Neverending Fractions
|
|
- Olivia Haynes
- 5 years ago
- Views:
Transcription
1 Neverending Fractions An Introduction to Continued Fractions ( c March 13, 2014: do not circulate) Jonathan Borwein, Alf van der Poorten, Jeffrey Shallit, and Wadim Zudilin
2
3
4
5 Contents Preface page ix 1 Some preliminaries from number theory Divisibility in Z. Euclidean algorithm Primes Fibonacci numbers and the complexity of the Euclidean algorithm Approximation of real numbers by rationals Farey sequences 16 Notes 21 2 Continued fractions A π-overview of the theory Finite continued fractions and the Euclidean algorithm Algebraic theory of continued fractions Relations for continuants of a continued fraction Continued fraction of a real number A taste of Diophantine approximation Equivalent numbers Continued fraction of a quadratic irrational The Euler Lagrange theorem Examples of non-periodic continued fractions The continued fraction for e The order of approximation of e by rationals Bounded partial quotients In the footsteps of Maillet 54 Notes 61 v
6 vi Contents 3 Metric theory of continued fractions Partial quotients of a number as functions of that number Growth of partial quotients of a typical real number Approximation of almost all real numbers by rationals Gauss Kuzmin statistics 75 Notes 77 4 Quadratic irrationals through a magnifier Continued fractions of algebraic numbers Quadratic irrationals revisited Units and Pell s equation Negative expansions and the many surprises of the number Notes 94 5 Hyperelliptic curves and Somos sequences Two surprising allegations Continued fractions in function fields Units and torsion The elliptic case Higher genus 112 Notes From folding to Fibonacci Folding Zaremba s conjecture On to Fibonacci Folding and rippling 120 Notes The integer part of qα + β Inhomogeneous Diophantine approximation Lambert series expansions of generating functions High-precision fraud 143 Notes The Erdős Moser equation Arithmetic and analysis of the equation Asymptotics and continued fraction for log Efficient ways of computing continued fractions Bounds for solutions 158 Notes 159
7 Contents vii 9 Irregular continued fractions General theory Euler continued fraction Gauss continued fraction for the hypergeometric function Ramanujan s AGM continued fraction An irregular continued fraction for ζ(2) = π 2 / The irrationality of π Notes 184 Appendix A Selected continued fractions 188 References 200 Index 209
8
9 Preface This book arose from many lectures the authors delivered independently at different locations to students of different levels. Theory is a scientific name for story. So, if the reader somehow feels uncomfortable about following a theory of continued fractions, he or she might be more content to read the story of neverending fractions. The queen of mathematics number theory remains one of the most accessible parts of significant mathematical knowledge. Continued fractions form a classical area within number theory, and there are many textbooks and monographs devoted to them. Despite their classical nature, continued fractions remain a neverending research field, many of whose results are elementary enough to be explained to a wide audience of graduates, postgraduates and researchers, as well as teachers and even amateurs in mathematics. These are the people to whom this book is addressed. After a standard introduction to continued fractions in the first three chapters, including generalisations such as continued fractions in function fields and irregular continued fractions, there are six topics chapters. In these we give various amazing applications of the theory (irrationality proofs, generating series, combinatorics on words, Somos sequences, Diophantine equations and many other applications) to seemingly unrelated problems in number theory. The main feature that we would like to make apparent through this book is the naturalness of continued fractions and of their expected appearance in mathematics. The book is a combination of formal and informal styles. The aforementioned applications of continued fractions are, for the most part, not to be found in earlier books but only in scattered scientific articles and lectures. We have included various remarks and exercises but have been sparing with the latter. In the topics chapters we do not always give full details. Needless to say, all topics can be followed up in the end notes for each chapter and through the references. ix
