DOCSLIB.ORG

~Umbers the BOO K O F Umbers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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. We have done this in a way which is free from the formality of textbooks and syllabuses, so that the professional mathematician can also glean important information in areas outside her own speciality, and correspondingly enrich her teaching. The uses of the word number are diverse, but we can identify at least three separate strands. The development of number (usually written in the singular), continually adapting and generalizing itself to meet the needs of both mathematics and its increasing variety of applications: the counting numbers, zero, fractions, negative num­ bers, quadratic surds, algebraic numbers, transcendental numbers, infinitesimal and transfinite numbers, surreal numbers, complex numbers, quatemions, octonions. Then there is the special study of the integers, Gauss's higher arithmetic, or the theory of numbers (usually written in the plural), which overlaps the more recent area of enumerative combinatorics. Special sets or sequences of numbers: the prime numbers, Mersenne v ~ THE BOOK OF NUMBERS and Fermat numbers, perfect numbers, Fibonacci and Catalan num­ bers, Euler and Eulerian numbers, Bernoulli numbers. Finally there is a host of special numbers: Ludolph's 'IT, Napier's e, Euler's 'Y, Feigenbaum's constant, algebraic numbers which arise in specific contexts ranging from the diagonal of a square or of a regular pentagon to the example of degree 71 which arises from the apparently simple "look and say" sequences. Although mathematics is traditionally arranged in logical se­ quences, that is not the way the human brain seems to work. While it is often useful to know parts of earlier chapters in the book when reading some of the later ones, the reader can browse at will, picking up nuggets of information, with no obligation to read steadily from cover to cover. Chapter 1 describes number words and symbols and Chapter 2 shows how many elementary but important facts can be discovered "without using any mathematics." Chapters 3 and 4 exhibit several sets of whole numbers which can manifest themselves in quite dif­ ferent contexts, while Chapter 5 is devoted to the "multiplicative building blocks," the prime numbers. Fractions, or "algebraic num­ bers of degree one," are dealt with in Chapter 6 and those of higher degree in Chapter 7. So-called "complex" and "transcendental" num­ bers are explained in Chapters 8 and 9. The final chapter is devoted to the infinite and the infinitesimal, to surreal numbers which con­ stitute an extremely large, yet infinitely small, subclass of the most recent development of number, the values of combinatorial games. The color graphics in Chapter 2 are the work of Kenny and Andy Guy, using Brian Wyvill's Graphicsland software at the Graphics Lab­ oratory in the Computer Science Department at The University of Calgary. Three of the figures in Chapter 4 are reproduced from Prze­ myslaw Prusinkiewicz and Aristid Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, New York 1990, by kind permis­ sion of the authors and publishers. The first figure is due to D.R. Fowler, the second to D.R.F. and P. Prusinkiewicz and the third to D.R.F. and A. Snider. We thank Andrew Odlyzko and Peter Renz for reading a much earlier draft of the book, and for making many useful suggestions. We PRE F ACE vii are also grateful for the polite and patient persistence of Jerry Lyons, liesl Gibson and Henry Krell, and for the general competence of the Springer staff. We are proud to be among the early contributors to what is yet another successful Springer series. Princeton and Calgary John H. Conway and Richard K. Guy Contents PREFACE v 1 THE ROMANCE OF NUMBERS 1 2 FIGURES FROM FIGURES: DOING ARITHMETIC AND ALGEBRA BY GEOMETRY 27 3 WHAT COMES NEXT? 63 4 FAMOUS FAMILIES OF NUMBERS 91 5 THE PRIMACY OF PRIMES 127 6 FURTHER FRUITFULNESS OF FRACTIONS 151 7 GEOMETRIC PROBLEMS AND ALGEBRAIC NUMBERS 181 8 IMAGINING IMAGINARY