DOCSLIB.ORG

25 Primes in Arithmetic Progression

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 SA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 K office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Ribenboim, Paulo. Title: Prime numbers, friends who give problems : a trialogue with Papa Paulo / by Paulo Ribenboim (Queen’s niversity, Canada). Description: New Jersey : World Scientific, 2016. | Includes indexes. Identifiers: LCCN 2016020705| ISBN 9789814725804 (hardcover : alk. paper) | ISBN 9789814725811 (softcover : alk. paper) Subjects: LCSH: Numbers, Prime. Classification: LCC QA246 .R474 2016 | DDC 512.7/23--dc23 LC record available at https://lccn.loc.gov/2016020705 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, SA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: [email protected] Printed in Singapore YingOi - Prime Numbers, Friends Who Give Problems.indd 1 22-08-16 9:11:29 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page v Qu’on ne me dise pas que je n’ai rien dit de nouveau; la disposition des mati`eres est nouvelle. Quand on jouealapaume,c’estunemˆ ` eme balle dont joue l’un et l’autre, mais l’un la place mieux. Blaise Pascal v b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page vii Preface Mathematicians are sometimes intelligent, but never magicians. Their ways of thinking are natural and intuitive, contorted or opaque as they may seem. Ideas, even when simple in essence, may have to be dressed up, supported by calculations and expressed in a jargon which repulses the layman. The challenge of explaining the theories and discoveries is difficult to surmount. As a result, so many are deprived of the music of mathematics. What you will read revolves around prime numbers. Hardly any kind of numbers is more attractive or mysterious. Not much of what follows will be supported by rigorous proof. But everything will be made credible, even irrefutable. You opened this book. You think that you are smart enough to understand all the discussions reported here. You may be surprised. This book should not be read by specialists, who discover much and explain little. This book is not for anyone lacking in intelligence. You, who are smart enough and even more, curious, must be warned: This is not a novel; what you read here should generate much reflection. vii October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page viii viii Prime Numbers, Friends Who Give Problems More than the facts described, the ways mathematicians think should captivate you. To read and to reflect: You must. To consult other books: You will (feel the urge). To reach a supreme intellectual enjoyment: I hope. P.R. October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page ix Acknowledgments? Do I have to thank Eric and Paulo? My brain sweats in replying to their incessant and inquisitive questions, prompted by a thirst for knowledge. A thirst for a guaran´a would be easier to satisfy. Do I have to thank Eric and Paulo who convinced me to write this book? Yes, but only if my readers will feel gratified, which remains to be seen. Acknowledgments! Once the crime was perpetrated, it was examined by my virtual friend Reny Montandon. Chosen because he allies the best of Brazil and Switzerland, rigor and friendliness, in whatever order. Thanks! ix b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page xi Contents Preface vii Acknowledgments? Acknowledgments! ix 1. What are Prime Numbers? 1 2. Division is Harder than Multiplication 5 3. Another Paulo! Is a Dialogue of Three Possible? 17 4. How Natural Numbers are Made Out of Primes 21 5. Tell Me: Which is the Largest Prime? 29 6. Trying Hard to Find Primes 33 7. A Formula, A Formula, Please 39 8. Paulo Came with a Lasso 47 9. Beautiful Old Elementary Arithmetic 51 xi October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page xii xii Prime Numbers, Friends Who Give Problems 10. The Old Man Still Knows 61 11. Can You Tell Me All About Congruences? 71 12. Homework Checked 85 13. Testing for Primality and Factorization 89 14. Fermat Numbers are Friendly. Are They Primes? 93 15. This World is Perfect 105 16. Unfriendly Numbers from a Friend of Fermat’s 111 17. Paying My Debt 123 18. Money and Primes 159 19. Secret Messages 175 20. New Numbers and Functions 185 21. Princeps Gauss 195 22. Gathering Forces 207 23. The After “Math” of Gauss 225 24. Primes After Dinner: Bad Dreams? 239 25. Primes in Arithmetic Progression 245 October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page xiii Contents xiii 26. Selling Primes 261 27. The Great Prime Mysteries 273 28. Mysteries in Sequences: More But Not All 291 29. The End and the Beginning 311 Name Index 317 Subject Index 319 October 4, 2016 8:33 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-ch01 page 1 1 What are Prime Numbers? Eric. Papa Paulo, I’ve heard about prime numbers. Some smart people talk about these numbers with voices full of mystery. I know nothing about them. Can you explain prime numbers to me? Papa Paulo (the friendly Papa Paulo). I will be happy to do it. I tell you, if I explain everything slowly, if you pay attention and understand all that I say, you will learn about prime numbers. Then it will be your turn to impress the people who appear to know it all. Let us agree that you’ll not be shy and that you will ask all the questions which come to your mind. If you have a question, my attitude is that you want to know the answer and I will provide it with patience and clarity. Eric. At least with you I don’t have to hide my ignorance. Papa Paulo. Ignorance calls for learning, so it is a privileged situation. Eric. After you learn, you are no longer ignorant, and the privileged position is lost. Why learn? Papa Paulo. It is a paradox. The more you learn, the more questions and problems appear requiring answers and solutions. If we continue our dialogue for a while, you’ll see how questions arise with any piece of knowledge acquired. Eric. Your game between ignorance and knowledge is nothing so new. What I want to know is: What is a prime number? 1 October 4, 2016 8:33 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-ch01 page 2 2 Prime Numbers, Friends Who Give Problems Papa Paulo took an almost solemn air and said in an irrefutable voice: Papa Paulo. A prime number is a natural number greater than 1 whose only factors are 1 and the number itself. Eric. You promised to explain everything. What is a natural number? Papa Paulo. The natural numbers are 0, 1, 2, 3, 4,...,1093,..., 301258,.... Eric. And what are the factors of a number? Papa Paulo. I will give an example and you will understand right away: The factors of 10 are 1, 2, 5 and 10. These factors are also called divisors. Another example: the factors or divisors of 12 are 1, 2, 3, 4, 6 and 12. You see that 12 divided by 5 is more than 2 and less than 3, so 12 divided by 5 is not an integer. Therefore 5 is not a divisor (= factor) of 12. Eric. I understand what you said. I can see that every natural number n greater than 1 has the divisors 1 and n, and sometimes has other divisors. Papa Paulo. You are right — 1 and n are called the trivial divisors (= factors) of n.AsIsaid,thenumbern greater than 1 is called a prime number if it has only the trivial divisors. Eric. If one thinks just a little, making simple mental calculations, it is easy to see that 2, 3, 5, 7, 11, 13, 17, 19 are prime numbers. Papa Paulo. Yes, Eric. Let us see how one has to proceed to find out if 19 is a prime number. 19 19 • Does 2 divide 19? No, because 9 < 2 < 10, so 2 is not an integer. 19 • Does 3 divide 19? No, because 6 < 3 < 7.
Recommended publications
  • Gaussian Prime Labeling of Super Subdivision of Star Graphs

