The Proving Ground – an Introduction to Mathematical Proof Forty Easy-To-Understand Mathematics Problems - Twenty Have Been Solved and Twenty Haven’T 2
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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
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. -
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. -
A Survey on the Square Peg Problem
A survey on the Square Peg Problem Benjamin Matschke∗ March 14, 2014 Abstract This is a short survey article on the 102 years old Square Peg Problem of Toeplitz, which is also called the Inscribed Square Problem. It asks whether every continuous simple closed curve in the plane contains the four vertices of a square. This article first appeared in a more compact form in Notices Amer. Math. Soc. 61(4), 2014. 1 Introduction In 1911 Otto Toeplitz [66] furmulated the following conjecture. Conjecture 1 (Square Peg Problem, Toeplitz 1911). Every continuous simple closed curve in the plane γ : S1 ! R2 contains four points that are the vertices of a square. Figure 1: Example for Conjecture1. A continuous simple closed curve in the plane is also called a Jordan curve, and it is the same as an injective map from the unit circle into the plane, or equivalently, a topological embedding S1 ,! R2. Figure 2: We do not require the square to lie fully inside γ, otherwise there are counter- examples. ∗Supported by Deutsche Telekom Stiftung, NSF Grant DMS-0635607, an EPDI-fellowship, and MPIM Bonn. This article is licensed under a Creative Commons BY-NC-SA 3.0 License. 1 In its full generality Toeplitz' problem is still open. So far it has been solved affirmatively for curves that are \smooth enough", by various authors for varying smoothness conditions, see the next section. All of these proofs are based on the fact that smooth curves inscribe generically an odd number of squares, which can be measured in several topological ways. -
Interschool Test 2010 MAΘ National Convention
Interschool Test 2010 MAΘ National Convention Section I – Trivia 101 For this section, determine the answer to each question, then change each letter in the answer to a number by using A=1, B=2, C=3, etc. Add these numbers, then report the sum as your team’s answer. For example, if the question is “Who plays Batman in the most recent Batman movies?”, the answer is Christian Bale, and the answer your team will submit is 121 (3+8+18+9+19+20+9+1+14+2+1+12+5=121). 1. What band originally sang the song “Pinball Wizard” as part of their rock opera Tommy? (2 words, 1 point) 2. What 1980s fantasy action film’s tagline was “There can be only one.”? (1 word, 2 points) 3. What sitcom features a regional manager whose name is David Brent in the British version, Gilles Triquet in the French version, and Bernd Stromberg in the German version? (2 words, 2 points) 4. What book, written in 1950 by C.S. Lewis, was released as a movie in December 2005? (7 words, 3 points) 5. What magazine published its 1000th issue in 2006 with a cover that was an homage to the Beatles’ album Sgt. Pepper’s Lonely Hearts Club Band? (2 words, 3 points) 6. What fake news organization features editorials from such personalities as Smoove B, a smooth‐ talking ladies man; Jackie Harvey, a clueless celebrity spotter; Gorzo the Mighty, Emperor of the Universe, a 1930s style sci‐fi villain; and Herbert Kornfeld, an accounts receivable supervisor who speaks in ebonics? (2 words, 5 points) 7. -
Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis -
~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. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Fermat's Last Theorem
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions.[1] The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death.[2] After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[3] It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.[4] Contents The 1670 edition of Diophantus's Arithmetica includes Fermat's Overview commentary, referred to as his "Last Pythagorean origins Theorem" (Observatio Domini Petri Subsequent developments and solution de Fermat), posthumously published Equivalent statements of the theorem by his son. -
A Combinatorial Approach to the Inscribed Square Problem
A Combinatorial Approach to the Inscribed Square Problem Elizabeth Kelley Francis Edward Su, Advisor Rob Thompson, Reader Department of Mathematics May, 2015 Copyright c 2015 Elizabeth Kelley. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract The inscribed square conjecture, also known as Toeplitz’ conjecture or the square peg problem, asserts that every Jordan curve in the Euclidean plane admits an inscribed square. Although there exists no proof of the general conjecture, there are affirmative proofs of the conjecture subject to addi- tional local smoothness conditions. The weakest of these local smooth- ness conditions include the special trapezoid criterion, due to Matschke, and local monotonicity, due to Stromquist. We develop several combinato- rial approaches to the inscribed square problem, explicitly locate inscribed squares for conic sections, and explore the existence of inscribed squares in the Koch snowflake, which we prove to be not locally monotone. Contents Abstract iii 1 Historical Development of the Inscribed Square Problem 1 1.1 Statement of the Problem . .1 1.2 Past Work . .2 1.3 Related Problems . .6 2 Conic Sections 9 3 Koch Snowflake 17 3.1 Inscribed Squares in the Koch Snowflake . 20 4 Discretizing the Parameter Space 35 4.1 Pivot vertex and side length . 36 4.2 Two pivot vertices . -
13 Lectures on Fermat's Last Theorem
I Paulo Ribenboim 13 Lectures on Fermat's Last Theorem Pierre de Fermat 1608-1665 Springer-Verlag New York Heidelberg Berlin Paulo Ribenboim Department of Mathematics and Statistics Jeffery Hall Queen's University Kingston Canada K7L 3N6 Hommage a AndrC Weil pour sa Leqon: goat, rigueur et pCnCtration. AMS Subiect Classifications (1980): 10-03, 12-03, 12Axx Library of Congress Cataloguing in Publication Data Ribenboim, Paulo. 13 lectures on Fermat's last theorem. Includes bibliographies and indexes. 1. Fermat's theorem. I. Title. QA244.R5 512'.74 79-14874 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. @ 1979 by Springer-Verlag New York Inc. Printed in the United States of America. 987654321 ISBN 0-387-90432-8 Springer-Verlag New York ISBN 3-540-90432-8 Springer-Verlag Berlin Heidelberg Preface Fermat's problem, also called Fermat's last theorem, has attracted the attention of mathematicians for more than three centuries. Many clever methods have been devised to attack the problem, and many beautiful theories have been created with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topic is presented in the form of lectures, where I survey the main lines of work on the problem. In the first two lectures, there is a very brief description of the early history, as well as a selection of a few of the more representative recent results. In the lectures which follow, I examine in suc- cession the main theories connected with the problem. -
Collection Volume I
Collection volume I PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Thu, 29 Jul 2010 21:47:23 UTC Contents Articles Abstraction 1 Analogy 6 Bricolage 15 Categorization 19 Computational creativity 21 Data mining 30 Deskilling 41 Digital morphogenesis 42 Heuristic 44 Hidden curriculum 49 Information continuum 53 Knowhow 53 Knowledge representation and reasoning 55 Lateral thinking 60 Linnaean taxonomy 62 List of uniform tilings 67 Machine learning 71 Mathematical morphology 76 Mental model 83 Montessori sensorial materials 88 Packing problem 93 Prior knowledge for pattern recognition 100 Quasi-empirical method 102 Semantic similarity 103 Serendipity 104 Similarity (geometry) 113 Simulacrum 117 Squaring the square 120 Structural information theory 123 Task analysis 126 Techne 128 Tessellation 129 Totem 137 Trial and error 140 Unknown unknown 143 References Article Sources and Contributors 146 Image Sources, Licenses and Contributors 149 Article Licenses License 151 Abstraction 1 Abstraction Abstraction is a conceptual process by which higher, more abstract concepts are derived from the usage and classification of literal, "real," or "concrete" concepts. An "abstraction" (noun) is a concept that acts as super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Abstractions may be formed by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behavior, eliminating the characteristics of that particular ball.