NUMBERS 211 9 SOME TRANSCENDENTAL NUMBERS 237 10 INFINITE AND INFINITESIMAL NUMBERS 265 INDEX 301 ix v.J \ ...: The Romance of Numbers NUMBER WORDS Throughout history, number and numbers have had a tremendous influence on our culture and on our language. Thousands of words are obviously associated with numbers, for example, a monologue is a speech by 1 person; a bicycle has 2 wheels; a tripod is a stool with 3 legs; a quadruped is an animal with 4 legs; a pentathlon consists of 5 athletic events; a sextet is a piece for 6 musicians; a heptagon is a 7-comered figure; an octopus has 8 "feet"; a nonagenarian has lived for 9 decades; and decimal means counting in lOs; but in many other cases the connections, once just as vivid, have been obscured by the passage of time and changes in meaning. Have you ever realized just how many words are associated with "number" itself? This comes from an Indo-European root meaning "share" or "portion" and seems to have been originally associated with the division of land. "Nimble" refers to one who is quick to take his share; your "nemesis" was originally your portion of Fate; and 1 2 THE BOOK OF NUMBERS "numb" means "seized" or "taken." A "nomad" is one who wanders about in search of pasture land. There are many technical "nom" words: "binomial" (two-numbered), "astronomy" (numbering or al­ lotting the stars), "economy," "autonomy," and so on. In German we still have the word nebmen, to take, with imper­ ative "nimm." English once had much the same verb, but by Shake­ speare's day the word "nim" usually referred to illegal taking or pil­ fering. In Henry V, Corporal Nym is a doubtful character that one might well suspect of nimming. In our century "nim" has become the name of a take-away game. Latin and Greek nomisma was a coin, and this has given us "nu­ mismatics," the love of coins, and "nummulite," a coin-shaped fossil. "Supernumerary" now often means "redundant," but originally su­ pernumerary officials were those in addition to the number laid down by the regulations. HICKORY, DICKORY, DOCK­ EENY, MEENY, MINY, Mo There are several systems of words for numbers that survived for special purposes in out-of-the-way places. The particular sequence of words varies both with the part of the country and the purpose for which they were used. One such system is wan, twan, tetbera, metbera, pimp, setbera, letbera, bovera, dovera, dick, wanadick, twanadick, tetberadick, metberadick, pimpdick, setberadick, letberadick, boveradick, doveradick, bumjit, wanabumjit, .... Such rustic sequences appear in many countries. They are usually highly corrupted versions of the standard number systems of ancient languages. Hickory, dickory, dock. The mouse ran up tbe clock. The clock struck one; The mouse ran down. Hickory, dickory, dock. THE ROMANCE OF NUMBERS 3 Probably "hickory," "dickory," and "dock" are the words for "eight," "nine," and "ten" in one of these systems (compare "hov­ era, dovera, dick"), while "eeny, meeny, miny, mo" mean "one, two, three, four" in another. Most languages have interesting features about their number words, which repay study. We content ourselves with exhibiting those in Welsh. The alternative forms are for masculine and feminine, while the parenthetical forms are elisions in front of certain letters. There are separate ways of saying 50, 70, and 90. 1 un 16 un ar bymtheg 2 dau, dwy 17 dau, dwy ar bymtheg 3 toi, tair 18 deunaw 4 pedwar, pedair 19 pedwar, pedair ar bymtheg 5 pump (pum) 20 ugain 6 chwech (chwe) 30 deg ar hugain 7 saith 40 dugain 8 wyth 50 deg ar deugain; hanner cant 9 naw 60 trigain 10 deg 70 trigain a deg; deg ar thrigain 11 un ar ddeg 80 pedwar ugain 12 deuddeg 90 pedwar ugain a deg; deg a 13 toi, tair ar ddeg phedwar ugain 14 pedwar, pedair ar ddeg 100 cant 15 pymtheg Note that it is less duodecimal than many languages, which have sep­ arate words for eleven and twelve. It thus lends itself easily to modem decimalization. There are signs of bases 5 and 20, as well as 10.
Recommended publications
  • The 'Crisis of Noosphere'