    Gaussian Prime Labeling of Super Subdivision of Star Graphs

    of Math al em rn a u ti o c J s l A a Int. J. Math. And Appl., 8(4)(2020), 35{39 n n d o i i t t a s n A ISSN: 2347-1557 r e p t p n l I i c • Available Online: http://ijmaa.in/ a t 7 i o 5 n 5 • s 1 - 7 4 I 3 S 2 S : N International Journal of Mathematics And its Applications Gaussian Prime Labeling of Super Subdivision of Star Graphs T. J. Rajesh Kumar1,∗ and Antony Sanoj Jerome2 1 Department of Mathematics, T.K.M College of Engineering, Kollam, Kerala, India. 2 Research Scholar, University College, Thiruvananthapuram, Kerala, India. Abstract: Gaussian integers are the complex numbers of the form a + bi where a; b 2 Z and i2 = −1 and it is denoted by Z[i]. A Gaussian prime labeling on G is a bijection from the vertices of G to [ n], the set of the first n Gaussian integers in the spiral ordering such that if uv 2 E(G), then (u) and (v) are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of super subdivision of star graphs. MSC: 05C78. Keywords: Gaussian Integers, Gaussian Prime Labeling, Super Subdivision of Graphs. © JS Publication. 1. Introduction The graphs considered in this paper are finite and simple. The terms which are not defined here can be referred from Gallian [1] and West [2]. A labeling or valuation of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy, a label depending upon the vertex labels f(x) and f(y).
  • Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11

    Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11

    q-PSEUDOPRIMALITY: A NATURAL GENERALIZATION OF STRONG PSEUDOPRIMALITY JOHN H. CASTILLO, GILBERTO GARC´IA-PULGAR´IN, AND JUAN MIGUEL VELASQUEZ-SOTO´ Abstract. In this work we present a natural generalization of strong pseudoprime to base b, which we have called q-pseudoprime to base b. It allows us to present another way to define a Midy’s number to base b (overpseudoprime to base b). Besides, we count the bases b such that N is a q-probable prime base b and those ones such that N is a Midy’s number to base b. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to q-probable prime to base b as with the concept of strong pseudoprimes to base b. 1. Introduction Recently, Grau et al. [7] gave a generalization of Pocklignton’s Theorem (also known as Proth’s Theorem) and Miller-Rabin primality test, it takes as reference some works of Berrizbeitia, [1, 2], where it is presented an extension to the concept of strong pseudoprime, called ω-primes. As Grau et al. said it is right, but its application is not too good because it is needed m-th primitive roots of unity, see [7, 12]. In [7], it is defined when an integer N is a p-strong probable prime base a, for p a prime divisor of N −1 and gcd(a, N) = 1. In a reading of that paper, we discovered that if a number N is a p-strong probable prime to base 2 for each p prime divisor of N − 1, it is actually a Midy’s number or a overpseu- doprime number to base 2.
  • Mathematical Circus & 'Martin Gardner