10 x Preface We would like to thank many colleagues for useful conversations during the development of this book, especially Mumtaz Hussain, Pieter Moree and James Wan. We are also deeply indebted to the copy-editor Susan Parkinson whose incisive and tireless work on the book has enhanced its appearance immeasurably. Finally, Alf van der Poorten ( ) died before this book could be brought to fruition. He was both a good friend and a fine colleague. We offer this book both in his memory and as a way of bringing to a more general audience some of his wonderful contributions to the area. Chapters 4, 5 and 6 originate in lectures Alf gave in the last few years of his life and, for matters of both taste and necessity, they are largely left as presentations in his unique and erudite style. Alfred Alf Jacobus van der Poorten (16 May 1942 in Amsterdam 9 October 2010 in Sydney) A full mathematical biography of Alf is to be found in the 2013 volume dedicated to his memory [31]. Jon Borwein, Jeffrey Shallit, and Wadim Zudilin Newcastle and Waterloo
Mathematics and the Real World
Mathematics and the Real World Zvi Artstein Prometheus Books, 2014, ISBN 978-1-61614-091-5 Zvi Artstein is a distinguished Professor of Mathematics, specialising in Control Theory and Game Theory, at The
More informationNumber Theory and Algebraic Equations. Odile Marie-Thérèse Pons
Number Theory and Algebraic Equations Odile Marie-Thérèse Pons Published by Science Publishing Group 548 Fashion Avenue New York, NY 10018, U.S.A. http://www.sciencepublishinggroup.com ISBN: 978-1-940366-74-6
More informationCLOSED FORM CONTINUED FRACTION EXPANSIONS OF SPECIAL QUADRATIC IRRATIONALS
CLOSED FORM CONTINUED FRACTION EXPANSIONS OF SPECIAL QUADRATIC IRRATIONALS DANIEL FISHMAN AND STEVEN J. MILLER ABSTRACT. We derive closed form expressions for the continued fractions of powers of certain
More informationCryptography. Number Theory with AN INTRODUCTION TO. James S. Kraft. Lawrence C. Washington. CRC Press
AN INTRODUCTION TO Number Theory with Cryptography James S Kraft Gilman School Baltimore, Maryland, USA Lawrence C Washington University of Maryland College Park, Maryland, USA CRC Press Taylor & Francis
More informationIntroduction to Algebraic Number Theory Part I
Introduction to Algebraic Number Theory Part I A. S. Mosunov University of Waterloo Math Circles November 7th, 2018 Goals Explore the area of mathematics called Algebraic Number Theory. Specifically, we
More informationarxiv: v1 [math.nt] 13 Mar 2017
MAHLER TAKES A REGULAR VIEW OF ZAREMBA MICHAEL COONS arxiv:170304287v1 mathnt] 13 Mar 2017 Abstract In the theory of continued fractions, Zaremba s conjecture states that there is a positive integer M
More informationGENERALIZED PALINDROMIC CONTINUED FRACTIONS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 48, Number 1, 2018 GENERALIZED PALINDROMIC CONTINUED FRACTIONS DAVID M. FREEMAN ABSTRACT. In this paper, we introduce a generalization of palindromic continued
More informationTopics in Number Theory
Topics in Number Theory THE UNIVERSITY SERIES IN MATHEMATICS Series Editor: Joseph J. Kohn Princeton University THE CLASSIFICATION OF FINITE SIMPLE GROUPS Daniel Gorenstein VOLUME 1: GROUPS OF NONCHARACTERISTIC
More informationE.J. Barbeau. Polynomials. With 36 Illustrations. Springer
E.J. Barbeau Polynomials With 36 Illustrations Springer Contents Preface Acknowledgment of Problem Sources vii xiii 1 Fundamentals 1 /l.l The Anatomy of a Polynomial of a Single Variable 1 1.1.5 Multiplication
More informationAn Invitation to Modern Number Theory. Steven J. Miller and Ramin Takloo-Bighash PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD
An Invitation to Modern Number Theory Steven J. Miller and Ramin Takloo-Bighash PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Contents Foreword Preface Notation xi xiii xix PART 1. BASIC NUMBER THEORY
More informationPi: The Next Generation
