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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. -
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. -
Complete Padovan Sequences in Finite Fields Juan B. Gil
COMPLETE PADOVAN SEQUENCES IN FINITE FIELDS JUAN B. GIL Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601 MICHAEL D. WEINER Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601 CATALIN ZARA Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601 Abstract Given a prime p ≥ 5, and given 1 <κ<p − 1, we call a sequence (an)n in Fp a Φκ-sequence if it is periodic with period p − 1, and if it satisfies the linear recurrence an + an+1 = an+κ with a0 = 1. Such a sequence is said to be a complete Φκ-sequence if in addition {a0, a1,...,ap−2} = {1,...,p − 1}. For instance, every primitive root b mod p generates a complete Φκ-sequence n an = b for some (unique) κ. A natural question is whether every complete Φκ-sequence is necessarily defined by a primitive root. For κ = 2 the answer is known to be positive. In this paper we reexamine that case and investigate the case κ = 3 together with the associated cases κ = p − 2 and κ = p − 3. 1. Introduction For a prime number p ≥ 5 and a number κ ∈ {2,...,p − 2}, a sequence (an)n∈Z of elements of Fp is said to be a Φκ-sequence if a0 =1 and an+κ = an + an+1 for all n ∈ Z, (1) where “=” means (throughout this paper) equality in Fp. AΦκ-sequence is called complete if (an)n is periodic, with period p − 1, and (2) {a1,...,ap−2} = {2,...,p − 1}. (3) The case κ = 2 has been studied by Brison [1]. -
Completeness Properties of Perturbed Sequences Let S Be
JOURNAL OF NUMBER THEORY 13, 44 6-455 (1981) Completeness Properties of Perturbed Sequences STEFAN A . BURR Department of Computer Science, City Univerisity of New York, City College, New York, New York 10031 AND PAUL ERDŐS Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Communicated by J. W. S . Cassels Received January 18, 1980 If S is an arbitrary sequence of positive integers, let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if P(S) contains all large integers, and subcomplete if P(S) contains an infinite arithmetic progression . It is shown that any sequence can be perturbed in a rather moderate way into a sequence which is not subcomplete . On the other hand, it is shown that if S is any sequence satisfying a mild growth condition, then a surprisingly gentle perturbation suffices to make S complete in a strong sense . Various related questions are also considered . 1. INTRODUCTION Let S be an arbitrary sequence of positive integers . Define P(S) to be the set of all integers which are representable as a sum of dictinct terms of S . (Having distinct terms means having distinct indices, so that the values need not be distinct .) Call a sequence S complete if P(S) contains all sufficiently large integers. Often writers have called S complete only if P(S) contains all positive integers ; we will call such a sequence entirely complete . Considerable study, spanning thirty years, has been devoted to completeness and related properties . (See [1 ; 2, Chap. -
Florentin Smarandache
FLORENTIN SMARANDACHE SEQUENCES OF NUMBERS INVOLVED IN UNSOLVED PROBLEMS 1 141 1 1 1 8 1 1 8 40 8 1 1 8 1 1 1 1 12 1 1 12 108 12 1 1 12 108 540 108 12 1 1 12 108 12 1 1 12 1 1 1 1 16 1 1 16 208 16 1 1 16 208 1872 208 16 1 1 16 208 1872 9360 1872 208 16 1 1 16 208 1872 208 16 1 1 16 208 16 1 1 16 1 1 2006 Introduction Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. ( on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes/squares/cubes/factorials, almost primes, mobile periodicals, functions, tables, prime/square/factorial bases, generalized factorials, generalized palindromes, etc. ) have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, University of Craiova Library, and Arhivele Statului (Filiala Vâlcea, Romania). It is based on the old article “Properties of Numbers” (1975), updated many times. Special thanks to C. Dumitrescu & V. Seleacu from the University of Craiova (see their edited book "Some Notions and Questions in Number Theory", Erhus Univ. Press, Glendale, 1994), M. Perez, J. Castillo, M. Bencze, L. Tutescu, E, Burton who helped in collecting and editing this material. The Author 1 Sequences of Numbers Involved in Unsolved Problems Here it is a long list of sequences, functions, unsolved problems, conjectures, theorems, relationships, operations, etc. -
Expression of Fibonacci Sequences in Plants and Animals
Email:[email protected] EXPRESSION OF FIBONACCI SEQUENCES IN PLANTS AND ANIMALS Dr. K. MANIKYA KUMARI Head, Department of Botany St. Joseph’s College for Women (A) Visakhapatnam-530 004, Andhra Pradesh Email: [email protected] ABSTRACT The Fibonacci numbers are Nature's numbering system discovered by the Italian mathematician Leonardo Fibonacci. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. In the seeming randomness of the natural world, we can find many instances of mathematical order involving the Fibonacci numbers themselves and the closely related "Golden ratio" elements. The series of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on is known as the Fibonacci numbers or the Fibonacci sequence. The ratio between the numbers (1.618034) is frequently called the golden ratio or golden number. Fibonacci numbers appear in nature often enough to prove that they reflect some naturally occurring patterns which are commonly evident by studying the manner in which various plants and animals grow. The golden ratio is expressed in spiraling shells where areas of the shell's growth are mapped out in squares. Most of human body parts follow the numbers one, two, three and five. DNA molecules follow this sequence, measuring 34 angstroms long and 21 angstroms wide for each full cycle of the double helix. -
