Elementary Number Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Mathematics Is a Gentleman's Art: Analysis and Synthesis in American College Geometry Teaching, 1790-1840 Amy K
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2000 Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 Amy K. Ackerberg-Hastings Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Higher Education and Teaching Commons, History of Science, Technology, and Medicine Commons, and the Science and Mathematics Education Commons Recommended Citation Ackerberg-Hastings, Amy K., "Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 " (2000). Retrospective Theses and Dissertations. 12669. https://lib.dr.iastate.edu/rtd/12669 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margwis, and improper alignment can adversely affect reproduction. in the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. -
Prime Divisors in the Rationality Condition for Odd Perfect Numbers
Aid#59330/Preprints/2019-09-10/www.mathjobs.org RFSC 04-01 Revised The Prime Divisors in the Rationality Condition for Odd Perfect Numbers Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92037 Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 4m+1 ℓ 2αi 1) i=1 qi to establish that there do not exist any odd integers with equality (4k+1)4m+2−1 between σ(N) and 2N. The existence of distinct prime divisors in the repunits 4k , 2α +1 Q q i −1 i , i = 1,...,ℓ, in σ(N) follows from a theorem on the primitive divisors of the Lucas qi−1 sequences and the square root of the product of 2(4k + 1), and the sequence of repunits will not be rational unless the primes are matched. Minimization of the number of prime divisors in σ(N) yields an infinite set of repunits of increasing magnitude or prime equations with no integer solutions. MSC: 11D61, 11K65 Keywords: prime divisors, rationality condition 1. Introduction While even perfect numbers were known to be given by 2p−1(2p − 1), for 2p − 1 prime, the universality of this result led to the the problem of characterizing any other possible types of perfect numbers. It was suggested initially by Descartes that it was not likely that odd integers could be perfect numbers [13]. After the work of de Bessy [3], Euler proved σ(N) that the condition = 2, where σ(N) = d|N d is the sum-of-divisors function, N d integer 4m+1 2α1 2αℓ restricted odd integers to have the form (4kP+ 1) q1 ...qℓ , with 4k + 1, q1,...,qℓ prime [18], and further, that there might exist no set of prime bases such that the perfect number condition was satisfied. -
A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members. -
Appendix a Tables of Fermat Numbers and Their Prime Factors
Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard -
Grade 7 Mathematics Strand 6 Pattern and Algebra
GR 7 MATHEMATICS S6 1 TITLE PAGE GRADE 7 MATHEMATICS STRAND 6 PATTERN AND ALGEBRA SUB-STRAND 1: NUMBER PATTERNS SUB-STRAND 2: DIRECTED NUMBERS SUB-STRAND 3: INDICES SUB-STARND 4: ALGEBRA GR 7 MATHEMATICS S6 2 ACKNOWLEDGEMENT Acknowledgements We acknowledge the contributions of all Secondary and Upper Primary Teachers who in one way or another helped to develop this Course. Special thanks to the Staff of the mathematics Department of FODE who played active role in coordinating writing workshops, outsourcing lesson writing and editing processes, involving selected teachers of Madang, Central Province and NCD. We also acknowledge the professional guidance provided by the Curriculum Development and Assessment Division throughout the processes of writing and, the services given by the members of the Mathematics Review and Academic Committees. The development of this book was co-funded by GoPNG and World Bank. MR. DEMAS TONGOGO Principal- FODE . Written by: Luzviminda B. Fernandez SCO-Mathematics Department Flexible Open and Distance Education Papua New Guinea Published in 2016 @ Copyright 2016, Department of Education Papua New Guinea All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic, mechanical, photocopying, recording or any other form of reproduction by any process is allowed without the prior permission of the publisher. ISBN: 978 - 9980 - 87 - 250 - 0 National Library Services of Papua New Guinea Printed by the Flexible, Open and Distance Education GR 7 MATHEMATICS S6 3 CONTENTS CONTENTS Page Secretary‟s Message…………………………………….…………………………………......... 4 Strand Introduction…………………………………….…………………………………………. 5 Study Guide………………………………………………….……………………………………. 6 SUB-STRAND 1: NUMBER PATTERNS ……………...….……….……………..……….. -
