DOCSLIB.ORG

Chinasmconf5.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chinasmconf5.Pdf 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.
Load more

Recommended publications
  • Generalizing Benford's Law Using Power Laws
    Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 970284, 10 pages doi:10.1155/2009/970284 Research Article Generalizing Benford’s Law Using Power Laws: Application to Integer Sequences Werner Hurlimann¨ Feldstrasse 145, CH-8004 Zurich,¨ Switzerland Correspondence should be addressed to Werner Hurlimann,¨ [email protected] Received 25 March 2009; Revised 16 July 2009; Accepted 19 July 2009 Recommended by Kenneth Berenhaut Many distributions for first digits of integer sequences are not Benford. A simple method to derive parametric analytical extensions of Benford’s law for first digits of numerical data is proposed. Two generalized Benford distributions are considered, namely, the two-sided power Benford TSPB distribution, which has been introduced in Hurlimann¨ 2003, and the new Pareto Benford PB distribution. Based on the minimum chi-square estimators, the fitting capabilities of these generalized Benford distributions are illustrated and compared at some interesting and important integer sequences. In particular, it is significant that much of the analyzed integer sequences follow with a high P-value the generalized Benford distributions. While the sequences of prime numbers less than 1000, respectively, 10 000 are not at all Benford or TSPB distributed, they are approximately PB distributed with high P-values of 93.3% and 99.9% and reveal after a further deeper analysis of longer sequences a new interesting property. On the other side, Benford’s law of a mixing of data sets is rejected at the 5% significance level while the PB law is accepted with a 93.6% P-value, which improves the P-value of 25.2%, which has been obtained previously for the TSPB law.
    [Show full text]
  • Some Results on Benford's Law and Ulam Sequences
    Some Results on Benford's Law and Ulam Sequences Nicholas Alvarez,´ Andrew Hwang, Aaron Kriegman Mentor: Jayadev Athreya September 28, 2017 Abstract In this paper, we give an overview on the ubiquitous nature of Benford's Law and cover principles of equidistribution to answer questions concerning under which conditions Benford's Law is satisfied. The paper expands to cover Benford's law for different bases, exponential sequences, recursive sequences, regular sequences, and certain Ulam sequences. By gaining a deeper insight of the criteria for sequences to satisfy Beford's Law, we were able to draw out conclusions about the possible linear growth of certain Ulam Sequences, particularly the standard (1,2) Ulam sequence. Furthermore, we establish an undiscovered, greater structure found within Ulam sequences, which sheds insight on the possibility of a formula to calculate all Ulam numbers. 1 Introduction The mysterious phenomena that characterizes the essence of Benfords Law lies in an observation of the frequency distribution of the leading digits in both very abstract and many real-life sets of numerical data (the leading digit of a number is its leftmost digit). It is natural to seek the criteria a set or sequence must satisfy in order to obey this intriguing frequency distribution. Many distinctly defined sets and sequences satisfy Benfords Law, and our group was able to find infinitely many more examples of sequences that satisfy this law. As we discover more of these criteria, we begin to uncover structures in seemingly unnatural sets. By running extensive code, we can take sets of numerical data and calculate the frequency of leading digits and observe whether these sets satisfy Benfords Law for an arbitrarily large number of terms.