Pi: The Next Generation . David H. Bailey Jonathan M. Borwein Pi: The Next Generation A Sourcebook on the Recent History of Pi and Its Computation David H. Bailey Lawrence Berkeley National Laboratory
More informationMAHLER TAKES A REGULAR VIEW OF ZAREMBA
#A6 INTEGERS 8A (28) MAHLER TAKES A REGULAR VIEW OF ZAREMBA Michael Coons School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, New South Wales, Australia michael.coons@newcastle.edu.au
More informationComprehensive Introduction to Linear Algebra
Comprehensive Introduction to Linear Algebra WEB VERSION Joel G Broida S Gill Williamson N = a 11 a 12 a 1n a 21 a 22 a 2n C = a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn a m1 a m2 a mn Comprehensive
More informationIntroduction to the Mathematical and Statistical Foundations of Econometrics Herman J. Bierens Pennsylvania State University
Introduction to the Mathematical and Statistical Foundations of Econometrics 1 Herman J. Bierens Pennsylvania State University November 13, 2003 Revised: March 15, 2004 2 Contents Preface Chapter 1: Probability
More informationA Geometric Proof that e is Irrational and a New Measure of its Irrationality
A Geometric Proof that e is Irrational and a New Measure of its Irrationality Jonathan Sondow. INTRODUCTION. While there exist geometric proofs of irrationality for 2 [2], [27], no such proof for e, π,
More informationRandom walks in the plane
Tulane University, New Orleans August 2, 2010 Joint work with: Jon Borwein Dirk Nuyens James Wan U. of Newcastle, AU K.U.Leuven, BE U. of Newcastle, AU We study random walks in the plane consisting of
More informationContinued Fractions New and Old Results
Continued Fractions New and Old Results Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca https://www.cs.uwaterloo.ca/~shallit Joint
More information1 Prologue (X 111 X 14 )F = X 11 (X 100 X 3 )F = X 11 (X 200 X 6 ) = X 211 X 17.
Prologue In this chapter we give a couple of examples where the method of auxiliary polynomials is used for problems that have no diophantine character. Thus we are not following Sam Goldwyn s advice to
More informationPELL S EQUATION NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA, ODISHA
PELL S EQUATION A Project Report Submitted by PANKAJ KUMAR SHARMA In partial fulfillment of the requirements For award of the degree Of MASTER OF SCIENCE IN MATHEMATICS UNDER GUIDANCE OF Prof GKPANDA DEPARTMENT
More informationGeneralized Continued Logarithms and Related Continued Fractions
Generalized Continued Logarithms and Related Continued Fractions Jonathan M Borwein Kevin G Hare Jason G Lynch June 22 206 Abstract We study continued arithms as introduced by Bill Gosper and studied by
More informationSolutions to Assignment 1
Solutions to Assignment 1 Question 1. [Exercises 1.1, # 6] Use the division algorithm to prove that every odd integer is either of the form 4k + 1 or of the form 4k + 3 for some integer k. For each positive
More informationContents. Preface to the First Edition. Preface to the Second Edition. Preface to the Third Edition
Contents Preface to the First Edition Preface to the Second Edition Preface to the Third Edition i iii iv Glossary of Symbols A. Prime Numbers 3 A1. Prime values of quadratic functions. 7 A2. Primes connected
More informationContents Real Numbers Polynomials 2 0 Pair of Linear Equations in Two Variables 3 8 Quadratic Equations 7 0
Foreword Preface Contents 1. Real Numbers 1 1.1 Introduction 1 1. Euclid s Division Lemma 1.3 The Fundamental Theorem of Arithmetic 7 1.4 Revisiting Irrational Numbers 11 1.5 Revisiting Rational Numbers
More informationA Classical Introduction to Modern Number Theory
Kenneth Ireland Michael Rosen A Classical Introduction to Modern Number Theory Second Edition Springer Contents Preface to the Second Edition Preface v vii CHAPTER 1 Unique Factorization 1 1 Unique Factorization
More informationX 0 is of the form Ɣ\B 2 and the map B 2 X 0
Introduction Historical precedent for the results in this book can be found in the theory of Riemann surfaces. Every compact Riemann surface of genus g 2has representations both as a plane algebraic curve,
More informationx 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.