A Quantum Primality Test with Order Finding
A quantum primality test with order finding Alvaro Donis-Vela1 and Juan Carlos Garcia-Escartin1,2, ∗ 1Universidad de Valladolid, G-FOR: Research Group on Photonics, Quantum Information and Radiation and Scattering of Electromagnetic Waves. o 2Universidad de Valladolid, Dpto. Teor´ıa de la Se˜nal e Ing. Telem´atica, Paseo Bel´en n 15, 47011 Valladolid, Spain (Dated: November 8, 2017) Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat’s theorem. For an integer N, the test tries to find an element of the multiplicative group of integers modulo N with order N − 1. If one is found, the number is known to be prime. During the test, we can also show most of the times N is composite with certainty (and a witness) or, after log log N unsuccessful attempts to find an element of order N − 1, declare it composite with high probability. The algorithm requires O((log n)2n3) operations for a number N with n bits, which can be reduced to O(log log n(log n)3n2) operations in the asymptotic limit if we use fast multiplication. Prime numbers are the fundamental entity in number For a prime N, ϕ(N) = N 1 and we recover Fermat’s theory and play a key role in many of its applications theorem. − such as cryptography. Primality tests are algorithms that If we can find an integer a for which aN−1 1 mod N, determine whether a given integer N is prime or not. -
Newsletter 115, 2017
Cumann Oid´ıMatamaitice na hEireann´ N e w s l e t t e r Irish Mathematics Teachers' Association 2017 Number 115 Maths Development Team Important update From September 1st 2017, the Maths Development Team(MDT) will form part of the Professional Development Service for Teachers. We will continue to provide professional development to post-primary teachers of Mathematics through the following supports: ◦ The effective use of Lesson Study: All new participants are asked to sign up at (https://goo.gl/forms/GJRS7GLuHdehVZqz1) for one of our three-day Induction Programmes, which will be held in Dublin, Carrick-on-Shannon and Limerick on September 27th , 28th and 29th , 2017. Closing date for receipt of applications: Monday 11th September 2017. ◦ A series of six Lesson Study meetings to be held from October 2017 to February 2018. These meetings will be conducted in over 50 locations nationally. Contact Administrator Gráinne Haughney: email [email protected] Phone: 01- 8576422 for venues. ◦ Ongoing participation in Lesson Study for all mathematics teachers. ◦ On-line support and resources available at www.projectmaths.ie ◦ Half-day school visits to support the on-going professional development of teachers. The primary focus will be on teaching and learning using reflective approaches with particular emphasis on: i. Algebra through the Lens of Functions, ii. Geometry through the Lens of Enquiry, iii. School-identified needs. Please note that substitution will not be provided for teachers availing of these school visits. If you wish to apply for a member of the team to visit your Mathematics Department during the 2017-2018 school year, please contact [email protected] or [email protected] for an application form. -
The History of the Primality of One—A Selection of Sources
THE HISTORY OF THE PRIMALITY OF ONE|A SELECTION OF SOURCES ANGELA REDDICK, YENG XIONG, AND CHRIS K. CALDWELLy This document, in an altered and updated form, is available online: https:// cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html. Please use that document instead. The table below is a selection of sources which address the question \is the number one a prime number?" Whether or not one is prime is simply a matter of definition, but definitions follow use, context and tradition. Modern usage dictates that the number one be called a unit and not a prime. We choose sources which made the author's view clear. This is often difficult because of language and typographical barriers (which, when possible, we tried to reproduce for the primary sources below so that the reader could better understand the context). It is also difficult because few addressed the question explicitly. For example, Gauss does not even define prime in his pivotal Disquisitiones Arithmeticae [45], but his statement of the fundamental theorem of arithmetic makes his stand clear. Some (see, for example, V. A. Lebesgue and G. H. Hardy below) seemed ambivalent (or allowed it to depend on the context?)1 The first column, titled `prime,' is yes when the author defined one to be prime. This is just a raw list of sources; for an evaluation of the history see our articles [17] and [110]. Any date before 1200 is an approximation. We would be glad to hear of significant additions or corrections to this list. Date: May 17, 2016. 