Palindromic Prime Pyramids
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Mathematical Circus & 'Martin Gardner
MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n. -
12. Versum Primes Class of Automorphic Quadratic Forms Is Also Infinite
12. Versum Primes class of automorphic quadratic forms is also infinite. Most persons interested in numbers consider the primes to be the most fascinating of all. We’ve That sounds impressive, but we have no idea what touched on versum primes in previous articles; it means, and we’re inclined to believe authorita- now it’s time to take a direct look at them. tive references that say the conjecture is unproved. We’ve seen versum primes as factors, and it’s You’ll recall that all palindromes with an clear that there are a lot of them, but how many and even number of digits are versum numbers, while what kinds of properties do they have? palindromes with an odd number of digits are versum only if the middle digit is even [3]. The smallest versum prime is the infamous 11 we encountered earlier [1]. It has the distinction of A survey of small palindromic primes and all being the only palindromic prime with an even the known large ones shows about half have an number of digits. We’ll explore palindromic primes even middle digit. So it’s reasonable to conjecture later in this article, but what about versum primes that if there are an infinite number of palindromic in general? They exist; the smallest non-palindro- primes, then there are an infinite number of versum mic versum prime is 241. The “big” question is are palindromic primes, and hence an infinite number there an infinite number of versum primes? That of versum primes. seems very likely, but there are a number of conjec- Nonpalindromic Versum Primes tures about the infinitude of various kinds of num- bers among primes that have defied the efforts of But what about nonpalindromic versum the best mathematicians. -
New Yorker Article About Yitang Zhang
Solving an Unsolvable Math Problem - The New Yorker http://www.newyorker.com/magazine/2015/02/02/pursuit-beautySave paper and follow @newyorker on Twitter Profiles FEBRUARY 2, 2015 ISSUE TABLE OF CONTENTS The Pursuit of Beauty Yitang Zhang solves a pure-math mystery. BY ALEC WILKINSON don’t see what difference it can make Unable to get an now to reveal that I passed high-school academic position, Zhang kept the books for a math only because I cheated. I could Subway franchise. add and subtract and multiply and PHOTOGRAPH BY PETER Idivide, but I entered the wilderness when BOHLER words became equations and x’s and y’s. On test days, I sat next to Bob Isner or Bruce Gelfand or Ted Chapman or Donny Chamberlain—smart boys whose handwriting I could read—and divided my attention between his desk and the teacher’s eyes. Having skipped me, the talent for math concentrated extravagantly in one of my nieces, Amie Wilkinson, a professor at the University of Chicago. From Amie I first heard about Yitang Zhang, a solitary, part-time calculus teacher at the University of New Hampshire who received several prizes, including a MacArthur award in September, for solving a problem that had been open for more than a hundred and fifty years. The problem that Zhang chose, in 2010, is from number theory, a branch of pure mathematics. Pure mathematics, as opposed to applied mathematics, is done with no practical purposes in mind. It is as close to art and philosophy as it is to engineering. “My result is useless for industry,” Zhang said. -
25 Primes in Arithmetic Progression
b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 SA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 K office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Ribenboim, Paulo. Title: Prime numbers, friends who give problems : a trialogue with Papa Paulo / by Paulo Ribenboim (Queen’s niversity, Canada). Description: New Jersey : World Scientific, 2016. | Includes indexes. Identifiers: LCCN 2016020705| ISBN 9789814725804 (hardcover : alk. paper) | ISBN 9789814725811 (softcover : alk. paper) Subjects: LCSH: Numbers, Prime. Classification: LCC QA246 .R474 2016 | DDC 512.7/23--dc23 LC record available at https://lccn.loc.gov/2016020705 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, SA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: [email protected] Printed in Singapore YingOi - Prime Numbers, Friends Who Give Problems.indd 1 22-08-16 9:11:29 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page v Qu’on ne me dise pas que je n’ai rien dit de nouveau; la disposition des mati`eres est nouvelle. -
Sum of the Reciprocals of Famous Series: Mathematical Connections with Some Sectors of Theoretical Physics and String Theory
