For any positive integer n, let φ(n) be the Euler function, and am(n) denotes the Smarandache m-th power residues function of n. The main purpose of this paper is using the elementary method to study the number of the solutions of the equation φ(n) = am(n), and give all solutions for this equation.

Research on Smarandache Problems in Number Theory , Vol. II

本书主要介绍Smarandache 函数与伪Smarandache 函数、几类Smarandache 序列及其它相关问题的 最新研究进展, 同时还介绍了一些其它数论问题的研 究成果. 本书各部分内容编排相对独立,有兴趣的读者 可以从阅读本书任何一个章节开始, 开拓读者视野，激发读者对Smarandache 相关问题的研究兴趣. This book mainly introduces the latest research progress of Smarandache functions and pseudo-Smarandache functions, several kinds of Smarandache sequences and other related problems. At the same time it also introduces some other study in number theory. This book commits itself to relatively independent content arrangement with each chapter and section, every reader who is interested in this book can read any chapter for the beginning. It could open up readers’ perspective; arouse readers to study Smarandache problems.

/pdf/smarandache-han-shu-ji-qi-xiang-guan-wen-ti-yan-jiu-vol-8-456ribysrs.pdf

Smarandache 函数 及其相关问题研究 Vol.8 / Research on Smarandache Functions and Other Related Problems, Vol. 8 [in Chinese Language Only]

The main purpose of this paper is to study the arithmetical properties of the primitive numbers of power p by using the elementary method, and give some interesting identities and asymptotic formulae.

Some arithmetical properties of primitive numbers of power p

Research on smarandache problems in number theory (vol. ii)

For any Positive Integer N, LetΦ(n)andS(n)Denote the Euler Function and the Smarandache Function of the Integer N.In this Paper, we Use the Elementary Number Theory Methods to Get the Solutions of the Equation Φ(n)=S(nk) if the K=9, and Give its All Positive Integer Solutions.

An Equation Involving the Smarandache Function

Let S(n), for n ∈ N+ denote the Smarandache function, then S(n) is defined as the smallest m ∈ N+, with n|m!. From the definition one can easily deduce that if n = p1 1 p α2 2 . . . p αk k is the canonical prime factorization of n, then S(n) = max{S(p αi i )}, where the maximum is taken over the i’s from 1 to k. This observation illustrates the importance of being able to calculate the Smarandache function for prime powers. This paper will be considering that process. We will give an upper and lower bound for S(pα) in Theorem 1.4. A recursive procedure of calculating S(pα) is then given in Proposition 1.8. Before preceeding we offer these trivial observations:

Bounding the Smarandache function

This observation illustrates the importance of being able to calculate the Smarandache function for prime powers. This paper will be considering that process.

/pdf/bounding-the-smarandache-function-2b59h4thh4.pdf

Bounding the Smarandache Function

This case study explores the adaptation of an online course to a hybrid course for honors students and provides a roadmap to help instructors prepare for teaching an honors course for the first time.. An examination of student perceptions of the hybrid course format reveals reports of more interaction and a higher quality of interaction with the students and instructor of this course than with students and instructors of other courses. Benefits and challenges of the hybrid format are discussed.

Internet and Society: A Hybrid Honors Course

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Bounding the Smarandache function

Bounding the Smarandache Function

Internet and Society: A Hybrid Honors Course