    [Show full text]
  • Integer Sequences
    UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular
    [Show full text]
  • The Ulam Numbers up to One Trillion
    The Ulam Numbers up to One Trillion Philip Gibbs and Judson McCranie All Ulam numbers up to one trillion are computed using an efficient linear-time algorithm. We report on the distribution of the numbers including the positions of the largest gaps and we provide an explanation for the previously observed period of clustering. Introduction The Ulam numbers were introduced in 1964 by Stanislaw Ulam as a sequence of integers with a simple additive definition that provide an apparently pseudo-random behaviour [1]. The sequence of Ulam numbers begins with the two integers 푈1 = 1 and 푈2 = 2. Each subsequent Ulam number 푈푛 is the smallest integer that is the sum of two distinct smaller Ulam numbers in exactly one way. It continues 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, … In this study we have calculated all Ulam numbers less than one trillion. There are 73,979,274,540 Ulam numbers up to this limit. The Ulam numbers form an infinite sequence. In the worst case an Ulam number 푈푛 is the sum of 푈푛−1 and 푈푛−3 Therefore 푈푛 ≤ 푁푛−2 where {푁푛} are the numbers in Narayana’s cows sequence: 1,1,1,2,3,4,6,9,13,19,… with the recurrence relation 푁푛 = 푁푛−1 + 푁푛−3 [2]. A lower bound can be set by observing that given five consecutive numbers greater than five, at most two can be Ulam 푈 numbers so 푈 ≥ 2.5푛 − 7. These limits imply that the asymptotic density 퐷 = lim 푛 of Ulam 푛 푛→∞ 푛 numbers in the positive integers is constrained by 0 ≤ 퐷 ≤ 0.4.
    [Show full text]
  • Computing Integer Sequences: Filtering Vs Generation (Functional Pearl)
    Computing Integer Sequences: Filtering vs Generation (Functional Pearl) Ivano Salvo†∗, Agnese Pacificoz y Departement of Computer Science, Sapienza University of Rome z Departement of Mathematics, Sapienza University of Rome ∗ Corresponding author: [email protected] Abstract As a report of a teaching experience, we analyse Haskell programs computing two integer sequences: the Hamming sequence and the Ulam sequence. For both of them, we investigate two strategies of computation: the first is based on filtering out those natural numbers that do not belong to the sequence, whereas the second is based on the direct generation of num- bers that belong to the sequence. Advocating cross-fertilisation among ideas emerging when programming in different programming paradigms, in the background, we sketch out some considerations about corresponding C programs solving the same two problems. 1 Prologue Prof. Ivano Salvo (writing to a colleague): [. ] Teaching can be very boring. Starting to teach a new course, the first year, one decides what to do according to the course program. The second year, one tries to arrange things that did not work so well during the previous year, probably taking into account comments in student evaluation forms, or working on students' lack of preparation as highlighted by exams. The third year, one makes it perfect. At least from his or her point of view. And then. ? How one can save himself, and of course his students, from tedium? I think that you will agree with me that one possible therapy is transforming teaching into learning and/or \research". Here, \research" can mean “find new solutions to old problems", or just “find some amusing new examples", sometimes just searching throughout books or the web and adapt or re-think their solution for your course.
    [Show full text]
  • Module 2 Primes
    Module 2 Primes “Looking for patterns trains the mind to search out and discover the similarities that bind seemingly unrelated information together in a whole. A child who expects things to ‘make sense’ looks for the sense in things and from this sense develops understanding. A child who does not expect things to make sense and sees all events as discrete, separate, and unrelated.” Mary Bratton-Lorton Math Their Way 1 Video One Definition Abundant, Deficient, and Perfect numbers, and 1, too. Formulas for Primes – NOT! Group? Relatively Prime numbers Fundamental Theorem of Arithmetic Spacing in the number line Lucky numbers Popper 02 Questions 1 – 7 2 Definition What exactly is a prime number? One easy answer is that they are the natural numbers larger than one that are not composites. So the prime numbers are a proper subset of the natural numbers. A prime number cannot be factored into some product of natural numbers greater than one and smaller than itself. A composite number can be factored into smaller natural numbers than itself. So here is a test for primality: can the number under study be factored into more numbers than 1 and itself? If so, then it is composite. If not, then it’s prime A more positive definition is that a prime number has only 1 and itself as divisors. Composite numbers have divisors in addition to 1 and themselves. These additional divisors along with 1 are called proper divisors. For example: The divisors of 12 are {1, 2, 3, 4, 6, 12}. { 1, 2, 3, 4, 6} are the proper divisors.
