Appendix a Elementary Number Theory A.1 Introduction Let Denote the Set of Natural Numbers, the Set of Integers, and the Set of Real Numbers
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Number Theory, Dover Publications, Inc., New York, 1994
Theory of Numbers Divisibility Theory in the Integers, The Theory of Congruences, Number-Theoretic Functions, Primitive Roots, Quadratic Residues Yotsanan Meemark Informal style based on the course 2301331 Theory of Numbers, offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University Second version August 2016 Any comment or suggestion, please write to [email protected] Contents 1 Divisibility Theory in the Integers 1 1.1 The Division Algorithm and GCD . 1 1.2 The Fundamental Theorem of Arithmetic . 5 1.3 The Euclidean Algorithm and Linear Diophantine Equations . 8 2 The Theory of Congruences 13 2.1 Basic Properties of Congruence . 13 2.2 Linear Congruences . 16 2.3 Reduced Residue Systems . 18 2.4 Polynomial Congruences . 21 3 Number-Theoretic Functions 25 3.1 Multiplicative Functions . 25 3.2 The Mobius¨ Inversion Formula . 28 3.3 The Greatest Integer Function . 30 4 Primitive Roots 33 4.1 The Order of an Integer Modulo n ............................ 33 4.2 Integers Having Primitive Roots . 35 4.3 nth power residues . 39 4.4 Hensel’s Lemma . 41 5 Quadratic Residues 45 5.1 The Legendre Symbol . 45 5.2 Quadratic Reciprocity . 48 Bibliography 53 Index 54 Chapter 1 Divisibility Theory in the Integers Let N denote the set of positive integers and let Z be the set of integers. 1.1 The Division Algorithm and GCD Theorem 1.1.1. [Well-Ordering Principle] Every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a in S such that a ≤ b for all b 2 S. -
The Complete Splittings of Finite Abelian Groups
The complete splittings of finite abelian groups Kevin Zhaoa aDepartment of Mathematics, South China normal university, Guangzhou 510631, China Abstract Let G be a finite group. We will say that M and S form a complete split- ting (splitting) of G if every element (nonzero element) g of G has a unique representation of the form g = ms with m ∈ M and s ∈ S, and 0 has a such representation (while 0 has no such representation). In this paper, we determine the structures of complete splittings of finite abelian groups. In particular, for complete splittings of cyclic groups our de- scription is more specific. Furthermore, we show some results for existence and nonexistence of complete splittings of cyclic groups and find a relationship between complete splittings and splittings for finite groups. Keywords: complete splittings, splittings, cyclic groups, finite abelian groups, relationship. 1. Introduction The splittings of finite abelian groups are closely related to lattice tilings and are easy to be generated to lattice packings. Let K0 be a polytope composed of unit cubes and v + K0 be a translate of K0 for some vector v. A family of translations {v + K0 : v ∈ H} is called an integer lattice packing if H is an n-dimensional subgroup of Zn and, for any two vectors v and w in H, the interiors of v + K0 and w + K0 are disjoint; furthermore, if the n-dimensional Euclidean space Zn is contained in the family of these translations, then we call it an integer lattice tiling. The splitting problem can be traceable to a geometric problem posed by H. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Primes in Arithmetical Progression
Colby College Digital Commons @ Colby Honors Theses Student Research 2019 Primes in Arithmetical Progression Edward C. Wessel Colby College Follow this and additional works at: https://digitalcommons.colby.edu/honorstheses Part of the Analysis Commons, and the Number Theory Commons Colby College theses are protected by copyright. They may be viewed or downloaded from this site for the purposes of research and scholarship. Reproduction or distribution for commercial purposes is prohibited without written permission of the author. Recommended Citation Wessel, Edward C., "Primes in Arithmetical Progression" (2019). Honors Theses. Paper 935. https://digitalcommons.colby.edu/honorstheses/935 This Honors Thesis (Open Access) is brought to you for free and open access by the Student Research at Digital Commons @ Colby. It has been accepted for inclusion in Honors Theses by an authorized administrator of Digital Commons @ Colby. Primes in Arithmetical Progression Edward (Teddy) Wessel April 2019 Contents 1 Message to the Reader 3 2 Introduction 3 2.1 Primes . 3 2.2 Euclid . 3 2.3 Euler and Dirichlet . 4 2.4 Shapiro . 4 2.5 Formal Statement . 5 3 Arithmetical Functions 6 3.1 Euler’s Totient Function . 6 3.2 The Mobius Function . 6 3.3 A Relationship Between j and m ................ 7 3.4 The Mangoldt Function . 8 4 Dirichlet Convolution 9 4.1 Definition . 10 4.2 Some Basic Properties of Dirichlet Multiplication . 10 4.3 Identity and Inverses within Dirichlet Multiplication . 11 4.4 Multiplicative Functions and their Relationship with Dirich- let Convolution . 13 4.5 Generalized Convolutions . 15 4.6 Partial Sums of Dirichlet Convolution . -
