Or Sequences of Polygonal Numbers) of Order R (Where R Is an Integer, R > 3) May Be Defined Recursively By
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Special Kinds of Numbers
Chapel Hill Math Circle: Special Kinds of Numbers 11/4/2017 1 Introduction This worksheet is concerned with three classes of special numbers: the polyg- onal numbers, the Catalan numbers, and the Lucas numbers. Polygonal numbers are a nice geometric generalization of perfect square integers, Cata- lan numbers are important numbers that (one way or another) show up in a wide variety of combinatorial and non-combinatorial situations, and Lucas numbers are a slight twist on our familiar Fibonacci numbers. The purpose of this worksheet is not to give an exhaustive treatment of any of these numbers, since that is just not possible. The goal is to, instead, introduce these interesting classes of numbers and give some motivation to explore on your own. 2 Polygonal Numbers The first class of numbers we'll consider are called polygonal numbers. We'll start with some simpler cases, and then derive a general formula. Take 3 points and arrange them as an equilateral triangle with side lengths of 2. Here, I use the term \side-length" to mean \how many points are in a side." The total number of points in this triangle (3) is the second triangular number, denoted P (3; 2). To get the third triangular number P (3; 3), we take the last configuration and adjoin points so that the outermost shape is an equilateral triangle with a side-length of 3. To do this, we can add 3 points along any one of the edges. Since there are 3 points in the original 1 triangle, and we added 3 points to get the next triangle, the total number of points in this step is P (3; 3) = 3 + 3 = 6. -
Mathematics 2016.Pdf
UNIVERSITY INTERSCHOLASTIC LEAGUE Making a World of Diference Mathematics Invitational A • 2016 DO NOT TURN THIS PAGE UNTIL YOU ARE INSTRUCTED TO DO SO! 1. Evaluate: 1 qq 8‚ƒ 1 6 (1 ‚ 30) 1 6 (A) qq 35 (B) 11.8 (C) qqq 11.2 (D) 8 (E) 5.0333... 2. Max Spender had $40.00 to buy school supplies. He bought six notebooks at $2.75 each, three reams of paper at $1.20 each, two 4-packs of highlighter pens at $3.10 per pack, twelve pencils at 8¢ each, and six 3-color pens at 79¢ each. How much money did he have left? (A) $4.17 (B) $5.00 (C) $6.57 (D) $7.12 (E) $8.00 3. 45 miles per hour is the same speed as _________________ inches per second. (A) 792 (B) 3,240 (C) 47,520 (D) 880 (E) 66 4. If P = {p,l,a,t,o}, O = {p,t,o,l,e,m,y}, and E = {e,u,c,l,i,d} then (P O) E = ? (A) { g } (B) {p,o,e} (C) {e, l } (D) {p,o,e,m} (E) {p,l,o,t} 5. An equation for the line shown is: (A) 3x qq 2y = 1 (B) 2x 3y = 2 (C) x y = q 1 (D) 3x 2y = qq 2 (E) x y = 1.5 6. Which of the following relations describes a function? (A) { (2,3), (3,3) (4,3) (5,3) } (B) { (qq 2,0), (0, 2) (0,2) (2,0) } (C) { (0,0), (2, q2) (2,2) (3,3) } (D) { (3,3), (3,4) (3,5) (3,6) } (E) none of these are a function 7. -
Newsletter 91
Newsletter 9 1: December 2010 Introduction This is the final nzmaths newsletter for 2010. It is also the 91 st we have produced for the website. You can have a look at some of the old newsletters on this page: http://nzmaths.co.nz/newsletter As you are no doubt aware, 91 is a very interesting and important number. A quick search on Wikipedia (http://en.wikipedia.org/wiki/91_%28number%29) will very quickly tell you that 91 is: • The atomic number of protactinium, an actinide. • The code for international direct dial phone calls to India • In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar) • The ISBN Group Identifier for books published in Sweden. In more mathematically related trivia, 91 is: • the twenty-seventh distinct semiprime. • a triangular number and a hexagonal number, one of the few such numbers to also be a centered hexagonal number, and it is also a centered nonagonal number and a centered cube number. It is a square pyramidal number, being the sum of the squares of the first six integers. • the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): 91 = 6 3+(-5) 3 = 43+33. • the smallest positive integer expressible as a sum of six distinct squares: 91 = 1 2+2 2+3 2+4 2+5 2+6 2. -