1Torino, 14/04/2016 Sum of the reciprocals of famous series: mathematical connections with some sectors of theoretical physics and string theory 1,2 Ing. Pier Franz Roggero, Dr. Michele Nardelli , P.i. Francesco Di Noto 1Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 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 Abstract In this paper it has been calculated the sums of the reciprocals of famous series. The sum of the reciprocals gives fundamental information on these series. The higher this sum and larger numbers there are in series and vice versa. Furthermore we understand also what is the growth factor of the series and that there is a clear link between the sums of the reciprocal and the "intrinsic nature" of the series. We have described also some mathematical connections with some sectors of theoretical physics and string theory 2Torino, 14/04/2016 Index: 1. KEMPNER SERIES ........................................................................................................................................................ 3 2. SEXY PRIME NUMBERS .............................................................................................................................................. 6 3. TWIN PRIME 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) -
A Crossnumber Meta: Nose Functions
A CROSSNUMBER META: NOSE FUNCTIONS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 NOSE FUNCTIONS: A CROSSNUMBER: PAGE 1/3 In the clues below, n represents the number being clued and k and m represent unspecified positive integers. In neither case is there a connection between these values in different clues. Across 1. the smallest conductor of an elliptic curve over the rational numbers 3. the sum of the first k Sophie Germain primes 8. (the number of primes between two consecutive powers of 2) + (the floor of (k=e)k=e) 10. a divisor of 99999999 11. an even non-totient number 12. (a tribonacci number) + (the number of conjugacy classes in GL8(F2)) 13. 2n contains no zeros 14. (the average of the divisors of n is itself a divisor of n) + (a semiprime tribonacci number) 15. a palindrome with exactly 5 palindromic prime factors (counted with multiplicity) 16. a sum of four consecutive cubes 17. the kissing number in dimension 4 19. the number of permutations of length k with exactly 2 distinct cycle lengths (e.g. any transposition) 21. a palindromic even semiprime not expressible as the sum of two squares 22. a repdigit in base 10 that is also a repdigit when expressed in some smaller base larger than 1. Down 1. (a Sophie Germain prime equal to the sum of the first k Sophie Germain primes) · (the number of directed graphs with m unlabeled vertices and no self-loops or multiple edges in the same direction) 2. -
Riemann Shatters the Gordian Knot Thierno M
Riemann Shatters The Gordian Knot Thierno M. Sow To cite this version: Thierno M. Sow. Riemann Shatters The Gordian Knot. 2018. hal-01852157 HAL Id: hal-01852157 https://hal.archives-ouvertes.fr/hal-01852157 Preprint submitted on 31 Jul 2018 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. Copyright Riemann Shatters The Gordian Knot Thierno M. SOW∗ July 7, 2018 “Riemann plays dice.” Abstract Solving the soaring Riemann Hypothesis is equivalent to face the over- awing challenge of Alexander the Great to slicing up or to unravel the conundrum of the Gordian Knot. In this article, we will discuss about the sufficient conditions to prove the Riemann Hypothesis. Likewise, we shed the light on some related problems in number theory and Physics. Thereby, we will vet the proof by revealing how many conjectures are sewn on the edges of the Riemann Hypothesis like the abc-conjecture, the twin prime conjecture, the prime number theorem, the Legendre conjec- ture and beyond. Ultimately, what is peculiar in our research is the one sentence proof. Mathematics Subject Classification 2010 codes: Primary 11M26; Secondary 11A41 1 THE RIEMANN HYPOTHESIS PROOF All sciences lead to Riemann and the most elegant proof of the Riemann Hy- pothesis can be expressed as follows: there are infinitely many nontrivial zeros and they all have real part 1/2. -
Number Gossip About 10 Years Ago and at first I Uploaded It on My Personal Website [6]