1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the
More information#A5 INTEGERS 18A (2018) EXPLICIT EXAMPLES OF p-adic NUMBERS WITH PRESCRIBED IRRATIONALITY EXPONENT
#A5 INTEGERS 8A (208) EXPLICIT EXAMPLES OF p-adic NUMBERS WITH PRESCRIBED IRRATIONALITY EXPONENT Yann Bugeaud IRMA, UMR 750, Université de Strasbourg et CNRS, Strasbourg, France bugeaud@math.unistra.fr
More informationMthEd/Math 300 Williams Fall 2011 Midterm Exam 3
Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 3 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first
More informationInstructor s suggestions for use of Number Theory notes
Number Theory Instructor s suggestions for use of Number Theory notes Goal. Our goal is to inspire students to be actively engaged in mathematical thought. We want them to discover ideas on their own,
More informationMATHEMATICS 6180, SPRING 2017 SOME MOTIVATIONAL PROBLEMS IN NUMBER THEORY. p k
MATHEMATICS 680, SPRING 207 SOME MOTIVATIONAL PROBLEMS IN NUMBER THEORY KATHERINE E. STANGE Number theory may be loosely defined as the study of the integers: in particular, the interaction between their
More informationNOTES ON IRRATIONALITY AND TRANSCENDENCE
NOTES ON IRRATIONALITY AND TRANSCENDENCE Frits Beukers September, 27 Introduction. Irrationality Definition.. Let α C. We call α irrational when α Q. Proving irrationality and transcendence of numbers
More informationContinued Fractions: Introduction and Applications
PROCEEDINGS OF THE ROMAN NUMBER THEORY ASSOCIATION Volume 2, Number, March 207, pages 6-8 Michel Waldschmidt Continued Fractions: Introduction and Applications written by Carlo Sanna The continued fraction
More informationFoundations of Analysis. Joseph L. Taylor. University of Utah
Foundations of Analysis Joseph L. Taylor University of Utah Contents Preface vii Chapter 1. The Real Numbers 1 1.1. Sets and Functions 2 1.2. The Natural Numbers 8 1.3. Integers and Rational Numbers 16
More informationAnalytic. Number Theory. Exploring the Anatomy of Integers. Jean-Marie. De Koninck. Florian Luca. ffk li? Graduate Studies.
Analytic Number Theory Exploring the Anatomy of Integers Jean-Marie Florian Luca De Koninck Graduate Studies in Mathematics Volume 134 ffk li? American Mathematical Society Providence, Rhode Island Preface
More informationRECURRENT SEQUENCES IN THE EQUATION DQ 2 = R 2 + N INTRODUCTION
RECURRENT SEQUENCES IN THE EQUATION DQ 2 = R 2 + N EDGAR I. EMERSON Rt. 2, Box 415, Boulder, Colorado INTRODUCTION The recreational exploration of numbers by the amateur can lead to discovery, or to a
More informationSundaram's Sieve. by Julian Havil. Sundaram's Sieve
1997 2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non commercial use. For other uses, including electronic redistribution,
More informationPartial Differential Equations with Numerical Methods
Stig Larsson Vidar Thomée Partial Differential Equations with Numerical Methods May 2, 2003 Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo Preface Our purpose in this
More informationChakravala - a modern Indian method. B.Sury
Chakravala - a modern Indian method BSury Indian Statistical Institute Bangalore, India sury@isibangacin IISER Pune, India Lecture on October 18, 2010 1 Weil Unveiled What would have been Fermat s astonishment
More informationAnalytic Number Theory
American Mathematical Society Colloquium Publications Volume 53 Analytic Number Theory Henryk Iwaniec Emmanuel Kowalski American Mathematical Society Providence, Rhode Island Contents Preface xi Introduction
More informationMATHEMATICS X l Let x = p q be a rational number, such l If p, q, r are any three positive integers, then, l that the prime factorisation of q is of t
CHAPTER 1 Real Numbers [N.C.E.R.T. Chapter 1] POINTS FOR QUICK REVISION l Euclid s Division Lemma: Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r
More informationSYMMETRY AND SPECIALIZABILITY IN THE CONTINUED FRACTION EXPANSIONS OF SOME INFINITE PRODUCTS
SYMMETRY AND SPECIALIZABILITY IN THE CONTINUED FRACTION EXPANSIONS OF SOME INFINITE PRODUCTS J MC LAUGHLIN Abstract Let fx Z[x] Set f 0x = x and for n 1 define f nx = ff n 1x We describe several infinite
More informationMathematics Without Calculations It s a Beautiful Thing!
