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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. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
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. -
On Ternary Cubic Equation
ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 9, March 2014 ON TERNARY CUBIC EQUATION P. Thirunavukarasu, S. Sriram Assistant Professor -P.G & Research Department of Mathematics, Periyar E.V.R College Tiruchirappalli – 620 023, Tamilnadu, India Assistant Professor–P.G & Research Department of Mathematics, National College, Tiruchirappalli – 620 001, Tamilnadu, India equations are analyzed for the non-trivial integral Abstract we obtain the non-trivial integral solutions for the solutions. These results have motivated us to search for ternary cubic equation non-trivial integral solutions of their varieties of ternary cubic Diophantine equation. This paper concerns with the problem of determining non-trivial integral solutions of . the equation with three unknowns given by A few interesting relations among the solutions are presented. Index Terms: Ternary Cubic, integral solutions, Pell’s explicit integral solutions of the above equation are form, nasty numbers presented. A few interesting relations among the solutions Notations are obtained. Oblong number of rank n obln n n 1 II. METHOD OF ANALYSIS Tetrahedral number of rank The ternary cubic equation under consideration is n n12 n n Tet n 6 Triangular number of rank (1) Taking (2) Polygonal number of rank n with sides nm12 (3) m tnmn, 1 2 We get Square pyramidal number of rank n n1 2 n 1 (4) Again taking the transformation n Sqpn 6 Pentagonal pyramidal number of rank Star number = and apply in (4) we get (5) 2 Stella Octangula number = St. oct n 2 n 1 n It is well known that the general form of the integral solutions of the Pellian equation. -
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used. -
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 • , , •• , , , . -
International Journal of Engineering Research-Online a Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles Available Online
International journal of Engineering Research-Online A Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles available online http://www.ijoer.in RESEARCH ARTICLE ISSN: 2321-7758 AN INTERESTING TRANSCENDENTAL EQUATION WITH SIX UNKNOWNS 3 2 x2 y 2 xy X 2 Y 2 z 2 w 2 M.A.GOPALAN, S.VIDHYALAKSHMI, K.LAKSHMI Department of Mathematics, Shrimati Indira Gandhi College,Trichy-620002. Article Received: 11/11/2013 Article Revised on: 21/11/2013 Article Accepted on: 22/11/2013 ABSTRACT The transcendental equation with six unknowns involving surds represented by the 3 equation 2 x2 y 2 xy X 2 Y 2 z 2 w 2 is analyzed for its patterns of non-zero distinct integral solutions. Infinitely many non-zero integer sextuple (,,,,,)x y X Y z w satisfying the above equation are obtained. Three different patterns for finding the solution to the above problem are discussed. The relations between the solutions and the Polygonal numbers, Pyramidal numbers, Pronic number, Jacobsthal number, Jacobsthal-Lucas number, Octahedral number, kynea K.LAKSHMI number, Centered pyramidal numbers and Four Dimensional Figurative numbers are presented. KEYWORDS: Transcendental equation, integral solutions, the Polygonal numbers, Pyramidal numbers, Pronic number, Jacobsthal number, Jacobsthal-Lucas number, Octahedral number, kynea number, Centered pyramidal numbers and Four Dimensional Figurative numbers. M.Sc 2000 mathematics subject classification: 11D99 NOTATIONS: KYn -kynea number of rank Tmn, -Polygonal number of rank n with size m CPn,3 - Centered Triangular pyramidal number of m Pn - Pyramidal number of rank with size rank CP - Centered hexagonal pyramidal number of PRn - Pronic number of rank n n,6 rank OHn - Octahedral number of rank n F4,n ,3 - Four Dimensional Figurative number of SOn -Stella octangular number of rank rank whose generating polygon is a triangle S -Star number of rank n F4,n ,5 - Four Dimensional Figurative number of Jn -Jacobsthal number of rank of rank whose generating polygon is a pentagon. -
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 . -
Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. Pp. 71-80, 2011 Applied Mathematics
Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. pp. 71-80, 2011 Applied Mathematics On the Periods of Some Figurate Numbers Omur Deveci1, Erdal Karaduman2 1Department of Mathematics, Faculty of Science and Letters, Kafkas University, 36100 Kars, Turkiye e-mail: [email protected] Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkiye e-mail: [email protected] Received Date:December 7, 2010 Accepted Date: October 19, 2011 Abstract. The number which can be represented by a regular geometrical arrangement of equally spaced point is defined as figurate number. Each of polygonal, centered polygonal and pyramidal numbers is a class of the series of figurate numbers. In this paper, we obtain the periods of polygonal, centered polygonal and pyramidal numbers by reducing each element of these numbers modulo m. Key words: Period, polygonal number, centered polygonal number, pyramidal number. 2000 Mathematics Subject Classification: 11B75, 11B50. 1. Introduction The figurate numbers have a very important role to solve some problems in num- ber theory and to determine speciality of some numbers, see for example, [8,9]. The polygonal numbers, the centered polygonal numbers, the pyramidal num- bers and their properties have been studied by some authors, see for example, [1,3,6,15]. The study of Fibonacci numbers by reducing modulo m began with the earlier work of Wall [13] where the periods of Fibonacci numbers according to modulo m were obtained. The theory is expanded to 3-step Fibonacci se- quence by Özkan, Aydin and Dikici [11]. Lü and Wang [10] contributed to study of the Wall number for the k step Fibonacci sequence. -
On the Integer Solutions of the Pell Equation
International Journal of Engineering Science Invention ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org || Volume 2 || Issue 12 || December 2013 || PP.01-03 On the integer solutions of the Pell equation M.A.Gopalan1,V.Sangeetha2, Manju Somanath3 1Professor,Dept.of Mathematics,Srimathi Indira Gandhi College,Trichy-620002,India. 2Asst.Professor,Dept.of Mathematics,National College,Trichy-620001,India. 3Asst.Professor,Dept.of Mathematics,National College,Trichy-620001,India. ABSTRACT: The binary quadratic diophantine equation represented by is considered. A method for obtaining infinitely many non-zero distinct integer solutions of the Pell equation considered above is illustrated. A few interesting relations among the solutions and special figurate numbers are presented.Recurrence relations on the solutions are given. KEYWORDS - Pell equation, binary quadratic diophantine equation, integer solutions. I. INTRODUCTION It is well known that the Pell equation (D > 0 and square free) has always positive integer solutions.When , the Pell equation may not have any positive integer solutions.For example, the equations and have no integer solutions. When k is a positive integer and , positive integer solutions of the equations and have been investigated by Jones in [1].In [2-11], some specific Pell equation and their integer solutions are considered.In [12], the integer solutions of the Pell equation has been considered. In [13], the Pell equation is analysed for the integer solutions. This communication concerns with the Pell equation and infinitely many positive integer solutions are obtained.A few interesting relations among the solutions and special figurate numbers are presented.Recurrence relations on the solutions are given. -
Observations on Icosagonal Pyramidal Number
International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 2, Issue 7 (July 2013), PP. 32-37 www.irjes.com Observations On Icosagonal Pyramidal Number 1M.A.Gopalan, 2Manju Somanath, 3K.Geetha, 1Department of Mathematics, Shrimati Indira Gandhi college, Tirchirapalli- 620 002. 2Department of Mathematics, National College, Trichirapalli-620 001. 3Department of Mathematics, Cauvery College for Women, Trichirapalli-620 018. ABSTRACT:- We obtain different relations among Icosagonal Pyramidal number and other two, three and four dimensional figurate numbers. Keyword:- Polygonal number, Pyramidal number, Centered polygonal number, Centered pyramidal number, Special number MSC classification code: 11D99 I. INTRODUCTION Fascinated by beautiful and intriguing number patterns, famous mathematicians, share their insights and discoveries with each other and with readers. Throughout history, number and numbers [2,3,7-15] have had a tremendous influence on our culture and on our language. Polygonal numbers can be illustrated by polygonal designs on a plane. The polygonal number series can be summed to form “solid” three dimensional figurate numbers called Pyramidal numbers [1,4,5 and 6] that can be illustrated by pyramids. In this communication we 32 20 65n n n deal with Icosagonal Pyramidal numbers given by p and various interesting relations n 2 among these numbers are exhibited by means of theorems involving the relations. Notation m pn = Pyramidal number of rank n with sides m tmn, = Polygonal number of rank n with sides m jaln = Jacobsthal Lucas number ctmn, = Centered Polygonal number of rank n with sides m m cpn = Centered Pyramidal number of rank n with sides m gn = Gnomonic number of rank n with sides m pn = Pronic number carln = Carol number mern = Mersenne number, where n is prime culn = Cullen number Than = Thabit ibn kurrah number II.