    [Show full text]
  • Download PDF # Integer Sequences
    4DHTUDDXIK44 ^ eBook ^ Integer sequences Integer sequences Filesize: 8.43 MB Reviews Extensive information for ebook lovers. It typically is not going to expense too much. I discovered this book from my i and dad recommended this pdf to learn. (Prof. Gerardo Grimes III) DISCLAIMER | DMCA DS4ABSV1JQ8G > Kindle » Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular number, Multiplicative partition, Perrin number, Smooth number, Ulam number, Primorial, Lambek Moser theorem,
    [Show full text]
  • 2N for All N ≥ 4. 1.2. Prove That for Any Integ
    PUTNAM TRAINING PROBLEMS, 2013 (Last updated: December 3, 2013) Remark. This is a list of problems discussed during the training sessions of the NU Putnam team and arranged by subjects. The document has three parts, the first one contains the problems, the second one hints, and the solutions are in the third part. |Miguel A. Lerma Exercises 1. Induction. 1.1. Prove that n! > 2n for all n ≥ 4. 1.2. Prove that for any integer n ≥ 1, 22n − 1 is divisible by 3. 1.3. Let a and b two distinct integers, and n any positive integer. Prove that an − bn is divisible by a − b. 1.4. The Fibonacci sequence 0; 1; 1; 2; 3; 5; 8; 13;::: is defined as a sequence whose two first terms are F0 = 0, F1 = 1 and each subsequent term is the sum of the two n previous ones: Fn = Fn−1 + Fn−2 (for n ≥ 2). Prove that Fn < 2 for every n ≥ 0. 1.5. Let r be a number such that r + 1=r is an integer. Prove that for every positive integer n, rn + 1=rn is an integer. 1.6. Find the maximum number R(n) of regions in which the plane can be divided by n straight lines. 1.7. We divide the plane into regions using straight lines. Prove that those regions can be colored with two colors so that no two regions that share a boundary have the same color. 1.8. A great circle is a circle drawn on a sphere that is an \equator", i.e., its center is also the center of the sphere.
    [Show full text]
  • The Math Encyclopedia of Smarandache Type Notions [Vol. I
    Vol. I. NUMBER THEORY MARIUS COMAN THE MATH ENCYCLOPEDIA OF SMARANDACHE TYPE NOTIONS Vol. I. NUMBER THEORY Educational Publishing 2013 Copyright 2013 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599732527 ISBN: 978-1-59973-252-7 2 INTRODUCTION About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name), it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. Because this is too vast to be covered in one book, we divide encyclopedia in more volumes. In this first volume of encyclopedia we try to synthesize his work in the field of number theory, one of the great Smarandache’s passions, a surfer on the ocean of numbers, to paraphrase the title of the book Surfing on the ocean of numbers – a few Smarandache notions and similar topics, by Henry Ibstedt.