Sacred Heart University DigitalCommons@SHU Mathematics Faculty Publications Mathematics Department 8-4-2016 Mathematics Without Calculations It s a Beautiful Thing! Jason J. Molitierno Sacred Heart University,
More informationStudents will be able to simplify numerical expressions and evaluate algebraic expressions. (M)
Morgan County School District Re-3 August What is algebra? This chapter develops some of the basic symbolism and terminology that students may have seen before but still need to master. The concepts of
More informationCONTENTS. Preface Preliminaries 1
Preface xi Preliminaries 1 1 TOOLS FOR ANALYSIS 5 1.1 The Completeness Axiom and Some of Its Consequences 5 1.2 The Distribution of the Integers and the Rational Numbers 12 1.3 Inequalities and Identities
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009 Preface The title of the book sounds a bit mysterious. Why should anyone read this
More informationElementary Number Theory
Elementary Number Theory 21.8.2013 Overview The course discusses properties of numbers, the most basic mathematical objects. We are going to follow the book: David Burton: Elementary Number Theory What
More informationStatistical Investigations as a Tool in Undergraduate Mathematics Research
TEACHING OF STATISTICS HONOLULU, JUNE 5, 2003 Statistical Investigations as a Tool in Undergraduate Mathematics Research Steven J. Miller, Leo Goldmakher, Atul Pokharel Department of Mathematics, Princeton
More informationSUMS OF SQUARES WUSHI GOLDRING
SUMS OF SQUARES WUSHI GOLDRING 1. Introduction Here are some opening big questions to think about: Question 1. Which positive integers are sums of two squares? Question 2. Which positive integers are sums
More informationZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS
ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental
More informationAn Unusual Continued Fraction
An Unusual Continued Fraction arxiv:505.00667v [math.nt] 4 May 205 Dzmitry Badziahin Department of Mathematical Sciences Durham University Lower Mountjoy Stockton Rd Durham, DH 3LE United Kingdom dzmitry.badziahin@durham.ac.uk
More informationarxiv: v2 [math.nt] 25 Jan 2017
GENERALIZED PALINDROMIC CONTINUED FRACTIONS DAVID M FREEMAN arxiv:160406755v2 [mathnt] 25 Jan 2017 ABSTRACT In this paper we introduce a generalization of palindromic continued fractions as studied by
More informationContinued Fractions New and Old Results
Continued Fractions New and Old Results Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca https://www.cs.uwaterloo.ca/~shallit Joint
More informationNatural Numbers: Also called the counting numbers The set of natural numbers is represented by the symbol,.
