DOCSLIB.ORG

Eureka Issue 62 | a Journal of the Archimedeans

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Eureka 62 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Jack Williams Cover by Andrew Ostrovsky, Inner Cover by George Hart © The Archimedeans, see page 95 December 2012 Editorial Eureka 62 hen the Archimedeans asked me to edit Eureka for the third Editors time, I was a bit sceptical. Issue 60 was the first to get a pa- Philipp Legner (St John’s) perback binding and issue 61 was the first to be published in Jack Williams (Clare) Wfull colour and with a new design. How could we make this issue special – not just a repeat of the previous one? Assistant Editors Stacey Law (Trinity) Eureka has always been a magazine for students, not a research journal. Carina Negreanu (Queens’) Articles should be interesting and entertaining to read, and often they Katarzyna Kowal (Churchill) are a stepping stone into particular problems or areas of mathematics Douglas Bourbert (Churchill) which the reader would not usually have encountered. Ram Sarujan (Corpus Christi) Every year we receive many great articles by students and mathemati- Subscriptions cians. Our task as editors is often to make them more visually appealing Wesley Mok (Trinity) – and we can do so using images, diagrams, fonts or colours. What we wanted to add in this issue was interactivity, such as videos, slideshows, animations or games. Unfortunately this still is quite dif- ficult on paper, so we decided to publish a second version of Eureka as interactive eBook for mobile devices like iPad. And we hope that this will make reading mathematics even more engaging and fun. The digital version will, for the first time, make Eureka available to a large number of students outside Cambridge. And therefore we have reprinted some of the best articles from previous issues. We spent many hours in the library archives, reading old copies of Eureka, though of course there are many more great articles we could have included. The articles in this issue are on a wide range of topics – from num- ber theory to cosmology, from statistics to geometry. Some are very technical while others are more recreational, but we hope that there is something interesting for everyone. I want to thank the editorial team for all their work, and the authors for their excellent articles. We hope you will enjoy reading Eureka 62! Philipp Legner and Jack Williams 10 Pentaplexity 4 The Archimedeans 32 Fractals, Compression and Contraction Mapping † 6 Talking to Computers Alexander Shannon, Christ’s Stephen Wolfram 38 Stein’s Paradox 10 Pentaplexity † Richard J Samworth, Statslab Cambridge Sir Roger Penrose, Oxford University 42 Optimal Card Shuffling † 16 Squared Squares Martin Mellish Philipp Kleppmann, Corpus Christi 44 Harmonic Game Theory † 20 Tricky Teacups Blanche Descartes Vito Videtta, Trinity Hall 46 The Logic of Logic 24 Annual Problems Drive Zoe Wyatt, Newnham David Phillips and Alec Barnes-Graham 48 Timeline 2012 28 Pi in Fours † Stacey Law and Philipp Legner John Conway and Michael Guy 50 Hopes and Fears † 30 Surface Differences Matter! Paul Dirac Arran Fernandez, Cambridge University First– magic number in nuclear physics. 2 √2 was the first known irrational number. 72 Glacier Dynamics 58 Multiverses 32 Fractals, Compression and Contraction Mapping 54 Quantum Gravity † 88 Archimedes Stephen Hawking, DAMTP Cambridge Tom Körner, DPMMS Cambridge 58 Multiverses 90 Book Reviews Georg Ellis, University of Cape Town 92 Christmas Catalogue † 64 Primes and Particles Jack Williams, Clare 93 Call My Bluff † 68 M-Theory, Duality and Art 94 Solutions David Berman, QMUL 95 Copyright Notices 72 Glacier Dynamics Indranil Banik and Justas Dauparas 76 Finding Order in Randomness Maithra Raghu, Trinity 80 Mathematics in Wartime † G H Hardy 84 Consecutive Integers † Paul Erdős Items marked † are reprinted from past issues of Eureka. Three of the five platonic solids have triangular faces. Trisecting angles is an impossible construction. 3 The Archimedeans Yuhan Gao, President 2012 − 2013 his year was yet another highly successful from as far afield as Oxford came to take part in an one for The Archimedeans. The society engaging and entertaining mathematics competi- welcomed over 150 new members, courtesy tion. Prizes were awarded not only for the teams Tof a very popular Freshers’ Squash. We hosted a with the highest scores, but also for particularly number of talks given by speakers from the uni- creative team names. The questions given can be versity over the course of Michaelmas and Lent. found in this journal, and we welcome you to try These covered a number of different topics, cater- them yourself. ing for those with interests in pure, applied and applicable mathematics. Highlights included The year finished on a high in May Week, courtesy talks by Prof. Grae Worster on Ice, and Prof. Imre of the Science and Engineering Garden Party. Six Leader on Games of Pursuit and Evasion. societies from the university joined together to host a brilliant