Modular Arithmetic Some Basic Number Theory: CS70 Summer 2016 - Lecture 7A Midterm 2 Scores Out
Announcements Agenda Modular Arithmetic Some basic number theory: CS70 Summer 2016 - Lecture 7A Midterm 2 scores out. • Modular arithmetic Homework 7 is out. Longer, but due next Wednesday before class, • GCD, Euclidean algorithm, and not next Monday. multiplicative inverses David Dinh • Exponentiation in modular There will be no homework 8. 01 August 2016 arithmetic UC Berkeley Mathematics is the queen of the sciences and number theory is the queen of mathematics. -Gauss 1 2 Modular Arithmetic Motivation: Clock Math Congruences Modular Arithmetic If it is 1:00 now. What time is it in 2 hours? 3:00! Theorem: If a c (mod m) and b d (mod m), then a + b c + d x is congruent to y modulo m, denoted “x y (mod m)”... ≡ ≡ ≡ What time is it in 5 hours? 6:00! ≡ (mod m) and a b = c d (mod m). · · What time is it in 15 hours? 16:00! • if and only if (x y) is divisible by m (denoted m (x y)). Proof: Addition: (a + b) (c + d) = (a c) + (b d). Since a c − | − − − − ≡ Actually 4:00. • if and only if x and y have the same remainder w.r.t. m. (mod m) the first term is divisible by m, likewise for the second 16 is the “same as 4” with respect to a 12 hour clock system. • x = y + km for some integer k. term. Therefore the entire expression is divisible by m, so Clock time equivalent up to to addition/subtraction of 12. a + b c + d (mod m). What time is it in 100 hours? 101:00! or 5:00. -
Zaneville Testimonial Excerpts and Reminisces
Greetings Dean & Mrs. “Jim” Fonseca, Barbara & Abe Osofsky, Surender Jain, Pramod Kanwar, Sergio López-Permouth, Dinh Van Huynh& other members of the Ohio Ring “Gang,” conference speakers & guests, Molly & I are deeply grateful you for your warm and hospitable welcome. Flying out of Newark New Jersey is always an iffy proposition due to the heavy air traffic--predictably we were detained there for several hours, and arrived late. Probably not coincidently, Barbara & Abe Osofsky, Christian Clomp, and José Luis Gómez Pardo were on the same plane! So we were happy to see Nguyen Viet Dung, Dinh Van Huynh, and Pramod Kanwar at the Columbus airport. Pramod drove us to the Comfort Inn in Zanesville, while Dinh drove Barbara and Abe, and José Luis & Christian went with Nguyen. We were hungry when we arrived in Zanesville at the Comfort Inn, Surender and Dinh pointed to nearby restaurants So, accompanied by Barbara & Abe Osofsky and Peter Vámos, we had an eleventh hour snack at Steak and Shake. (Since Jose Luis had to speak first at the conference, he wouldn’t join us.) Molly & I were so pleased Steak and Shake’s 50’s décor and music that we returned the next morning for breakfast. I remember hearing“You Are So Beautiful,” “Isn’t this Romantic,” and other nostalgia-inducing songs. Rashly, I tried to sing a line from those two at the Banquet, but fell way short of Joe Cocker’s rendition of the former. (Cocker’s is free to hear on You Tube on the Web.) Zanesville We also enjoyed other aspects of Zanesville and its long history. -
Group Properties of the Residue Classes of Certain Kroneckermodular Systems and Some Related Generalizationsin Number Theory*
GROUP PROPERTIES OF THE RESIDUE CLASSES OF CERTAIN KRONECKERMODULAR SYSTEMS AND SOME RELATED GENERALIZATIONSIN NUMBER THEORY* BT EDWARD KIRCHER The object of this paper is to study the groups formed by the residue classes of a certain type of Kronecker modular system and some closely related gen- eralizations of well-known theorems in number theory. The type of modular system to be studied is of the form 9JÎ = (mn, m„_i, • • •, mi, m). Here m, defined by ( mi, m2, ■• • , m* ), is an ideal in the algebraic domain Q of degree k. Each term m,, i = 1, 2, • • • , n, belongs to the domain of in- tegrity of Í2¿ = (ß, xi, x2, • • • , Xi), and is defined by the fundamental system ( ^'i0 ) typ, • - - > typ ) • The various typ, j = 1, 2, • • • , j,-, are rational integral functions of Xj with coefficients that are in turn rational integral func- tions of a:i, x2, • • •, Xi-i, with coefficients that are algebraic integers in ß. In every case the coefficient of the highest power of a-¿in each of the typ shall be equal to 1. We shall see later that the developments of this paper also apply to modular systems where the last restriction here cited is omitted, being re- placed by another admitting more systems, these new systems in every case being equivalent to a system in the standard form as here defined. Any expression that fulfills all of the conditions placed upon each typ with the possible ex- ception of the last one, we shall call a polynomial, and no other expression shall be so designated. -
Scientific Workplace· • Mathematical Word Processing • LATEX Typesetting Scientific Word· • Computer Algebra
Scientific WorkPlace· • Mathematical Word Processing • LATEX Typesetting Scientific Word· • Computer Algebra (-l +lr,:znt:,-1 + 2r) ,..,_' '"""""Ke~r~UrN- r o~ r PooiliorK 1.931'J1 Po6'lf ·1.:1l26!.1 Pod:iDnZ 3.881()2 UfW'IICI(JI)( -2.801~ ""'"""U!NecteoZ l!l!iS'11 v~ 0.7815399 Animated plots ln spherical coordln1tes > To make an anlm.ted plot In spherical coordinates 1. Type an expression In thr.. variables . 2 WMh the Insertion poilt In the expression, choose Plot 3D The next exampfe shows a sphere that grows ftom radius 1 to .. Plot 3D Animated + Spherical The Gold Standard for Mathematical Publishing Scientific WorkPlace and Scientific Word Version 5.5 make writing, sharing, and doing mathematics easier. You compose and edit your documents directly on the screen, without having to think in a programming language. A click of a button allows you to typeset your documents in LAT£X. You choose to print with or without LATEX typesetting, or publish on the web. Scientific WorkPlace and Scientific Word enable both professionals and support staff to produce stunning books and articles. Also, the integrated computer algebra system in Scientific WorkPlace enables you to solve and plot equations, animate 20 and 30 plots, rotate, move, and fly through 3D plots, create 3D implicit plots, and more. MuPAD' Pro MuPAD Pro is an integrated and open mathematical problem solving environment for symbolic and numeric computing. Visit our website for details. cK.ichan SOFTWARE , I NC. Visit our website for free trial versions of all our products. www.mackichan.com/notices • Email: info@mac kichan.com • Toll free: 877-724-9673 It@\ A I M S \W ELEGRONIC EDITORIAL BOARD http://www.math.psu.edu/era/ Managing Editors: This electronic-only journal publishes research announcements (up to about 10 Keith Burns journal pages) of significant advances in all branches of mathematics. -
Joseph Wedderburn
Joseph Wedderburn Joseph Wedderburn Maclagan Henry (1882-1948) 02 februarie 1882 Născut Forfar , Angus , Scoţia 09 octombrie 1948 (cu vârsta 66) Decedat Princeton, New Jersey , Statele Unite ale Americii Reşedinţă Statele Unite ale Americii Cetăţenie American Naţionalitate Scoţian Domenii Matematician Instituţii Universitatea Princeton Alma Mater Universitatea din Edinburgh De doctorat George Chrystal Merrill Flood Doctoranzi Nathan Jacobson Ernst Snapper Cunoscut pentru Artin-Wedderburn teorema premii MacDougall-Brisbane Medalia de aur notabile Joseph Henry Maclagan Wedderburn (02 februarie 1882 Forfar, Angus , Scoţia - 9 octombrie 1948, Princeton, New Jersey ) a fost un matematician scoţian, care a predat la Universitatea Princeton pentru cea mai mare din cariera sa. Algebrist de prestigiu , el a dovedit că orice corp finit este comutativ , şi o parte a teoremei Artin-Wedderburn referitoare la algebre. De asemenea, el a lucrat in teoria grupurilor si algebra matricilor . Viaţa si opera Joseph Wedderburn a fost al zecelea din cei 14 copii ai medicului Alexander Wedderburn şi ai Ogilvie Anne. În 1898, el a intrat la Universitatea din Edinburgh . In 1903, el a publicat primele trei lucrari, a lucrat ca asistent la Laboratorul de Fizică a Universităţii, şi a obţinut o diplomă MA cu onoruri deosebite în matematică. A studiat apoi timp scurt la Universitatea din Leipzig şi Universitatea din Berlin, unde s-a întâlnit cu algebriştii Frobenius şi Schur. O Bursa Carnegie i-a permis să-şi petreacă anul universitar 1904-1905 la Universitatea din Chicago, unde a lucrat cu Oswald Veblen , EH Moore , şi cel mai important, Leonard Dickson , care avea să devină cel mai important algebrist american al acelor timpuri. -