    The 'Crisis of Noosphere'

    The ‘crisis of noosphere’ as a limiting factor to achieve the point of technological singularity Rafael Lahoz-Beltra Department of Applied Mathematics (Biomathematics). Faculty of Biological Sciences. Complutense University of Madrid. 28040 Madrid, Spain. [email protected] 1. Introduction One of the most significant developments in the history of human being is the invention of a way of keeping records of human knowledge, thoughts and ideas. The storage of knowledge is a sign of civilization, which has its origins in ancient visual languages e.g. in the cuneiform scripts and hieroglyphs until the achievement of phonetic languages with the invention of Gutenberg press. In 1926, the work of several thinkers such as Edouard Le Roy, Vladimir Ver- nadsky and Teilhard de Chardin led to the concept of noosphere, thus the idea that human cognition and knowledge transforms the biosphere coming to be something like the planet’s thinking layer. At present, is commonly accepted by some thinkers that the Internet is the medium that brings life to noosphere. Hereinafter, this essay will assume that the words Internet and noosphere refer to the same concept, analogy which will be justified later. 2 In 2005 Ray Kurzweil published the book The Singularity Is Near: When Humans Transcend Biology predicting an exponential increase of computers and also an exponential progress in different disciplines such as genetics, nanotechnology, robotics and artificial intelligence. The exponential evolution of these technologies is what is called Kurzweil’s “Law of Accelerating Returns”. The result of this rapid progress will lead to human beings what is known as tech- nological singularity.
  • On Variations of Nim and Chomp Arxiv:1705.06774V1 [Math.CO] 18

    On Variations of Nim and Chomp Arxiv:1705.06774V1 [Math.CO] 18

    On Variations of Nim and Chomp June Ahn Benjamin Chen Richard Chen Ezra Erives Jeremy Fleming Michael Gerovitch Tejas Gopalakrishna Tanya Khovanova Neil Malur Nastia Polina Poonam Sahoo Abstract We study two variations of Nim and Chomp which we call Mono- tonic Nim and Diet Chomp. In Monotonic Nim the moves are the same as in Nim, but the positions are non-decreasing numbers as in Chomp. Diet-Chomp is a variation of Chomp, where the total number of squares removed is limited. 1 Introduction We study finite impartial games with two players where the same moves are available to both players. Players alternate moves. In a normal play, the person who does not have a move loses. In a misère play, the person who makes the last move loses. A P-position is a position from which the previous player wins, assuming perfect play. We can observe that all terminal positions are P-positions. An N-position is a position from which the next player wins given perfect play. When we play we want to end our move with a P-position and want to see arXiv:1705.06774v1 [math.CO] 18 May 2017 an N-position before our move. Every impartial game is equivalent to a Nim heap of a certain size. Thus, every game can be assigned a non-negative integer, called a nimber, nim- value, or a Grundy number. The game of Nim is played on several heaps of tokens. A move consists of taking some tokens from one of the heaps. The game of Chomp is played on a rectangular m by n chocolate bar with grid lines dividing the bar into mn squares.
  • Elementary Functions: Towards Automatically Generated, Efficient

    Elementary Functions: Towards Automatically Generated, Efficient

    Elementary functions : towards automatically generated, efficient, and vectorizable implementations Hugues De Lassus Saint-Genies To cite this version: Hugues De Lassus Saint-Genies. Elementary functions : towards automatically generated, efficient, and vectorizable implementations. Other [cs.OH]. Université de Perpignan, 2018. English. NNT : 2018PERP0010. tel-01841424 HAL Id: tel-01841424 https://tel.archives-ouvertes.fr/tel-01841424 Submitted on 17 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. Délivré par l’Université de Perpignan Via Domitia Préparée au sein de l’école doctorale 305 – Énergie et Environnement Et de l’unité de recherche DALI – LIRMM – CNRS UMR 5506 Spécialité: Informatique Présentée par Hugues de Lassus Saint-Geniès [email protected] Elementary functions: towards automatically generated, efficient, and vectorizable implementations Version soumise aux rapporteurs. Jury composé de : M. Florent de Dinechin Pr. INSA Lyon Rapporteur Mme Fabienne Jézéquel MC, HDR UParis 2 Rapporteur M. Marc Daumas Pr. UPVD Examinateur M. Lionel Lacassagne Pr. UParis 6 Examinateur M. Daniel Menard Pr. INSA Rennes Examinateur M. Éric Petit Ph.D. Intel Examinateur M. David Defour MC, HDR UPVD Directeur M. Guillaume Revy MC UPVD Codirecteur À la mémoire de ma grand-mère Françoise Lapergue et de Jos Perrot, marin-pêcheur bigouden.
  • Catalan Numbers Modulo 2K