    Mathematical Circus & 'Martin Gardner

    MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3.
  • A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)

    A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)

    A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n.
  • Connected Components of Complex Divisor Functions

    Connected Components of Complex Divisor Functions

    CONNECTED COMPONENTS OF COMPLEX DIVISOR FUNCTIONS COLIN DEFANT Princeton University Fine Hall, 304 Washington Rd. Princeton, NJ 08544 X c Abstract. For any complex number c, define the divisor function σc : N ! C by σc(n) = d . djn Let σc(N) denote the topological closure of the range of σc. Extending previous work of the current author and Sanna, we prove that σc(N) has nonempty interior and has finitely many connected components if <(c) ≤ 0 and c 6= 0. We end with some open problems. 2010 Mathematics Subject Classification: Primary 11B05; Secondary 11N64. Keywords: Divisor function; connected component; spiral; nonempty interior; perfect number; friendly number. 1. Introduction One of the most famous and perpetually mysterious mathematical contributions of the ancient Greeks is the notion of a perfect number, a number that is equal to the sum of its proper divisors. Many early theorems in number theory spawned from attempts to understand perfect numbers. Although few modern mathematicians continue to attribute the same theological or mystical sig- nificance to perfect numbers that ancient people once did, these numbers remain a substantial inspiration for research in elementary number theory [2, 3, 5, 11, 14{16, 18]. arXiv:1711.04244v3 [math.NT] 4 May 2018 P A positive integer n is perfect if and only if σ1(n)=n = 2, where σ1(n) = djn d. More generally, for any complex number c, we define the divisor function σc by X c σc(n) = d : djn These functions have been studied frequently in the special cases in which c 2 {−1; 0; 1g.
  • Number Theory Course Notes for MA 341, Spring 2018

    Number Theory Course Notes for MA 341, Spring 2018

    Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads .
  • The Prime Number Theorem a PRIMES Exposition

    The Prime Number Theorem a PRIMES Exposition

    The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1.
  • The RSA Algorithm Clifton Paul Robinson

    The RSA Algorithm Clifton Paul Robinson

    Bridgewater State University Virtual Commons - Bridgewater State University Honors Program Theses and Projects Undergraduate Honors Program 5-1-2018 The Key to Cryptography: The RSA Algorithm Clifton Paul Robinson Follow this and additional works at: http://vc.bridgew.edu/honors_proj Part of the Computer Sciences Commons Recommended Citation Robinson, Clifton Paul. (2018). The Key ot Cryptography: The RSA Algorithm. In BSU Honors Program Theses and Projects. Item 268. Available at: http://vc.bridgew.edu/honors_proj/268 Copyright © 2018 Clifton Paul Robinson This item is available as part of Virtual Commons, the open-access institutional repository of Bridgewater State University, Bridgewater, Massachusetts. The Key to Cryptography: The RSA Algorithm Clifton Paul Robinson Submitted in Partial Completion of the Requirements for Commonwealth Interdisciplinary Honors in Computer Science and Mathematics Bridgewater State University May 1, 2018 Dr. Jacqueline Anderson Thesis Co-Advisor Dr. Michael Black, Thesis Co-Advisor Dr. Ward Heilman, Committee Member Dr. Haleh Khojasteh, Committee Member BRIDGEWATER STATE UNIVERSITY UNDERGRADUATE THESIS The Key To Cryptography: The RSA Algorithm Author: Advisors: Clifton Paul ROBINSON Dr. Jackie ANDERSON Dr. Michael BLACK Submitted in Partial Completion of the Requirements for Commonwealth Honors in Computer Science and Mathematics Dr. Ward Heilman, Reading Committee Dr. Haleh Khojasteh, Reading Committee ii Dedicated to Mom, Dad, James, and Mimi iii Contents Abstractv 1 Introduction1 1.1 The Project Overview........................1 2 Theorems and Definitions2 2.1 Definitions..............................2 2.2 Theorems...............................5 3 The History of Cryptography6 3.1 Origins................................6 3.2 A Transition.............................6 3.3 Cryptography at War........................7 3.4 The Creation and Uses of RSA...................7 4 The Mathematics9 4.1 What is a Prime Number?.....................9 4.2 Factoring Numbers........................
  • Related Distribution of Prime Numbers