Name Period Date: Topic: Real Numbers and Their Graphs Standard: 9-12.A.1.3 Objective: Essential Question: What is the significance of a point on a number line? Determine the relative position on the number
More informationNumber Theory, Algebra and Analysis. William Yslas Vélez Department of Mathematics University of Arizona
Number Theory, Algebra and Analysis William Yslas Vélez Department of Mathematics University of Arizona O F denotes the ring of integers in the field F, it mimics Z in Q How do primes factor as you consider
More informationLesson 3.5 Exercises, pages
Lesson 3.5 Exercises, pages 232 238 A 4. Calculate the value of the discriminant for each quadratic equation. a) 5x 2-9x + 4 = 0 b) 3x 2 + 7x - 2 = 0 In b 2 4ac, substitute: In b 2 4ac, substitute: a 5,
More informationLecture Historical introduction to irrationality
The University of Arizona The Southwest Center for Arithmetic Geometry 2008 Arizona Winter School, March 5-9, 2008 Special Functions and Transcendence http://swc.math.arizona.edu/aws/08/index.html Updated:
More informationElliptic Nets With Applications to Cryptography
Elliptic Nets With Applications to Cryptography Katherine Stange Brown University http://www.math.brown.edu/~stange/ Elliptic Divisibility Sequences: Seen In Their Natural Habitat Example Elliptic Divisibility
More informationAdv. Algebra/Algebra II Pacing Calendar August 2016
Pacing Calendar August 2016 Unit 1: Quadratics Revisited Students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist, analogous to
More informationENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS. Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS Preface page xiii 1 The Gamma and Beta Functions 1 1.1 The Gamma
More informationAbdul-Majid Wazwaz. Linear and Nonlinear Integral Equations. Methods and Applications
Abdul-Majid Wazwaz Linear and Nonlinear Integral Equations Methods and Applications Abdul-Majid Wazwaz Linear and Nonlinear Integral Equations Methods and Applications With 4 figures ~ -:tr It i >j: Pt.~l
More informationMath 3012 Applied Combinatorics Lecture 4
August 27, 2015 Math 3012 Applied Combinatorics Lecture 4 William T. Trotter trotter@math.gatech.edu The Principle of Math Induction Postulate If S is a set of positive integers, 1 is in S, and k + 1 is
More informationElliptic Curves. Kenneth A. Ribet. PARC Forum October 17, UC Berkeley
UC Berkeley PARC Forum October 17, 2008 In talking about elliptic curves, one can do no better than to quote the Yale mathematician Serge Lang (1927 2005), who began one of his many monographs as follows:
More informationShi Feng Sheng Danny Wong
Exhibit C A Proof of the Fermat s Last Theorem Shi Feng Sheng Danny Wong Abstract: Prior to the Diophantine geometry, number theory (or arithmetic) was to study the patterns of the numbers and elementary
More informationx Preface for such an audience as well as for undergraduates who desire to look at a topic they may have unwittingly skipped over, or anyone mathemati
I have also omitted here the demonstration of most of my statements, because... if you take the trouble to examine them systematically the demonstrations will present themselves to you and it will be of
More informationIntegrated Arithmetic and Basic Algebra
211 771 406 III T H I R D E D I T I O N Integrated Arithmetic and Basic Algebra Bill E. Jordan Seminole Community College William P. Palow Miami-Dade College Boston San Francisco New York London Toronto
More informationHow to Discover the Rogers Ramanujan Identities
How to Discover the Rogers Ramanujan Identities Gaurav Bhatnagar We examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers..