Handbook of Number Theory Ii
HANDBOOK OF NUMBER THEORY II by J. Sandor´ Babes¸-Bolyai University of Cluj Department of Mathematics and Computer Science Cluj-Napoca, Romania and B. Crstici formerly the Technical University of Timis¸oara Timis¸oara Romania KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 1-4020-2546-7 (HB) ISBN 1-4020-2547-5 (e-book) Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved C 2004 Kluwer Academic Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands. Contents PREFACE 7 BASIC SYMBOLS 9 BASIC NOTATIONS 10 1 PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES 15 1.1 Introduction .............................. 15 1.2 Some historical facts ......................... 16 1.3 Even perfect numbers ......................... 20 1.4 Odd perfect numbers ......................... 23 1.5 Perfect, multiperfect and multiply perfect numbers ......... 32 1.6 Quasiperfect, almost perfect, and pseudoperfect numbers ............................... -
Non-Desarguian Geometries and the Foundations of Geometry from David Hilbert to Ruth Moufang
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 31 (2004) 320–336 www.elsevier.com/locate/hm Non-Desarguian geometries and the foundations of geometry from David Hilbert to Ruth Moufang Cinzia Cerroni Dipartimento di Matematica ed Applicazioni, Università degli Studi di Palermo, Palermo, Italy Available online 19 November 2003 Abstract In this work, we study the development of non-Desarguian geometry from David Hilbert to Ruth Moufang. We will see that a geometric model became a complicated interrelation between algebra and geometry. 2003 Elsevier Inc. All rights reserved. Sommario In questo articolo analizziamo lo sviluppo della geometria non-Desarguesiana, da David Hilbert a Ruth Moufang. Come vedremo, da un iniziale modello geometrico si arriverà ad una sottile interrelazione tra proprietà algebriche e proprietà geometriche di queste geometrie. 2003 Elsevier Inc. All rights reserved. MSC: 01A70; 01A60; 51A35; 05B35; 17D05 Keywords: David Hilbert; Forest Ray Moulton; Joseph H.M. Wedderburn; Oswald Veblen; Max Dehn; Ruth Moufang; Non-Desarguesian geometry; Quasifield; Alternative ring 1. Introduction In 1899, David Hilbert published the Grundlagen der Geometrie, a book that opened up research in the foundations of geometry. In fact, the Grundlagen took the axiomatic method both as a culmination of geometry and as the beginning of a new phase of research. In that new phase, the links between the postulates were not seen as the cold expression of their logical relations or interdependence, but as the creation of new geometries having equal importance at the research level. For example, the independence E-mail address: [email protected]. -
Noether's Contribution to the Rise of the Structural Approach in Algebra
Noether’s Contribution to the Rise of the Structural Approach in Algebra Leo Corry ‚ Tel Aviv University To appear in: Proving It Her Way. Emmy Noether, a Life in Mathematics. Edited by David Rowe and Mechthild Koreuber. Springer (2020). Contents Introduction...................................................................................................1 What is that thing called “structural algebra”? ....................................................3 What is, then, that thing called „classical algebra“: ..............................................4 Dedekind’s Ideals............................................................................................7 Polynomials and Factorization ........................................................................ 10 From p-adic numbers to abstract fields ............................................................ 12 Noether’s Abstract Rings................................................................................ 13 Other works in Algebra in the early 20th Century .............................................. 17 Concluding Remarks ...................................................................................... 19 References................................................................................................... 20 Introduction The centrality of the notion of a mathematical structure was a prominent motif in mathematical discourse throughout the twentieth century. A representative instance of this appears in a well- known article of 1979 by Jean Dieudonné (1906-1922),