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

There are two current trends on-going in the teaching of mathematics, incorporating technology and involving undergraduates in research. In this paper we will combine the two trends. Most of the examples given will be from the field of number theory, however the methods described could easily be adapted to other branches of mathematics.

Technology a bridge to experimentation with open problems

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Patrick Mitchell

Author Tools

Chat about Author

Papers

Bounding the Smarandache function

Bounding the Smarandache Function

Technology a bridge to experimentation with open problems