RESEARCH ON NUMBER THEORY AND SMARANDACHE NOTIONS ˄PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON NUMBER THEORY AND SMARANDACHE NOTIONS˅ Edited by ZHANG WENPENG Department of Mathematics Northwest University Xi'an, P.R.China Hexis 2009 i This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 North Zeeb Road P.O. Box 1346, Ann Arbor, MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/search/basic Peer Reviewer: Shigreru Kanemitsu, Graduate School of Advanced Technology of Kinki, 11-6 Kayanomori, Iizuka 820-8555, Japan. Zhai Wenguang, Department of Mathematics, Shandong Teachers’ University, Jinan, Shandong, P.R.China. Guo Jinbao, College of Mathematics and Computer Science, Yanan University, Shaanxi, P.R.China. Copyright 2009 By Hexis and Zhang Wenpeng, and all authors for their own articles Many books can be downloaded from the following E-Libraray of Science: http: // www. gallup.unm.edu /~smarandache/eBooks-otherformats.htm ISBN: 9781599730882 Standard Address Number : 297-5092 Printed in the United States of America ii Preface This Book is devoted to the proceedings of the fifth International Conference on Number Theory and Smarandache Notions held in Shangluo during March 27-30, 2009. The organizers were myself and Professor Chao Li from Shangluo University. The conference was supported by Shangluo University and there were more than 100 participants. We had three foreign guests, Professor K.Chakraborty from India, Professor Imre Katai from Hungary, Professor S. Kanemitsu from Japan. The conference was a great success and will give a strong impact on the development of number theory in general and Smarandache Notions in particular. We hope this will become a tradition in our country and will continue to grow. And indeed we are planning to organize the sixth conference in coming March which will be held in Tianshui, a beautiful city of Gansu . 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 or other mathematical 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. Y. Wang, Smarandache sequence of ulam numbers; 2. H. Gunarto and A. A. K. Majumdar, On numerical values of Z(n); 3. K. Nagarajan, A. Nagarajan and S. Somasundaram, M-graphoidal path covers of a graph; etc. 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. Researchers can download books on the Smarandache notions from the following open source Digital Library of Science: www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm. Wenpeng Zhang iii Contents J. 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-algebras 98 W. Zhu : An identity related to Dedekind sums 102 X. Yuan : An equation involving the Smarandache dual function and Smarandache ceil function 106 Y. Wang : On the quadratic mean value of the Smarandache dual function S¤¤(n) 109 P. Xi and Y. Yi : Distribution of quadratic residues over short intervals 116 Y. Liu : On an equation involving Smarandache power function 120 S. Shang and G. Chen : A new Smarandache multiplicative function and its mean value formula 123 X. Li : The mean value of the k-th Smarandache dual function 128 iv The Fifth International Conference on Number Theory and Smarandache Notions The opening ceremony of the conference is occurred in Shangluo University (http://www.slxy.cn), which is attracting more and more people from other countries to study Chinese Calligraphy, Chinese Drawing and Chinese Culture. Professor Chao Li v Professor Nianliang Wang Professor Shigeru Kanemitsu vi Professor Wenpeng Zhang Professor Yongzhuang Chen vii viii An equation related to the Smarandache power function Jianping Wang College of Science, Chang'an University, Xi'an, 710064, P.R.China Abstract For any positive integer x, let '(x) and f(x) denote the Euler totient function and the Smarandache power function of x respectively. Let n be a ¯xed positive integer. Using elementary method, we prove that the equation f(xn) = '(x) has at least two positive integer solutions x, but it has only ¯nitely many positive integer solutions. Keywords Euler totient function, Smarandache power function, equation, solution. x1. Introduction and Results Let N + be the set of all positive integers. For any positive integer x, let P (x) denotes the set of distinct prime divisors of x, and let © ª f(x) = min m j m 2 N +; x j mm;P (m) = P (x) : (1) Then f(x) is called the Smarandache power function of x. Many properties of this arithmetical function are discussed, see references [1], [2], [3] and [4]. Let n be a positive integer. Let '(x) denotes the Euler totient function of x. In this paper, we consider the equation f(xn) = '(x); x 2 N +: (2) Recently, Chengliang Tian and Xiaoyan Li [4] gave all positive integer solutions of equation (2) for n = 1; 2 and 3. Simultaneously, they proposed the following: Conjecture. For any ¯xed positive integer n, the equation (2) has only ¯nitely many positive integer solutions x. In this paper, we use the elementary method to prove the following general results. Theorem 1. For any ¯xed positive integer n, let ½ µ ¶¾ 1 r = min m j m 2 N +; n · 2m¡1 1 ¡ : (3) m Then x = 2r is a solution of equation (2). Since x = 1 is a solution of (2), by Theorem 1, we get the following Corollary immediately. Corollary 1. For any ¯xed positive integer n, the equation (2) has at least two positive integer solutions. 2 Jianping Wang On the other hand, we can also obtain an explicit upper bound estimate for the solutions of equation (2) as follows: Theorem 2. All positive integer solutions x of equation (2) satisfy x < 256n2. From Theorem 2, we may immediately deduce that the above Conjecture is correct. That is, we have the following: Corollary 2. For any ¯xed positive integer n, the equation (2) has only ¯nite many positive integer solutions. x2. Proof of the theorem 1 For any positive integer m, let µ ¶ µ ¶ 1 2 g(m) = 2m¡1 1 ¡ ; h(m) = 2m¡2 1 ¡ : (4) m m Lemma 1. If m > 1, then h(m + 1) < g(m) < h(m + 2). 2 1 Proof. By (4), if h(m + 1) ¸ g(m), then 1 ¡ ¸ 1 ¡ . But, since m ¸ 2, it m + 1 m is impossible. So we have h(m + 1) < g(m). Similarly, we can prove that g(m) < h(m + 2). Lemma 1 is proved. Lemma 2. Let s be a positive integer. Then x = 2s is a solution of (2) if and only if s satis¯es h(s) < n · g(s): (5) Proof. Let x = 2s. Then we have '(x) = 2s¡1. By (4), if n · g(s), then sn · 2s¡1(s ¡ 1), whence we get xn j '(x)'(x): (6) On the other hand, if h(s) < n, then 2s¡2(s ¡ 2) < sn. It implies that xn - mm; m = 2t; t = 0; 1; ¢ ¢ ¢ ; s ¡ 2: (7) Therefore, by (1), we see from (6) and (7) that f(xn) = '(x) for x = 2s, namely, it is a positive integer solution of equation (2). By the same method, we can prove that if x = 2s is a solution of (2), then s satis¯es (5). This proves Lemma 1. Proof of Theorem 1. We see from (3) and (4) that g(r ¡ 1) < n · g(r). If n · h(r), then we have g(r ¡ 1) < h(r). However, since r ¸ 2, by Lemma 1, it is impossible. So we have h(r) < n · g(r). Thus, by Lemma 2, x = 2r is a solution of equation (2). This completes the proof of Theorem 1. An equation related to the Smarandache power function 3 x2. Proof of the theorem 2 Lemma 3. For any ¯xed positive integer n, the equation (2) has at most two positive integer solutions with the form x = 2s; s 2 N +: (8) Moreover, if (2) has exactly two solutions with the form (8), then x = 2r and 2r+1. Proof. We now assume that equation (2) has three solutions x = 2si (i = 1; 2; 3) with s1 < s2 < s3.