Arxiv:Math/0605348V1
COMPLETE PADOVAN SEQUENCES IN FINITE FIELDS JUAN B. GIL Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601 MICHAEL D. WEINER Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601 CATALIN ZARA Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601 Abstract Given a prime p ≥ 5, and given 1 <κ<p − 1, we call a sequence (an)n in Fp a Φκ-sequence if it is periodic with period p − 1, and if it satisfies the linear recurrence an + an+1 = an+κ with a0 = 1. Such a sequence is said to be a complete Φκ-sequence if in addition {a0, a1, . , ap−2} = {1,...,p − 1}. For instance, every primitive root b mod p generates a complete Φκ-sequence n an = b for some (unique) κ. A natural question is whether every complete Φκ-sequence is necessarily defined by a primitive root. For κ = 2 the answer is known to be positive. In this paper we reexamine that case and investigate the case κ = 3 together with the associated cases κ = p − 2 and κ = p − 3. 1. Introduction For a prime number p ≥ 5 and a number κ ∈ {2,...,p − 2}, a sequence (an)n∈Z of elements of Fp is said to be a Φκ-sequence if a0 =1 and an+κ = an + an+1 for all n ∈ Z, (1) arXiv:math/0605348v1 [math.NT] 12 May 2006 where “=” means (throughout this paper) equality in Fp. AΦκ-sequence is called complete if (an)n is periodic, with period p − 1, and (2) {a1, . , ap−2} = {2,...,p − 1}. -
Square Pyramidal Number 1 Square Pyramidal Number
Square pyramidal number 1 Square pyramidal number In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid. Formula Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30. The first few square pyramidal numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819 (sequence A000330 in OEIS). These numbers can be expressed in a formula as This is a special case of Faulhaber's formula, and may be proved by a straightforward mathematical induction. An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12). In modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L(P,t) of a polyhedron P is a polynomial that counts the number of integer points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid whose base is a unit square with integer coordinates, and whose apex is an integer point at height one above the base plane, is (t + 1)(t + 2)(2t + 3)/6 = P .[1] t + 1 Relations to other figurate numbers The square pyramidal numbers can also be expressed as sums of binomial coefficients: The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive triangular numbers. -
The General Formula to Find the Sum of First N Kth Dimensional S Sided Polygonal Numbers and a Simple Way to Find the N-Th Term of Polynomial Sequences
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 8, Issue 3 (Sep. - Oct. 2013), PP 01-10 www.iosrjournals.org The General formula to find the sum of first n Kth Dimensional S sided polygonal numbers and a simple way to find the n-th term of polynomial sequences Arjun. K Vth Semester,B.Sc Physics (Student), PTM Government College Perinthalmanna, University of Calicut, Kerala Abstract: Here a particular method is made to generate a A Single Formula to find the nth term and sum of n terms of first n Kth dimensional S sided Polygonal numbers. Keywords: Dimensional Polygonal Numbers, Polygonal Numbers,Square Numbers, Triangular Numbers, 3Dimensional Polygonal Numbers, I. Introduction In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots were thought of as alphas (units). A triangular number or triangle number, numbers the objects that can form an equilateral triangle. The nth term of Triangle number is the number of dots in a triangle with n dots on a side The sequence of triangular numbers are 1,3,6,10,15,21,28,36,45,55,.... A Square number , numbers the objects that can form a square. The nth square number is the number of dots in a square with n dots on a side.1,4,9,16,25,36,49,64,81.... A Pentagonal number , numbers the objects that can form a regular pentagon. The nth pentagonal number is the number of dots in a pentagon with n dots on a side. -
Square Pyramidal Number and Polynomials
Square Pyramidal Number and Polynomials Pawel Jan Piskorz The square pyramidal number represents the number of stacked spheres in a pyramid with a square base. The number of spheres in each layer of the pyramid is just a squared number of the length of the layer edge, and the square pyramidal number is the sum of the numbers of spheres in each layer. The first few square pyramidal numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240 appropriately for the sum of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 square pyramid layers. It follows from the definition of square pyramidal number P (N) where N is the number of layers of spheres in the square base pyramid that the square pyramidal number is equal to the sum of the squares of the consecutive natural numbers N X P (N) = k2 (1) k=1 In [1] the author presented a method of computing square pyramidal number as a function of the number N of the layers of square pyramid using the properties of the multiplication table. In this article we will present a simpler method of computing the square pyramidal number suggested in [2] in a footnote on page 81. Authors notice that (k + 1)3 = k3 + 3k2 + 3k + 1 (2) We may subtract k3 from the left side of the equation (2) and we receive (k + 1)3 − k3 = 3k2 + 3k + 1 (3) 1 We did that just to leave on the right side the term with k2. -
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)