More informationContext-free languages in Algebraic Geometry and Number Theory
Context-free languages in Algebraic Geometry and Number Theory Ph.D. student Laboratoire de combinatoire et d informatique mathématique (LaCIM) Université du Québec à Montréal (UQÀM) Presentation at the
More informationRassias Conjecture. For any prime p with p>2therearetwoprimesp 1, p 2, with p 1 <p 2 such that p = p 1 + p 2 +1 p 1 . (1)
The International Mathematics Olympiad (IMO), in the last two decades, has become an international institution with an impact in most countries throughout the world, fostering young mathematical talent
More informationMA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra
0.) Real Numbers: Order and Absolute Value Definitions: Set: is a collection of objections in mathematics Real Numbers: set of numbers used in arithmetic MA 80 Lecture Chapter 0 College Algebra and Calculus
More informationA Note on Jacobi Symbols and Continued Fractions
Reprinted from: The American Mathematical Monthly Volume 06, Number, pp 52 56 January, 999 A Note on Jacobi Symbols and Continued Fractions A. J. van der Poorten and P. G. Walsh. INTRODUCTION. It is well
More informationElliptic Nets and Points on Elliptic Curves
Department of Mathematics Brown University http://www.math.brown.edu/~stange/ Algorithmic Number Theory, Turku, Finland, 2007 Outline Geometry and Recurrence Sequences 1 Geometry and Recurrence Sequences
More informationUniversity of Rochester Topology Seminar. Doug Ravenel
Beyond elliptic cohomology and TMF: where number theory and stable homotopy theory meet in mortal combat. University of Rochester Topology Seminar Doug Ravenel March 10, 2006 1 2 1. Introduction This talk
More informationTrinity Christian School Curriculum Guide
Course Title: Calculus Grade Taught: Twelfth Grade Credits: 1 credit Trinity Christian School Curriculum Guide A. Course Goals: 1. To provide students with a familiarity with the properties of linear,
More information(i) 2-5 (ii) (3 + 23) - 23 (v) 2π
Number System - Worksheet Question 1: Express the following in the form p/q, where p and q are integers and q 0. Question 2: Express 0.99999... in the form p/q. Are you surprised by your answer? With your
More informationBLOOM PUBLIC SCHOOL Vasant Kunj, New Delhi Lesson Plan Subject: MATHEMATICS Class: X. Month: April No of Periods: 12 TTT: 4 Periods WT: 8 Periods
BLOOM PUBLIC SCHOOL Vasant Kunj, New Delhi Lesson Plan Subject: MATHEMATICS Class: X Month: April No of Periods: 12 TTT: 4 Periods WT: 8 Periods Chapter-: 1 Learning Objectives Chapter s Name :REAL NUMBERS
More informationTHESIS. Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University
The Hasse-Minkowski Theorem in Two and Three Variables THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By
More informationOn the Cardinality of Mersenne Primes
On the Cardinality of Mersenne Primes Garimella Rama Murthy, Associate Professor, International Institute of Information Technology (IIIT), Gachibowli, Hyderabad-32,AP,INDIA ABSTRACT In this research paper,
More informationApplications of Number Theory in Statistics
Bonfring International Journal of Data Mining, Vol. 2, No., September 202 Applications of Number Theory in Statistics A.M.S. Ramasamy Abstract--- There have been several fascinating applications of Number
More informationMATH 0409: Foundations of Mathematics COURSE OUTLINE
MATH 0409: Foundations of Mathematics COURSE OUTLINE Spring 2016 CRN91085 MW 5:30-7:30pm AM209 Professor Sherri Escobar sherri.escobar@hccs.edu 281-620-1115 Catalog Description: Foundations of Mathematics.
More informationINTRODUCTORY ALGEBRAIC NUMBER THEORY
INTRODUCTORY ALGEBRAIC NUMBER THEORY Algebraic number theory is a subject that came into being through the attempts of mathematicians to try to prove Fermat s last theorem and that now has a wealth of
More informationWaterloo, ON & Lincoln, NE March, Kenneth R. Davidson Allan P. Donsig
Preface This book provides an introduction both to real analysis and to a range of important applications that depend on this material. Three-fifths of the book is a series of essentially independent chapters
More informationConference on Diophantine Analysis and Related Fields 2006 in honor of Professor Iekata Shiokawa Keio University Yokohama March 8, 2006
Diophantine analysis and words Institut de Mathématiques de Jussieu + CIMPA http://www.math.jussieu.fr/ miw/ March 8, 2006 Conference on Diophantine Analysis and Related Fields 2006 in honor of Professor
More informationSums and products. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA
Sums and products Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA International Number Theory Conference in Memory of Alf van der Poorten, AM 12 16 March, 2012 CARMA, the University of Newcastle
More informationNORTH MAC MIDDLE SCHOOL CURRICULUM GUIDE
Teacher Sara Floyd Grade Level 7 th and 8 th Grade Course Pre-Algebra Course Aims To prepare students for Algebra I. Course Description This course is designed for students who need the basic skills required
More informationALGEBRA. COPYRIGHT 1996 Mark Twain Media, Inc. ISBN Printing No EB
ALGEBRA By Don Blattner and Myrl Shireman COPYRIGHT 1996 Mark Twain Media, Inc. ISBN 978-1-58037-826-0 Printing No. 1874-EB Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing Company,
More informationA Simple Left-to-Right Algorithm for Minimal Weight Signed Radix-r Representations
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. XX, NO. X, MONTH 2007 1 A Simple Left-to-Right Algorithm for Minimal Weight Signed Radix-r Representations James A. Muir Abstract We present a simple algorithm
More informationSolving x 2 Dy 2 = N in integers, where D > 0 is not a perfect square. Keith Matthews
Solving x 2 Dy 2 = N in integers, where D > 0 is not a perfect square. Keith Matthews Abstract We describe a neglected algorithm, based on simple continued fractions, due to Lagrange, for deciding the
More information1 Continued Fractions
Continued Fractions To start off the course, we consider a generalization of the Euclidean Algorithm which has ancient historical roots and yet still has relevance and applications today.. Continued Fraction
More informationTable of Contents. 2013, Pearson Education, Inc.