afternoon of fun, aided by a jazz The society expanded the range of events which band. Finger food and Pimm’s was served, and we offered to our members this year. We held a there was even a cheese bar on offer. board games evening, which proved to be a thor- oughly enjoyable night for all those who attended. We would like to thank our members for contrib- One of our most anticipated events was the black- uting to an excellent year for the society. I would tie Annual Dinner in the delightful surroundings also like to thank the committee for all of their of the Crowne Plaza Hotel. hard work, and Philipp Kleppmann, last years’ President, along with the previous committee, for A tradition of the Archimedeans is to hold an an- everything which they have done for the society. nual Problems Drive. This time around, teams We look forward to another exciting year ahead. The Committee 2012 – 2013 President Treasurer Yuhan Gao (Trinity) Colin Egan (Gonville and Caius) Vice-Presidents Events Managers Sean Moss (Trinity) Pawel Rzemieniecki (Fitzwilliam) Dana Ma (Newnham) Yuming Mei (Emmanuel) Corporate Officer Publicity Officer Joseph Briggs (Trinity) James Bell (Gonville and Caius) Secretary Webmaster Jacquie Hu (Jesus) Ben Millwood (Downing) Smallest composite number. Number of 4 Nucleobase types in the DNA: A, G, C and T. Archimedeans Garden Party Professor Archimedeans David Tong Problems Drive Archimedeans Annual Dinner Archimedeans Talk in the CMS Archimedeans Problems Drive Archimedeans Garden Party K5 is the smallest complete non-planar graph. The cycle of fifths underlies musical harmonies. 5 Talking to Computers Stephen Wolfram love computer languages. In fact, I’ve spent Wolfram|Alpha responds by computing and roughly half my life nurturing one particular presenting whatever knowledge is requested. But I very rich computer language: Mathematica. programming is different. It is not about gen- erating static knowledge, but about generating But do we really need computer languages to tell programs that can take a range of inputs, and our computers what to do? Why can’t we just use dynamically perform operations. natural human languages, like English, instead? The first question is: how might we represent If you had asked me a few years ago, I would these programs? In principle we could use pretty have said it was hopeless. That perhaps one could much any programming language. But to make make toy examples, but that ultimately natural things practical, particularly at the beginning, language just wouldn’t be up to the task of creat- ing useful programs. we need a programming language with a couple of key characteristics. But then along came Wolfram|Alpha in which we’ve been able to make free-form linguistics The most important is that programs a user might work vastly better than I ever thought possible. specify with short pieces of natural language must typically be short – and readable – in the But still, in Wolfram|Alpha the input is essen- computer language. Because otherwise the user tially just set up to request knowledge – and won’t be able to tell – at least not easily – whether Smallest perfect number. Hexagonal tilings give 6 the densest ‘sphere’ packing in two dimensions. the program that’s been produced actually does The linguistic capabilities of Wolfram|Alpha give what they want. one the idea that one might be able to under- stand free-form natural language specifications A second, somewhat related criterion is that it of programs. Mathematica is what gives one the must be possible for arbitrary program frag- idea that there might be a reasonable target for ments to stand alone – so that large programs can realistically be built up incrementally, much programs generated automatically from natural like a description in natural language is built up language. incrementally with sentences and the like. For me, there was also a third motivating idea – To get the first of these characteristics requires that came from my work on A New Kind of Sci- a very high-level language, in which there are ence. One might have thought that to perform already many constructs already built in to the any kind of complex task would always require language – and well enough designed that they a complex program. But what I learned in A New all fit together without messy “glue” code. Kind of Science is that simple programs can often do highly complex things. And to get the second characteristic essentially requires a symbolic language, in which any piece And the result of this is that it’s often possible to of any program is always a meaningful symbolic find useful programs just by searching for them expression. in the computational universe of possible pro- Conveniently enough, there is one language that grams – a technique that we use with increas- satisfies rather well both these requirements: ing frequency in the actual development of both Mathematica! Wolfram|Alpha and Mathematica.
Recommended publications
  • Fast Tabulation of Challenge Pseudoprimes Andrew Shallue and Jonathan Webster