    Catalan Numbers Modulo 2K

    1 2 Journal of Integer Sequences, Vol. 13 (2010), 3 Article 10.5.4 47 6 23 11 Catalan Numbers Modulo 2k Shu-Chung Liu1 Department of Applied Mathematics National Hsinchu University of Education Hsinchu, Taiwan [email protected] and Jean C.-C. Yeh Department of Mathematics Texas A & M University College Station, TX 77843-3368 USA Abstract In this paper, we develop a systematic tool to calculate the congruences of some combinatorial numbers involving n!. Using this tool, we re-prove Kummer’s and Lucas’ theorems in a unique concept, and classify the congruences of the Catalan numbers cn (mod 64). To achieve the second goal, cn (mod 8) and cn (mod 16) are also classified. Through the approach of these three congruence problems, we develop several general properties. For instance, a general formula with powers of 2 and 5 can evaluate cn (mod k 2 ) for any k. An equivalence cn ≡2k cn¯ is derived, wheren ¯ is the number obtained by partially truncating some runs of 1 and runs of 0 in the binary string [n]2. By this equivalence relation, we show that not every number in [0, 2k − 1] turns out to be a k residue of cn (mod 2 ) for k ≥ 2. 1 Introduction Throughout this paper, p is a prime number and k is a positive integer. We are interested in enumerating the congruences of various combinatorial numbers modulo a prime power 1Partially supported by NSC96-2115-M-134-003-MY2 1 k q := p , and one of the goals of this paper is to classify Catalan numbers cn modulo 64.
  • An Very Brief Overview of Surreal Numbers for Gandalf MM 2014

    An Very Brief Overview of Surreal Numbers for Gandalf MM 2014

    An very brief overview of Surreal Numbers for Gandalf MM 2014 Steven Charlton 1 History and Introduction Surreal numbers were created by John Horton Conway (of Game of Life fame), as a greatly simplified construction of an earlier object (Alling’s ordered field associated to the class of all ordinals, as constructed via modified Hahn series). The name surreal numbers was coined by Donald Knuth (of TEX and the Art of Computer Programming fame) in his novel ‘Surreal Numbers’ [2], where the idea was first presented. Surreal numbers form an ordered Field (Field with a capital F since surreal numbers aren’t a set but a class), and are in some sense the largest possible ordered Field. All other ordered fields, rationals, reals, rational functions, Levi-Civita field, Laurent series, superreals, hyperreals, . , can be found as subfields of the surreals. The definition/construction of surreal numbers leads to a system where we can talk about and deal with infinite and infinitesimal numbers as naturally and consistently as any ‘ordinary’ number. In fact it let’s can deal with even more ‘wonderful’ expressions 1 √ 1 ∞ − 1, ∞, ∞, ,... 2 ∞ in exactly the same way1. One large area where surreal numbers (or a slight generalisation of them) finds application is in the study and analysis of combinatorial games, and game theory. Conway discusses this in detail in his book ‘On Numbers and Games’ [1]. 2 Basic Definitions All surreal numbers are constructed iteratively out of two basic definitions. This is an wonderful illustration on how a huge amount of structure can arise from very simple origins.
  • Cauchy, Infinitesimals and Ghosts of Departed Quantifiers 3

