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This 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 Details
Abstract : 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 Details
J. 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...

Description Details
In the 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. There are a few papers which are not directly related to but should fall within the scope of Smarandache type problems. They are 1. L. Liu and W. Zhou, On conjectures about the class number of binary quadratic forms; 2. W. Liang, An identity for Stirling numbers of the second kind; 3. Y. Wang and Z. Sheng, Two formulas for x^n in terms of Chebyshev polynomials. 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 Details
On Algebraic Multi-Vector Spaces Abstract A Smarandache multi-space is a union of n spaces A1;A2;An with some additional conditions hold. Combining these Smarandache multi-spaces with linear vector spaces in classical linear algebra, the conception of multi-vector spaces is introduced. Some characteristics of multi-vector spaces are obtained in this paper. Keywords Vector, multi-space, multi-vector space, dimension of a space. x1. Introduction These multi-spaces was introduced by Smarandache in under his idea of hybrid mathematics: combining different fields into a unifying field, which can be formally defined with mathematical words by the next definition. ...

Table of Contents Details
L. Mao : On Algebraic Multi-Vector Spaces 1 L. Liu and W. Zhou : On conjectures concerning class number of binary quadratic forms 7 W. Zhai and H. Liu : On square-free primitive roots mod p 15 X. Pan : A new limit theorem involving the Smarandache LCM sequence 20 M. Le : The Smarandache Perfect Numbers 24 N. Yuan : On the solutions of an equation involving the Smarandache dual function 27 J. Wang : Mean value of a Smarandache-Type Function 31 H. Yang and R. Fu : On the mean value of the Near Pseudo Smarandache Function 35 W. Liang : An Identity of Stirling Numbers of the Second Kind 40 R. Ma : On the F.Smarandache LCM Ratio Sequence 44 L. Mao : On Algebraic Multi-Ring Spaces 48 Y. Han : On the Product of the Square-free Divisor of a Natural Number 55 P. Zhang : Some identities on k-power complement 60 Y. Wang and Z. Sheng : Two Formulas for xn being Represented by Chebyshev Polynomials 64 X. Chen : Two Problems About 2-Power Free Numbers 70 X. Ma : The Asymptotic Formula of P n·x log Pad(n) 72 Q. Tian : On the K-power free number sequence 77 C. Lv : On a generalized equation of Smarandache and its integer sol...