    Fast Tabulation of Challenge Pseudoprimes Andrew Shallue and Jonathan Webster

    THE OPEN BOOK SERIES 2 ANTS XIII Proceedings of the Thirteenth Algorithmic Number Theory Symposium Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster msp THE OPEN BOOK SERIES 2 (2019) Thirteenth Algorithmic Number Theory Symposium msp dx.doi.org/10.2140/obs.2019.2.411 Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW-challenge pseudoprimes with two or three prime factors up to 280. In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods. 1. Introduction Pomerance, Selfridge, and Wagstaff famously offered $620 for a composite n that satisfies (1) 2n 1 1 .mod n/ so n is a base-2 Fermat pseudoprime, Á (2) .5 n/ 1 so n is not a square modulo 5, and j D (3) Fn 1 0 .mod n/ so n is a Fibonacci pseudoprime, C Á or to prove that no such n exists. We call composites that satisfy these conditions PSW-challenge pseudo- primes. In[PSW80] they credit R. Baillie with the discovery that combining a Fermat test with a Lucas test (with a certain specific parameter choice) makes for an especially effective primality test[BW80].
  • Prime Divisors in the Rationality Condition for Odd Perfect Numbers

    Prime Divisors in the Rationality Condition for Odd Perfect Numbers

    Aid#59330/Preprints/2019-09-10/www.mathjobs.org RFSC 04-01 Revised The Prime Divisors in the Rationality Condition for Odd Perfect Numbers Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92037 Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 4m+1 ℓ 2αi 1) i=1 qi to establish that there do not exist any odd integers with equality (4k+1)4m+2−1 between σ(N) and 2N. The existence of distinct prime divisors in the repunits 4k , 2α +1 Q q i −1 i , i = 1,...,ℓ, in σ(N) follows from a theorem on the primitive divisors of the Lucas qi−1 sequences and the square root of the product of 2(4k + 1), and the sequence of repunits will not be rational unless the primes are matched. Minimization of the number of prime divisors in σ(N) yields an infinite set of repunits of increasing magnitude or prime equations with no integer solutions. MSC: 11D61, 11K65 Keywords: prime divisors, rationality condition 1. Introduction While even perfect numbers were known to be given by 2p−1(2p − 1), for 2p − 1 prime, the universality of this result led to the the problem of characterizing any other possible types of perfect numbers. It was suggested initially by Descartes that it was not likely that odd integers could be perfect numbers [13]. After the work of de Bessy [3], Euler proved σ(N) that the condition = 2, where σ(N) = d|N d is the sum-of-divisors function, N d integer 4m+1 2α1 2αℓ restricted odd integers to have the form (4kP+ 1) q1 ...qℓ , with 4k + 1, q1,...,qℓ prime [18], and further, that there might exist no set of prime bases such that the perfect number condition was satisfied.
  • [Math.NT] 10 May 2005

    [Math.NT] 10 May 2005

    Journal de Th´eorie des Nombres de Bordeaux 00 (XXXX), 000–000 Restriction theory of the Selberg sieve, with applications par Ben GREEN et Terence TAO Resum´ e.´ Le crible de Selberg fournit des majorants pour cer- taines suites arithm´etiques, comme les nombres premiers et les nombres premiers jumeaux. Nous demontrons un th´eor`eme de restriction L2-Lp pour les majorants de ce type. Comme ap- plication imm´ediate, nous consid´erons l’estimation des sommes exponentielles sur les k-uplets premiers. Soient les entiers posi- tifs a1,...,ak et b1,...,bk. Ecrit h(θ) := n∈X e(nθ), ou X est l’ensemble de tous n 6 N tel que tousP les nombres a1n + b1,...,akn+bk sont premiers. Nous obtenons les bornes sup´erieures pour h p T , p > 2, qui sont (en supposant la v´erit´ede la con- k kL ( ) jecture de Hardy et Littlewood sur les k-uplets premiers) d’ordre de magnitude correct. Une autre application est la suivante. En utilisant les th´eor`emes de Chen et de Roth et un «principe de transf´erence », nous demontrons qu’il existe une infinit´ede suites arithm´etiques p1 < p2 < p3 de nombres premiers, telles que cha- cun pi + 2 est premier ou un produit de deux nombres premier. Abstract. The Selberg sieve provides majorants for certain arith- metic sequences, such as the primes and the twin primes. We prove an L2–Lp restriction theorem for majorants of this type. An im- mediate application is to the estimation of exponential sums over prime k-tuples.
  • FACTORING COMPOSITES TESTING PRIMES Amin Witno