    Cauchy, Infinitesimals and Ghosts of Departed Quantifiers 3

    CAUCHY, INFINITESIMALS AND GHOSTS OF DEPARTED QUANTIFIERS JACQUES BAIR, PIOTR BLASZCZYK, ROBERT ELY, VALERIE´ HENRY, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TARAS KUDRYK, SEMEN S. KUTATELADZE, THOMAS MCGAFFEY, THOMAS MORMANN, DAVID M. SCHAPS, AND DAVID SHERRY Abstract. Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction be- tween assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robin- son’s framework, while Leibniz’s law of homogeneity with the im- plied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide paral- lel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Eu- ler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infi- nite exponent. Such procedures have immediate hyperfinite ana- logues in Robinson’s framework, while in a Weierstrassian frame- work they can only be reinterpreted by means of paraphrases de- parting significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of de- arXiv:1712.00226v1 [math.HO] 1 Dec 2017 parted quantifiers in his work.
  • An Introduction to Conway's Games and Numbers

    An Introduction to Conway's Games and Numbers

    AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS DIERK SCHLEICHER AND MICHAEL STOLL 1. Combinatorial Game Theory Combinatorial Game Theory is a fascinating and rich theory, based on a simple and intuitive recursive definition of games, which yields a very rich algebraic struc- ture: games can be added and subtracted in a very natural way, forming an abelian GROUP (§ 2). There is a distinguished sub-GROUP of games called numbers which can also be multiplied and which form a FIELD (§ 3): this field contains both the real numbers (§ 3.2) and the ordinal numbers (§ 4) (in fact, Conway’s definition gen- eralizes both Dedekind sections and von Neumann’s definition of ordinal numbers). All Conway numbers can be interpreted as games which can actually be played in a natural way; in some sense, if a game is identified as a number, then it is under- stood well enough so that it would be boring to actually play it (§ 5). Conway’s theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go [Go]. There is a beautiful microcosmos of numbers and games which are infinitesimally close to zero (§ 6), and the theory contains the classical and complete Sprague-Grundy theory on impartial games (§ 7). The theory was founded by John H. Conway in the 1970’s. Classical references are the wonderful books On Numbers and Games [ONAG] by Conway, and Win- ning Ways by Berlekamp, Conway and Guy [WW]; they have recently appeared in their second editions.
  • Solutions of the Markov Equation Over Polynomial Rings

    Solutions of the Markov Equation Over Polynomial Rings

    Solutions of the Markov equation over polynomial rings Ricardo Conceição, R. Kelly, S. VanFossen 49th John H. Barrett Memorial Lectures University of Tennessee, Knoxville, TN Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation Introduction In his work on Diophantine approximation, Andrei Markov studied the equation x2 + y2 + z2 = 3xyz: The structure of the integral solutions of the Markov equation is very interesting. Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation Automorphisms x2 + y2 + z2 = 3xyz (x1; x2; x3) −! (xπ(1); xπ(2); xπ(3)) (x; y; z) −! (−x; −y; z) A Markov triple (x; y; z) is a solution of the Markov equation that satisfies 0 < x ≤ y ≤ z. (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) (3zy − x; y; z) Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x; y; z) (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) ··· (2,29,169) ··· (2,5,29) ··· (5,29,433) ··· (1,1,1) (1,1,2) (1,2,5) ··· (5,13,194) ··· (1,5,13) ··· (1,13,34) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x; y; z) (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) ··· (2,29,169) ··· (2,5,29) ··· (5,29,433) ··· (1,1,1) (1,1,2) (1,2,5) ··· (5,13,194) ··· (1,5,13) ··· (1,13,34) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x;
  • Continued Fraction: a Research on Markov Triples