A Study of .Perfect Numbers and Unitary Perfect
CORE Metadata, citation and similar papers at core.ac.uk Provided by SHAREOK repository A STUDY OF .PERFECT NUMBERS AND UNITARY PERFECT NUMBERS By EDWARD LEE DUBOWSKY /I Bachelor of Science Northwest Missouri State College Maryville, Missouri: 1951 Master of Science Kansas State University Manhattan, Kansas 1954 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of .the requirements fqr the Degree of DOCTOR OF EDUCATION May, 1972 ,r . I \_J.(,e, .u,,1,; /q7Q D 0 &'ISs ~::>-~ OKLAHOMA STATE UNIVERSITY LIBRARY AUG 10 1973 A STUDY OF PERFECT NUMBERS ·AND UNITARY PERFECT NUMBERS Thesis Approved: OQ LL . ACKNOWLEDGEMENTS I wish to express my sincere gratitude to .Dr. Gerald K. Goff, .who suggested. this topic, for his guidance and invaluable assistance in the preparation of this dissertation. Special thanks.go to the members of. my advisory committee: 0 Dr. John Jewett, Dr. Craig Wood, Dr. Robert Alciatore, and Dr. Vernon Troxel. I wish to thank. Dr. Jeanne Agnew for the excellent training in number theory. that -made possible this .study. I wish tc;, thank Cynthia Wise for her excellent _job in typing this dissertation •. Finally, I wish to express gratitude to my wife, Juanita, .and my children, Sondra and David, for their encouragement and sacrifice made during this study. TABLE OF CONTENTS Chapter Page I. HISTORY AND INTRODUCTION. 1 II. EVEN PERFECT NUMBERS 4 Basic Theorems • • • • • • • • . 8 Some Congruence Relations ••• , , 12 Geometric Numbers ••.••• , , , , • , • . 16 Harmonic ,Mean of the Divisors •. ~ ••• , ••• I: 19 Other Properties •••• 21 Binary Notation. • •••• , ••• , •• , 23 III, ODD PERFECT NUMBERS . " . 27 Basic Structure • , , •• , , , . -
Notations Used 1
NOTATIONS USED 1 NOTATIONS ⎡ (n −1)(m − 2)⎤ Tm,n = n 1+ - Gonal number of rank n with sides m . ⎣⎢ 2 ⎦⎥ n(n +1) T = - Triangular number of rank n . n 2 1 Pen = (3n2 − n) - Pentagonal number of rank n . n 2 2 Hexn = 2n − n - Hexagonal number of rank n . 1 Hep = (5n2 − 3n) - Heptagonal number of rank n . n 2 2 Octn = 3n − 2n - Octagonal number of rank n . 1 Nan = (7n2 − 5n) - Nanogonal number of rank n . n 2 2 Decn = 4n − 3n - Decagonal number of rank n . 1 HD = (9n 2 − 7n) - Hendecagonal number of rank n . n 2 1 2 DDn = (10n − 8n) - Dodecagonal number of rank n . 2 1 TD = (11n2 − 9n) - Tridecagonal number of rank n . n 2 1 TED = (12n 2 −10n) - Tetra decagonal number of rank n . n 2 1 PD = (13n2 −11n) - Pentadecagonal number of rank n . n 2 1 HXD = (14n2 −12n) - Hexadecagonal number of rank n . n 2 1 HPD = (15n2 −13n) - Heptadecagonal number of rank n . n 2 NOTATIONS USED 2 1 OD = (16n 2 −14n) - Octadecagonal number of rank n . n 2 1 ND = (17n 2 −15n) - Nonadecagonal number of rank n . n 2 1 IC = (18n 2 −16n) - Icosagonal number of rank n . n 2 1 ICH = (19n2 −17n) - Icosihenagonal number of rank n . n 2 1 ID = (20n 2 −18n) - Icosidigonal number of rank n . n 2 1 IT = (21n2 −19n) - Icositriogonal number of rank n . n 2 1 ICT = (22n2 − 20n) - Icositetragonal number of rank n . n 2 1 IP = (23n 2 − 21n) - Icosipentagonal number of rank n . -
Figurate Numbers: Presentation of a Book
Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Figurate Numbers: presentation of a book Elena DEZA and Michel DEZA Moscow State Pegagogical University, and Ecole Normale Superieure, Paris October 2011, Fields Institute Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview 1 Overview 2 Chapter 1. Plane figurate numbers 3 Chapter 2. Space figurate numbers 4 Chapter 3. Multidimensional figurate numbers 5 Chapter 4. Areas of Number Theory including figurate numbers 6 Chapter 5. Fermat’s polygonal number theorem 7 Chapter 6. Zoo of figurate-related numbers 8 Chapter 7. Exercises 9 Index Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter 0. Overview Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview Figurate numbers, as well as a majority of classes of special numbers, have long and rich history. They were introduced in Pythagorean school (VI -th century BC) as an attempt to connect Geometry and Arithmetic. Pythagoreans, following their credo ”all is number“, considered any positive integer as a set of points on the plane. Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview In general, a figurate number is a number that can be represented by regular and discrete geometric pattern of equally spaced points. It may be, say, a polygonal, polyhedral or polytopic number if the arrangement form a regular polygon, a regular polyhedron or a reqular polytope, respectively. -
The Secret Code of God
The Secret Code of God Abul Hassan Copy Rights and 19 19 – The Secret Code of God By: Abul Hassan 1st Edition: 2010 No Part of this book may be reproduced in any form or by any means electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher or author. However any one can download the electronic version of this book, i.e. free eBook from any website on the internet without our permission for the purposes of reading and research or from our website – www.ali-pi.com. Abul Hassan www.ali-pi.com [email protected] [email protected] Price: US$ 19.00/CAN$ 19.00 19 – The Secret Code of God Page 2 Contents and 19 Topics Page 1. Copy Rights and 19 2 2. Abul Hassan’s Desk and 19 5 3. Dedication and 19 7 4. Coding and 19 8 5. Religions of the world and 19 10 6. Bahai’s and 19 11 7. Hinduism and 19 12 8. Symbolism and 19 12 9. Judaism and 19 13 10. Other faiths and 19 16 11. Christianity and 19 17 12. Islam and 19 25 13. Security Code and 19 29 14. Holy Quran and 19 30 15. 99 Names of Allah and 19 73 16. Arabic Alphabets and 19 87 17. Everything and 19 90 18. Mathematics and 19 94 19. Mathematical Properties of 19 97 20. Prime Numbers and 19 102 21. Perfect Numbers and 19 111 22. Powerful Numbers and 19 115 23. Enneadecagon and 19 116 24. -
Perfect Numbers and Perfect Polynomials: Motivating Concepts from Kindergarten to College
Perfect Numbers and Perfect Polynomials: Motivating Concepts From Kindergarten to College A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Mathematical Science in the Graduate School of The Ohio State University By Hiba Abu-Arish, B.S. Graduate Program in Mathematics The Ohio State University 2016 Master's Examination Committee: Dr. Bart Snapp, Research Advisor Dr. Rodica Costin, Advisor Dr. Herb Clemens c Copyright by Hiba Abu-Arish 2016 Abstract In this thesis we will show that perfect numbers can be used as a medium for conveying fundamental ideas from school mathematics. Guided by the Common Core Standards, we present activities designed for students from pre-kindergarten through high school. Additionally, we show how perfect numbers can be used in college level courses. These activities aim for students to gain a deeper understanding of the different mathematical concepts that are related to perfect numbers. This will bring students closer to unsolved problems in mathematics. ii This is dedicated to all who helped and supported me during my college years, and who wished me success from the bottom of their hearts: my teachers, my colleagues, my friends and my family. In particular I would like to thank those who took care of my daughters and treated them with love in order for me to pass this stage of my life. iii Vita 2004 . .High School, Palestine 2004-2008 . B.C. Applied Mathematics, Palestine Polytechnic University. 2007-2008 . Teacher, University Graduates Union, Palestine 2014-present . .Graduate Teaching Associate, The Ohio State University. Fields of Study Major Field: Mathematics iv Table of Contents Page Abstract . -
Simulations Between Triangular and Hexagonal Number-Conserving Cellular Automata Katsunobu Imai, Bruno Martin