    FACTORING COMPOSITES TESTING PRIMES Amin Witno

    WON Series in Discrete Mathematics and Modern Algebra Volume 3 FACTORING COMPOSITES TESTING PRIMES Amin Witno Preface These notes were used for the lectures in Math 472 (Computational Number Theory) at Philadelphia University, Jordan.1 The module was aborted in 2012, and since then this last edition has been preserved and updated only for minor corrections. Outline notes are more like a revision. No student is expected to fully benefit from these notes unless they have regularly attended the lectures. 1 The RSA Cryptosystem Sensitive messages, when transferred over the internet, need to be encrypted, i.e., changed into a secret code in such a way that only the intended receiver who has the secret key is able to read it. It is common that alphabetical characters are converted to their numerical ASCII equivalents before they are encrypted, hence the coded message will look like integer strings. The RSA algorithm is an encryption-decryption process which is widely employed today. In practice, the encryption key can be made public, and doing so will not risk the security of the system. This feature is a characteristic of the so-called public-key cryptosystem. Ali selects two distinct primes p and q which are very large, over a hundred digits each. He computes n = pq, ϕ = (p − 1)(q − 1), and determines a rather small number e which will serve as the encryption key, making sure that e has no common factor with ϕ. He then chooses another integer d < n satisfying de % ϕ = 1; This d is his decryption key. When all is ready, Ali gives to Beth the pair (n; e) and keeps the rest secret.
  • On Sufficient Conditions for the Existence of Twin Values in Sieves

    On Sufficient Conditions for the Existence of Twin Values in Sieves

    On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers by Luke Szramowski Submitted in Partial Fulfillment of the Requirements For the Degree of Masters of Science in the Mathematics Program Youngstown State University May, 2020 On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers Luke Szramowski I hereby release this thesis to the public. I understand that this thesis will be made available from the OhioLINK ETD Center and the Maag Library Circulation Desk for public access. I also authorize the University or other individuals to make copies of this thesis as needed for scholarly research. Signature: Luke Szramowski, Student Date Approvals: Dr. Eric Wingler, Thesis Advisor Date Dr. Thomas Wakefield, Committee Member Date Dr. Thomas Madsen, Committee Member Date Dr. Salvador A. Sanders, Dean of Graduate Studies Date Abstract For many years, a major question in sieve theory has been determining whether or not a sieve produces infinitely many values which are exactly two apart. In this paper, we will discuss a new result in sieve theory, which will give sufficient conditions for the existence of values which are exactly two apart. We will also show a direct application of this theorem on an existing sieve as well as detailing attempts to apply the theorem to the Sieve of Eratosthenes. iii Contents 1 Introduction 1 2 Preliminary Material 1 3 Sieves 5 3.1 The Sieve of Eratosthenes . 5 3.2 The Block Sieve . 9 3.3 The Finite Block Sieve . 12 3.4 The Sieve of Joseph Flavius .
  • 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)

    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.
  • Appendix a Tables of Fermat Numbers and Their Prime Factors

    Appendix a Tables of Fermat Numbers and Their Prime Factors

    Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard
  • Monthly Science and Maths Magazine 01

    Monthly Science and Maths Magazine 01

    1 GYAN BHARATI SCHOOL QUEST….. Monthly Science and Mathematics magazine Edition: DECEMBER,2019 COMPILED BY DR. KIRAN VARSHA AND MR. SUDHIR SAXENA 2 IDENTIFY THE SCIENTIST She was an English chemist and X-ray crystallographer who made contributions to the understanding of the molecular structures of DNA , RNA, viruses, coal, and graphite. She was never nominated for a Nobel Prize. Her work was a crucial part in the discovery of DNA, for which Francis Crick, James Watson, and Maurice Wilkins were awarded a Nobel Prize in 1962. She died in 1958, and during her lifetime the DNA structure was not considered as fully proven. It took Wilkins and his colleagues about seven years to collect enough data to prove and refine the proposed DNA structure. RIDDLE TIME You measure my life in hours and I serve you by expiring. I’m quick when I’m thin and slow when I’m fat. The wind is my enemy. Hard riddles want to trip you up, and this one works by hitting you with details from every angle. The big hint comes at the end with the wind. What does wind threaten most? I have cities, but no houses. I have mountains, but no trees. I have water, but no fish. What am I? This riddle aims to confuse you and get you to focus on the things that are missing: the houses, trees, and fish. 3 WHY ARE AEROPLANES USUALLY WHITE? The Aeroplanes might be having different logos and decorations. But the colour of the aeroplane is usually white.Painting the aeroplane white is most practical and economical.
  • Second Grade Unit 1: Extending Base Ten Understanding