Table of Contents Chapter 1 What is Number Theory? 1 Chapter Pythagorean Triples 5 Chapter 3 Pythagorean Triples and the Unit Circle 11 Chapter 4 Sums of Higher Powers and Fermat s Last Theorem 16 Chapter
More informationIrregular continued fractions
9 Irregular continued fractions We finish our expedition with a look at irregular continued fractions and pass by some classical gems en route. 9. General theory There exist many generalisations of classical
More informationSOME IDEAS FOR 4400 FINAL PROJECTS
SOME IDEAS FOR 4400 FINAL PROJECTS 1. Project parameters You can work in groups of size 1 to 3. Your final project should be a paper of 6 to 10 double spaced pages in length. Ideally it should touch on
More informationMathematics for Engineers and Scientists
Mathematics for Engineers and Scientists Fourth edition ALAN JEFFREY University of Newcastle-upon-Tyne B CHAPMAN & HALL University and Professional Division London New York Tokyo Melbourne Madras Contents
More informationPMATH 340 Lecture Notes on Elementary Number Theory. Anton Mosunov Department of Pure Mathematics University of Waterloo
PMATH 340 Lecture Notes on Elementary Number Theory Anton Mosunov Department of Pure Mathematics University of Waterloo Winter, 2017 Contents 1 Introduction............................. 3 2 Divisibility.
More informationIntroduction. Question 1
Introduction Because I've written several expository articles about Fermat's Last Theorem, I get lots of mail (and e-mail) from amateur mathematicians. Today I'll discuss some questions that have been
More informationAndrzej Schinzel 80: an outstanding scientific career
1 / 18 Andrzej Schinzel 80: an outstanding scientific career Kálmán Győry Short biography 2 / 18 Professor Andrzej Schinzel world-famous number theorist, old friend of Hungarian mathematicians. Born on
More informationModern Computer Algebra
Modern Computer Algebra JOACHIM VON ZUR GATHEN and JURGEN GERHARD Universitat Paderborn CAMBRIDGE UNIVERSITY PRESS Contents Introduction 1 1 Cyclohexane, cryptography, codes, and computer algebra 9 1.1
More informationREAL NUMBERS. Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.
REAL NUMBERS Introduction Euclid s Division Algorithm Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Fundamental
More information#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC
#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania lkjone@ship.edu Received: 9/17/10, Revised:
More informationBeukers integrals and Apéry s recurrences
2 3 47 6 23 Journal of Integer Sequences, Vol. 8 (25), Article 5.. Beukers integrals and Apéry s recurrences Lalit Jain Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G CANADA lkjain@uwaterloo.ca
More informationCollection of problems in probability theory
Collection of problems in probability theory L. D. MESHALKIN Moscow State University Collection of problems in probability theory Translated from the Russian and edited by LEO F. BORON University of Idaho
More information