Simulations between triangular and hexagonal number-conserving cellular automata Katsunobu Imai, Bruno Martin To cite this version: Katsunobu Imai, Bruno Martin. Simulations between triangular and hexagonal number-conserving cellular automata. International Workshop on Natural Computing, Sep 2008, Yokohama, Japan. hal-00315932 HAL Id: hal-00315932 https://hal.archives-ouvertes.fr/hal-00315932 Submitted on 1 Sep 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Simulations between triangular and hexagonal number-conserving cellular automata Katsunobu Imai†‡ and Bruno Martin‡ †Graduate School of Engineering, Hiroshima University, Japan. ‡I3S, CNRS, University of Nice-Sophia Antipolis, France. [email protected],[email protected] Abstract. A number-conserving cellular automaton is a cellular au- tomaton whose states are integers and whose transition function keeps the sum of all cells constant throughout its evolution. It can be seen as a kind of modelization of the physical conservation laws of mass or energy. In this paper, we first propose a necessary condition for triangular and hexagonal cellular automata to be number-conserving. The local transi- tion function is expressed by the sum of arity two functions which can be regarded as ’flows’ of numbers. -
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number) -
Number 23 Is the Twelfth Odd Number, the Ninth Prime, and the Nineteenth Deficient Number
23 Twenty-Three XXIII ii ii iii iiii iiiii iiii iii Corresponding ordinal: twenty-third. The number 23 is the twelfth odd number, the ninth prime, and the nineteenth deficient number. The prime 23 remains prime when its digits are reversed in each of the first seven prime bases 2, 3, 5, 7, 11, 13, and 17. However 23 is 1419 in base 19 and 4119 = 77 is not a prime. The number 23 is the largest prime p less than a million that remains prime in each prime base less than p/2. The number 23 = 19 + 4 is the sum of a centered hexagonal number and a square, as you can see. As the sum of four or fewer squares: 23 = 12 + 22 + 32 + 32. As the sum of nine or fewer cubes: 23 = 7 13 + 2 23. This is the first number to · · require nine cubes. The only other such number is 239. No number requires more than nine cubes. As the difference of two squares: 23 = 122 112. The number 23 appears in only one Pythagorean Triple: [23, 264, 265]. As a sum of three odd primes: 23 = 3+3+17 = 3+7+13 = 5+5+13 = 5+7+11. The 23-digit number 11111111111111111111111 is a prime. A number whose digits are all 1 is called a repunit. The number 23 is the only prime p such that p! is p digits long. (Number Gossip and Prime Curios) Indeed, 23! = 25 852 016 738 884 976 640 000—count them. In fact, 23 is one of only four numbers n such that n! is n digits long. -
Chapter 1 Plane Figurate Numbers
December 6, 2011 9:6 9in x 6in Figurate Numbers b1260-ch01 Chapter 1 Plane figurate numbers In this introductory chapter, the basic figurate numbers, polygonal numbers, are presented. They generalize triangular and square numbers to any regular m-gon. Besides classical polygonal numbers, we consider in the plane cen- tered polygonal numbers, in which layers of m-gons are drown centered about a point. Finally, we list many other two-dimensional figurate numbers: pronic numbers, trapezoidal numbers, polygram numbers, truncated plane figurate numbers, etc. 1.1 Definitions and formulas 1.1.1. Following to ancient mathematicians, we are going now to consider the sets of points forming some geometrical figures on the plane. Starting from a point, add to it two points, so that to obtain an equilateral triangle. Six-points equilateral triangle can be obtained from three-points triangle by adding to it three points; adding to it four points gives ten-points triangle, etc. So, by adding to a point two, three, four etc. points, then organizing the points in the form of an equilateral triangle and counting the number of points in each such triangle, one can obtain the numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, . (Sloane’s A000217, [Sloa11]), which are called triangular numbers. 1 December 6, 2011 9:6 9in x 6in Figurate Numbers b1260-ch01 2 Figurate Numbers Similarly, by adding to a point three, five, seven etc. points and organizing them in the form of a square, one can obtain the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, .