2010 Mathematics Subject Classification. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration
Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration Bart de Keijzer∗ Tomas B. Klosy Yingqian Zhangz Abstract We study the power index voting game design problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distri- bution. Our goal is to find algorithms that solve this problem exactly. Thereto, we consider various subclasses of simple games, and their asso- ciated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions (also known as shift-maximal losing coalitions) of a game from its list of roof coalitions (also known as shift-minimal winning coalitions), and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the synthesis problem, we then improve on this naive algorithm, and we obtain an enumeration algorithm that runs in quadratic exponential time (that is, 2 O(2n · p(n)) for a polynomial p). Moreover, we show that this algorithm runs in output-polynomial time, making it the best possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime algorithm for the power index voting game design problem that runs in exponential time. -
View This Volume's Front and Back Matter
Titles in This Series Volume 8 Kare n Hunger Parshall and David £. Rowe The emergenc e o f th e America n mathematica l researc h community , 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moore 1994 7 Hen k J. M. Bos Lectures in the history of mathematic s 1993 6 Smilk a Zdravkovska and Peter L. Duren, Editors Golden years of Moscow mathematic s 1993 5 Georg e W. Mackey The scop e an d histor y o f commutativ e an d noncommutativ e harmoni c analysis 1992 4 Charle s W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Pete r L. Duren, editor, et al. A century of mathematics in America, part III 1989 2 Pete r L. Duren, editor, et al. A century of mathematics in America, part II 1989 1 Pete r L. Duren, editor, et al. A century of mathematics in America, part I 1988 This page intentionally left blank https://doi.org/10.1090/hmath/008 History of Mathematics Volume 8 The Emergence o f the American Mathematical Research Community, 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moor e Karen Hunger Parshall David E. Rowe American Mathematical Societ y London Mathematical Societ y 1991 Mathematics Subject Classification. Primary 01A55 , 01A72, 01A73; Secondary 01A60 , 01A74, 01A80. Photographs o n th e cove r ar e (clockwis e fro m right ) th e Gottinge n Mathematisch e Ges - selschafft, Feli x Klein, J. J. Sylvester, and E. H. Moore. -
The First 50 Million Prime Numbers*
7 sin ~F(s) The First 50 Million I - s ; sin -- Prime Numbers* s Don Zagier for the algebraist it is "the characteristic of a finite field" or "a point in Spec ~" or "a non-archimedean valuation"; To my parents the combinatorist defines the prime num- bers inductively by the recursion (I) 1 Pn+1 = [I - log2( ~ + n (-1)r )] r~1 ISil<'~'<irSn 2 pi1"''pir - I ([x] = biggest integer S x); and, finally, the logicians have recent- iv been defininq the primes as the posi- tive values of the polynomial (2) F(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q, r,s,t,u,v,w,x,y,z) = [k + 2] [1 - (wz+h+j-q) 2 - (2n+p+q+z-e) 2 - (a2y2-y2+1-x2)2 - ({e4+2e3}{a+1}2+1-o2) 2 - (16{k+1}3{k+2}{n+1}2+1-f2)2 - ({(a+u~-u2a) 2-1}{n+4dy}2+1-{x+cu}2) 2 - (ai+k+1-l-i) 2 - I would like to tell you today about a ({gk+2g+k+1}{h+j}+h-z) 2 - subject which, although I have not worked (16r2y~{a2-1}+1-u2) 2 - in it myself, has always extraordinarily (p-m+l{a-n-1}+b{2an+2a-n2-2n-2}) 2 - captivated me, and which has fascinated (z-pm+pla-p21+t{2ap-p2-1}) 2 - mathematicians from the earliest times (qTx+y{a-p-1}+s{2ap+2a-p2-2~-2})2 - until the present - namely, the question (a~12-12+1-m2) 2 - (n+l+v-y) ].