    [Show full text]
  • Subject Index
    Subject Index Many of these terms are defined in the glossary, others are defined in the Prime Curios! themselves. The boldfaced entries should indicate the key entries. γ 97 Arecibo Message 55 φ 79, 184, see golden ratio arithmetic progression 34, 81, π 8, 12, 90, 102, 106, 129, 136, 104, 112, 137, 158, 205, 210, 154, 164, 172, 173, 177, 181, 214, 219, 223, 226, 227, 236 187, 218, 230, 232, 235 Armstrong number 215 5TP39 209 Ars Magna 20 ASCII 66, 158, 212, 230 absolute prime 65, 146, 251 atomic number 44, 51, 64, 65 abundant number 103, 156 Australopithecus afarensis 46 aibohphobia 19 autism 85 aliquot sequence 13, 98 autobiographical prime 192 almost-all-even-digits prime 251 averaging sets 186 almost-equipandigital prime 251 alphabet code 50, 52, 61, 65, 73, Babbage 18, 146 81, 83 Babbage (portrait) 147 alphaprime code 83, 92, 110 balanced prime 12, 48, 113, 251 alternate-digit prime 251 Balog 104, 159 Amdahl Six 38 Balog cube 104 American Mathematical Society baseball 38, 97, 101, 116, 127, 70, 102, 196, 270 129 Antikythera mechanism 44 beast number 109, 129, 202, 204 apocalyptic number 72 beastly prime 142, 155, 229, 251 Apollonius 101 bemirp 113, 191, 210, 251 Archimedean solid 19 Bernoulli number 84, 94, 102 Archimedes 25, 33, 101, 167 Bernoulli triangle 214 { Page 287 { Bertrand prime Subject Index Bertrand prime 211 composite-digit prime 59, 136, Bertrand's postulate 111, 211, 252 252 computer mouse 187 Bible 23, 45, 49, 50, 59, 72, 83, congruence 252 85, 109, 158, 194, 216, 235, congruent prime 29, 196, 203, 236 213, 222, 227,
    [Show full text]
  • On Some Ramanujan Equations: Mathematical Connections with Various Topics Concerning Prime Numbers Theory, 흓, 휻(ퟐ) and Several Parameters of Particle Physics
    On some Ramanujan equations: mathematical connections with various topics concerning Prime Numbers Theory, 흓, 휻(ퟐ) and several parameters of Particle Physics. III Michele Nardelli1, Antonio Nardelli2 Abstract In this paper we have described and analyzed some Ramanujan equations. We have obtained several mathematical connections between some topics concerning Prime Numbers Theory, 휙, 휁(2) and various parameters of Particle Physics. 1 M.Nardelli studied at Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 - 80138 Napoli, Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” - Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy 2 A. Nardelli studies at the Università degli Studi di Napoli Federico II - Dipartimento di Studi Umanistici – Sezione Filosofia - scholar of Theoretical Philosophy 1 https://mobygeek.com/features/indian-mathematician-srinivasa-ramanujan-quotes-11012 We want to highlight that the development of the various equations was carried out according an our possible logical and original interpretation 2 We have already analyzed, in the part II of this paper, several Ramanujan formulas of the Manuscript Book 3. (page 6): Exact result: Decimal approximation: 0.0234624217108….. Series representations: 3 Integral representations: Page 6 Result: 1.0118002913779… And the following final expression: Result: 14.17407524806… 4 From (page 7) we obtain: Input interpretation: Result:
    [Show full text]
  • Chromatic Number of Graphs with Special Distance Sets, I
    Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 17 (2014). Number 1. pp. 135 – 160 c Journal “Algebra and Discrete Mathematics” Chromatic number of graphs with special distance sets, I V. Yegnanarayanan Communicated by D. Simson Dedicated to Prof. Dr. R. Balasubramanian, Director, IMSC, India on his 63rd Birth Day Abstract. Given a subset D of positive integers, an integer distance graph is a graph G(Z,D) with the set Z of integers as vertex set and with an edge joining two vertices u and v if and only if |u−v| ∈ D. In this paper we consider the problem of determining the chromatic number of certain integer distance graphs G(Z,D)whose distance set D is either 1) a set of (n + 1) positive integers for which the nth power of the last is the sum of the nth powers of the previous terms, or 2) a set of pythagorean quadruples, or 3) a set of pythagorean n-tuples, or 4) a set of square distances, or 5) a set of abundant numbers or deficient numbers or carmichael numbers, or 6) a set of polytopic numbers, or 7) a set of happy numbers or lucky numbers, or 8) a set of Lucas numbers, or 9) a set of Ulam numbers, or 10) a set of weird numbers. Besides finding the chromatic number of a few specific distance graphs we also give useful upper and lower bounds for general cases. Further, we raise some open problems. 1. Introduction The graphs considered here are mostly finite, simple and undirected. A k-coloring of a graph G is an assignment of k different colors to the vertices of G such that adjacent vertices receive different colors.
    [Show full text]