . 25 1.11 S(n) 函数及其均值. . . . . . . . . . . . . . . . . . 27 第二章Smarandache 对偶函数 . . . . . . . . . . . . . . . . .30 2.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Smarandache 对偶函数的渐近公式. . . . . . . . . . . . 30 2.3 关于Smarandache 对偶函数的一个方程. . . . . . . . . . 33 2.4 关于Smarandache 对偶函数S¤¤(n) . . . . . . . . . . . 37 2.5 一个包含SM(n) 函数的方程. . . . . . . . . . . . . . 40 2.6 一个包含Smarandache 对偶函数的方程. . . . . . . . . . 44 第三章关于SL(n) 函数及其对偶函数的性质 . . . . . . . . . . . . . . . . .48 3.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 SL(n) 函数的渐近公式. . . . . . . . . . . . . . ....
 Date: 2013, Volume 3, No. 1, 2007 Book Id: WPLBN0002828572 Subjects: Non Fiction, Science, Algebra ► Abstract Description DetailsThis issue of the journal is devoted to the proceedings of the third International Conference on Number Theory and Smarandache Problems. The conference was a great success and will give a strong impact on the development of number theory in general and Smarandache problems in particular. In this volume we assemble not only those papers which were presented at the conference but also those papers which were submitted later and are concerned with the Smarandache type problems or other mathematical problems. Other papers are concerned with the number-theoretic Smarandache problems and will enrich the already rich stock of results on them. Readers can learn various techniques used in number theory and will get familiar with the beautiful identities and sharp asymptotic formulas obtained in the volume.... Excerpt DetailsAbstract : Let k be any ¯xed positive integer, n be any positive integer, Sk(n) denotes the smallest positive integer m such that m! is divisible by kn: In this paper, we use the elementary methods to study the asymptotic properties of Sk(n), and give an interesting asymptotic formula for it. Keywords : F. Smarandache problem, primitive numbers, asymptotic formula. ... Table of Contents DetailsJ. Wang : Cube-free integers as sums of two squares 1 G. Liu and H. Li : Recurrences for generalized Euler numbers 9 H. Li and Q. Yang : Some properties of the LCM sequence 14 M. Liu : On the generalization of the primitive number function 18 Z. Lv : On the F. Smarandache LCM function and its mean value 22 Q. Wu : A conjecture involving the F. Smarandache LCM function 26 S. Xue : On the Smarandache dual function 29 N. Yuan : A new arithmetical function and its asymptotic formula 33 A. Muktibodh, etc. : Sequences of pyramidal numbers 39 R. Zhang and S. Ma : An e±cient hybrid genetic algorithm for continuous optimization problems 46 L. Mao : An introduction to Smarandache multi-spaces and mathematical combinatorics 54 M. Selariu : Smarandache stepped functions 81 X. Zhang and Y. Zhang : Sequences of numbers with alternate common di®erences 93 Y. Zhang : On the near pseudo Smarandache function 98 A. Muktibodh : Smarandache mukti-squares 102...
 Date: 2013, Volume: Second Edition Book Id: WPLBN0002828178 Subjects: Non Fiction, Education, Geometry ► Abstract Description DetailsIn The 2nd Conference on Combinatorics and Graph Theory of China (Aug. 16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathematical sciences (abbreviated to CC Conjecture), i.e., a mathematical science can be reconstructed from or made by combinatorialization, implicated in the foreword of Chapter 5 of my book Automorphism groups of Maps, Surfaces and Smarandache Geometries (USA, 2005). This conjecture is essentially a philosophic notion for developing mathematical sciences of 21st century, which means that we can combine different fields into a union one and then determines its behavior quantitatively. It is this notion that urges me to research mathematics and physics by combinatorics, i.e., mathematical combinatorics beginning in 2004 when I was a post-doctor of Chinese Academy of Mathematics and System Science. It finally brought about me one self-contained book, the first edition of this book, published by InfoQuest Publisher in 2009. This edition is a revisited edition, also includes the development of a few topics discussed in the first edition.... Excerpt Details1.5 ENUMERATION TECHNIQUES 1.5.1 Enumeration Principle. The enumeration problem on a finite set is to count and find closed formula for elements in this set. A fundamental principle for solving this problem in general is on account of the enumeration principle: For finite sets X and Y , the equality |X| = |Y | holds if and only if there is a bijection f : X → Y . Certainly, if the set Y can be easily countable, then we can find a closed formula for elements in X.... Table of Contents DetailsContents Preface to the Second Edition . . . . . . . . . . . . . . . . . . . i Chapter 1. Combinatorial Principle with Graphs . . . . . . . . . . 1 1.1 Multi-sets with operations. . . . . . . . . . . . . . . . . . . . .2 1.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Multi-Set . . . . . . . . . . . . . . . . . . . . . . . . . .8 1.2 Multi-posets . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Partially ordered set . . . . . . . . . . . . . . . . . . . . .11 1.2.2 Multi-Poset . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Countable sets . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Countable set . . . . . . . . . . . . . . . . . . . . 16 1.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.1 Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . .18 1.4.2 Subgraph . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.3 Labeled graph. . . . . . ...