Number Gossip Tanya Khovanova Department of Mathematics, MIT May 30, 2018 Abstract This article covers my talk at the Gathering for Gardner 2008, with some additions. 1 Introduction My pet project Number Gossip has its own website: http://www.numbergossip.com/, where you can plug in your favorite integer up to 9,999 and learn its properties. A behind-the-scenes program checks your number for 49 regular properties and also checks a database for unique properties I collected. 2 Eight The favorite number of this year’s Gathering is composite, deficient, even, odious, palindromic, powerful, practical and Ulam. It is also very cool as it has the rare properties of being a cake and a narcissistic number, as well as a cube and a Fibonacci number. And it also is a power of two. In addition, eight has the following unique properties: • 8 is the only composite cube in the Fibonacci sequence • 8 is the dimension of the octonions and is the highest possible dimension of a normed division algebra • 8 is the smallest number (except 1) which is equal to the sum of the digits of its cube 3 Properties There are 49 regular properties that I check for: arXiv:0804.2277v1 [math.CO] 14 Apr 2008 abundant evil odious Smith amicable factorial palindrome sociable apocalyptic power Fibonacci palindromic prime square aspiring Google pentagonal square-free automorphic happy perfect tetrahedral cake hungry power of 2 triangular Carmichael lazy caterer powerful twin Catalan lucky practical Ulam composite Mersenne prime undulating compositorial Mersenne prime primorial untouchable cube narcissistic pronic vampire deficient odd repunit weird even 1 I selected regular properties for their importance as well as their funny names, so your favorite number could be lucky and happy at the same time, as is the case for 7. -
Representations of Palindromic, Prime, and Fibonacci Sequence Patterns
Representations of Palindromic, Prime, and Fibonacci Sequence Patterns Inder J. Taneja1 Abstract This work brings representations of palindromic and number patterns in terms of single letter ”a”. Some examples of prime number patterns are also considered. Different classifications of palindromic patterns are considered, such as, palindromic decompositions, double symmetric patterns, number pattern decompositions, etc. Numbers patterns with power are also studied. Study towards Fibonacci sequence and its extensions is also made. 1 Palindromic and Numbers Patterns Before proceeding further, here below are some general ways of writing palindromic numbers. Odd and even orders are considered separately. 1.1 General Form of Palindromic Numbers We shall divide palindromic numbers in two parts, odd and even orders. 1.1.1 Odd Order Palindromes Odd order palindromes are those where the number of digits are odd, i.e, aba, abcba, etc. See the general form below: aba := (102 + 100) × a + 101 × b abcba := (104 + 100) × a + (103 + 101) × b + 102 × c abcdcba := (106 + 100) × a + (105 + 101) × b + (104 + 102) × c + 103 × d abcdedcba := (108 + 100) × a + (107 + 101) × b + (106 + 102) × c + (105 + 103) × d + 104 × e: equivalently, aba := 101 × a + 10 × b abcba := 10001 × a + 1010 × b + 100 × c abcdcba := 1000001 × a + 100010 × b + 10100 × c + 1000 × d abcdedcba := 100000001 × a + 10000010 × b + 1000100 × c + 101000 × d + 10000 × e: 1.1.2 Even Order Palindromes Even order palindromes are those where the number of digits are even, i.e, abba, abccba, etc. See the general form below: abba := (103 + 100) × a + (102 + 101) × b abccba := (105 + 100) × a + (104 + 101) × b + (103 + 102) × c abcddcba := (107 + 100) × a + (106 + 101) × b + (105 + 102) × c + (104 + 103) × d abcdeedcba := (109 + 100) × a + (108 + 101) × b + (107 + 102) × c + (106 + 103) × d + (105 + 104) × e: 1Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florian´opolis,SC, Brazil. -
A-Primer-On-Prime-Numbers.Pdf
A Primer on Prime Numbers Prime Numbers “Prime numbers are the very atoms of arithmetic. The primes are the jewels studded throughout the vast expanse of the infinite universe of numbers that mathematicians have studied down the centuries.” Marcus du Sautoy, The Music of the Primes 2 • Early Primes • Named Primes • Hunting for Primes • Visualizing Primes • Harnessing Primes 3 Ishango bone The Ishango bone is a bone tool, dated to the Upper Paleolithic era, about 18,000 to 20,000 BC. It is a dark brown length of bone, the fibula of a baboon, It has a series of tally marks carved in three columns running the length of the tool Note: image is 4 reversed A History and Exploration of Prime Numbers • In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could have come about only after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20,… Left column 5 https://en.wikipedia.org/wiki/Ishango_bone Euclid of Alexandria 325-265 B.C. • The only man to summarize all the mathematical knowledge of his times. • In Proposition 20 of Book IX of the Elements, Euclid proved that there are infinitely many prime numbers. https://en.wikipedia.org/wiki/Euclid 6 Eratosthenes of Cyrene 276-194 B.C.