    Second Grade Unit 1: Extending Base Ten Understanding

    Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Second Grade Unit 1: Extending Base Ten Understanding These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Georgia Department of Education Georgia Standards of Excellence Framework GSE Extending Base Ten Understanding Unit 1 Unit 1: Extending Base Ten Understanding TABLE OF CONTENTS Overview……………………………………………………………………………..... 3 Standards for Mathematical Practice …………………………………………………. 5 Standards for Mathematical Content …………………………………………………. 6 Big Ideas ………………………………………………………………………………. 6 Essential Questions ……………………………………………………………………….. 7 Concepts and Skills to Maintain ………………………………………………………. 7 Strategies for Teaching and Learning ……………………………………………. 8 Selected Terms and Symbols ………………………………………………..………… 10 Task Types ………………………………………………………………...………….. 12 Task Descriptions …………………………………………………………………….. 13 Intervention Table………………….………………………………………………….. 15 Straws! Straws! Straws!.........................................................................… 16 Where Am I on the Number Line? …………………………………… 23 I Spy a Number ………………………………………………………. 31 Number Hop …………………………………………………………… 38 Place Value Play …………………………………………………….... 44 The Importance of Zero ………………………………………………. 51 Base Ten Pictures …………………………………………………….. 62 Building Base Ten Numbers …………………………………………. 69 What's My Number? …………………………………………………. 74 Capture the Caterpillar ……………………………………………….. 79 Formative Assessment Lesson ……………………………………….. 88 Fill the Bucket ………………………………………………………..
  • ~Umbers the BOO K O F Umbers

    ~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 Family of Sequences Generating Smith Numbers

    A Family of Sequences Generating Smith Numbers

    1 2 Journal of Integer Sequences, Vol. 16 (2013), 3 Article 13.4.6 47 6 23 11 A Family of Sequences Generating Smith Numbers Amin Witno Department of Basic Sciences Philadelphia University 19392 Jordan [email protected] Abstract Infinitely many Smith numbers can be constructed using sequences which involve repunits. We provide an alternate method of construction by introducing a generaliza- tion of repunits, resulting in new Smith numbers whose decimal digits are composed of zeros and nines. 1 Introduction We work with natural numbers in their decimal representation. Let S(N) denote the “digital sum” (the sum of the base-10 digits) of the number N and let Sp(N) denote the sum of all the digital sums of the prime factors of N, counting multiplicity. For instance, S(2013) = 2+0+1+3 = 6 and, since 2013 = 3 × 11 × 61, we have Sp(2013) = 3+1+1+6+1 = 12. The quantity Sp(N) does not seem to have a name in the literature, so let us just call Sp(N) the p-digit sum of N. The natural number N is called a Smith number when N is composite and S(N)= Sp(N). For example, since 22 = 2 × 11, we have both S(22) = 4 and Sp(22) = 4. Hence, 22 is a Smith number—thus named by Wilansky [4] in 1982. Several different ways of constructing Smith numbers are already known. (See our 2010 article [5], for instance, which contains a brief historical account on the subject as well as a list of references.
  • POWERFUL AMICABLE NUMBERS 1. Introduction Let S(N) := ∑ D Be the Sum of the Proper Divisors of the Natural Number N. Two Disti

    POWERFUL AMICABLE NUMBERS 1. Introduction Let S(N) := ∑ D Be the Sum of the Proper Divisors of the Natural Number N. Two Disti

    POWERFUL AMICABLE NUMBERS PAUL POLLACK P Abstract. Let s(n) := djn; d<n d denote the sum of the proper di- visors of the natural number n. Two distinct positive integers n and m are said to form an amicable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first example of an amicable pair, known already to the ancients, is f220; 284g. We do not know if there are infinitely many amicable pairs. In the opposite direction, Erd}osshowed in 1955 that the set of amicable numbers has asymptotic density zero. Let ` ≥ 1. A natural number n is said to be `-full (or `-powerful) if p` divides n whenever the prime p divides n. As shown by Erd}osand 1=` Szekeres in 1935, the number of `-full n ≤ x is asymptotically c`x , as x ! 1. Here c` is a positive constant depending on `. We show that for each fixed `, the set of amicable `-full numbers has relative density zero within the set of `-full numbers. 1. Introduction P Let s(n) := djn; d<n d be the sum of the proper divisors of the natural number n. Two distinct natural numbers n and m are said to form an ami- cable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first amicable pair, 220 and 284, was known already to the Pythagorean school. Despite their long history, we still know very little about such pairs.