    Continued Fraction: a Research on Markov Triples

    Fort Hays State University FHSU Scholars Repository John Heinrichs Scholarly and Creative Activities 2020 SACAD Entrants Day (SACAD) 4-22-2020 Continued fraction: A research on Markov triples Daheng Chen Fort Hays State University, [email protected] Follow this and additional works at: https://scholars.fhsu.edu/sacad_2020 Recommended Citation Chen, Daheng, "Continued fraction: A research on Markov triples" (2020). 2020 SACAD Entrants. 13. https://scholars.fhsu.edu/sacad_2020/13 This Poster is brought to you for free and open access by the John Heinrichs Scholarly and Creative Activities Day (SACAD) at FHSU Scholars Repository. It has been accepted for inclusion in 2020 SACAD Entrants by an authorized administrator of FHSU Scholars Repository. Daheng Chen CONTINUED FRACTION: Fort Hays State University Mentor: Hongbiao Zeng A research on Markov triples Department of Mathematics INTRODUCTION STUDY PROCESS OUTCOMES Being one of the many studies in Number Theory, continued fractions is a Our study of Markov Triple aimed at a simple level since the main researcher, Daheng Chen who was still a When solving the second question, we saw the clear relationship between study of math that offer useful means of expressing numbers and functions. Mathematicians in early ages used algorithms and methods to express high school student, did not have enough professional knowledge in math. Professor Hongbiao Zeng left three Fibonacci sequence and Markov numbers: all odd indices of Fibonacci numbers. Many of these expressions turned into the study of continued questions as the mainstream of the whole study. sequence are a subset of Markov numbers. fractions. We did not reach our ultimate research goal at this time; however, both of us Since the start of the 20th century, mathematicians have been able to develop Professor Zeng came up with the following three questions regarding Markov triples: gained precious experience in this area where we were not familiar with.
  • Algebra & Number Theory Vol. 7 (2013)

    Algebra & Number Theory Vol. 7 (2013)

    Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Virginia, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University,
  • On the Ordering of the Markov Numbers

    On the Ordering of the Markov Numbers

    ON THE ORDERING OF THE MARKOV NUMBERS KYUNGYONG LEE, LI LI, MICHELLE RABIDEAU, AND RALF SCHIFFLER Abstract. The Markov numbers are the positive integers that appear in the solutions of the equation x2 + y2 + z2 = 3xyz. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points (q; p) in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. We give a partial answer in terms of the slope of the line segment that connects the two lattice points. We prove that the Markov number with the greater x-coordinate 8 is larger than the other if the slope is at least − 7 and that it is smaller than the other if the 5 slope is at most − 4 . As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book \Markov's theorem and 100 years of the uniqueness conjecture". 1. Introduction In 1879, Andrey Markov studied the equation (1.1) x2 + y2 + z2 = 3xyz; which is now known as the Markov equation. A positive integer solution (m1; m2; m3) of (1.1) is called a Markov triple and the integers that appear in the Markov triples are called Markov num- bers.
  • Impartial Games

    Impartial Games

    Combinatorial Games MSRI Publications Volume 29, 1995 Impartial Games RICHARD K. GUY In memory of Jack Kenyon, 1935-08-26 to 1994-09-19 Abstract. We give examples and some general results about impartial games, those in which both players are allowed the same moves at any given time. 1. Introduction We continue our introduction to combinatorial games with a survey of im- partial games. Most of this material can also be found in WW [Berlekamp et al. 1982], particularly pp. 81{116, and in ONAG [Conway 1976], particu- larly pp. 112{130. An elementary introduction is given in [Guy 1989]; see also [Fraenkel 1996], pp. ??{?? in this volume. An impartial game is one in which the set of Left options is the same as the set of Right options. We've noticed in the preceding article the impartial games = 0=0; 0 0 = 1= and 0; 0; = 2: {|} ∗ { | } ∗ ∗ { ∗| ∗} ∗ that were born on days 0, 1, and 2, respectively, so it should come as no surprise that on day n the game n = 0; 1; 2;:::; (n 1) 0; 1; 2;:::; (n 1) ∗ {∗ ∗ ∗ ∗ − |∗ ∗ ∗ ∗ − } is born. In fact any game of the type a; b; c;::: a; b; c;::: {∗ ∗ ∗ |∗ ∗ ∗ } has value m,wherem =mex a;b;c;::: , the least nonnegative integer not in ∗ { } the set a;b;c;::: . To see this, notice that any option, a say, for which a>m, { } ∗ This is a slightly revised reprint of the article of the same name that appeared in Combi- natorial Games, Proceedings of Symposia in Applied Mathematics, Vol. 43, 1991. Permission for use courtesy of the American Mathematical Society.