    Related Distribution of Prime Numbers

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CSCanada.net: E-Journals (Canadian Academy of Oriental and... ISSN 1923-8444 [Print] Studies in Mathematical Sciences ISSN 1923-8452 [Online] Vol. 6, No. 2, 2013, pp. [96{103] www.cscanada.net DOI: 10.3968/j.sms.1923845220130602.478 www.cscanada.org Related Distribution of Prime Numbers Kaifu SONG[a],* [a] East Lake High School, Yichang, Hubei, China. * Corresponding author. Address: East Lake High School, Yichang 443100, Hubei, China; E-Mail: skf08@ sina.com Received: March 6, 2013/ Accepted: May 10, 2013/ Published: May 31, 2013 Abstract: The surplus model is established, and the distribution of prime numbers are solved by using the surplus model. Key words: Surplus model; Distribution of prime numbers Song, K. (2013). Related Distribution of Prime Numbers. Studies in Mathematical Sci- ences, 6 (2), 96{103. Available from http://www.cscanada.net/index.php/sms/article/ view/j.sms.1923845220130602.478 DOI: 10.3968/j.sms.1923845220130602.478 1. SURPLUS MODEL In the range of 1 ∼ N, r1 represents the quantity of numbers which do not contain 1 a factor 2, r = [N · ] ([x] is the largest integer not greater than x); r represents 1 2 2 2 the quantity of numbers which do not contain factor 3, r = [r · ]; r represents 2 1 3 3 4 the quantity of numbers do not contain factor 5, r = [r · ]; ... 3 2 5 In the range of 1 ∼ N, ri represents the quantity of numbers which do not contain factor pi , the method of the surplus numbers ri−1 to ri is called the residual model of related distribution of prime numbers.
  • Generating Provable Primes Efficiently on Embedded Devices

    Generating Provable Primes Efficiently on Embedded Devices

    Generating Provable Primes Efficiently on Embedded Devices Christophe Clavier1, Benoit Feix1;2, Lo¨ıc Thierry2;?, and Pascal Paillier3 1 XLIM, University of Limoges, [email protected] 2 INSIDE Secure [email protected],[email protected] 3 CryptoExperts [email protected] Abstract. This paper introduces new techniques to generate provable prime numbers efficiently on embedded devices such as smartcards, based on variants of Pocklington's and the Brillhart-Lehmer-Selfridge-Tuckerman- Wagstaff theorems. We introduce two new generators that, combined with cryptoprocessor-specific optimizations, open the way to efficient and tamper-resistant on-board generation of provable primes. We also report practical results from our implementations. Both our theoretical and ex- perimental results show that constructive methods can generate provable primes essentially as efficiently as state-of-the-art generators for probable primes based on Fermat and Miller-Rabin pseudo-tests. We evaluate the output entropy of our two generators and provide techniques to ensure a high level of resistance against physical attacks. This paper intends to provide practitioners with the first practical solutions for fast and secure generation of provable primes in embedded security devices. Keywords: Prime Numbers, Pocklington's theorem, Public Key Cryp- tography, Embedded Software, Modular Exponentiation, Cryptographic Accelerators, Primality Proving. 1 Introduction Large prime numbers are a basic ingredient of keys in several standardized primi- tives such as RSA [21], Digital Signature Algorithm (DSA) [12] or Diffie-Hellman key exchange (DH) [10]. This paper precisely addresses the generation of prov- able prime numbers in embedded, crypto-enabled devices. When it comes to RSA key generation, two approaches coexist: key pairs may be generated off-board (i.e.
  • THE GAUSSIAN INTEGERS Since the Work of Gauss, Number Theorists

    THE GAUSSIAN INTEGERS Since the Work of Gauss, Number Theorists

    THE GAUSSIAN INTEGERS KEITH CONRAD Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. The example we will look at in this handout is the Gaussian integers: Z[i] = fa + bi : a; b 2 Zg: Excluding the last two sections of the handout, the topics we will study are extensions of common properties of the integers. Here is what we will cover in each section: (1) the norm on Z[i] (2) divisibility in Z[i] (3) the division theorem in Z[i] (4) the Euclidean algorithm Z[i] (5) Bezout's theorem in Z[i] (6) unique factorization in Z[i] (7) modular arithmetic in Z[i] (8) applications of Z[i] to the arithmetic of Z (9) primes in Z[i] 1. The Norm In Z, size is measured by the absolute value. In Z[i], we use the norm. Definition 1.1. For α = a + bi 2 Z[i], its norm is the product N(α) = αα = (a + bi)(a − bi) = a2 + b2: For example, N(2 + 7i) = 22 + 72 = 53. For m 2 Z, N(m) = m2. In particular, N(1) = 1. Thinking about a + bi as a complex number, its norm is the square of its usual absolute value: p ja + bij = a2 + b2; N(a + bi) = a2 + b2 = ja + bij2: The reason we prefer to deal with norms on Z[i] instead of absolute values on Z[i] is that norms are integers (rather than square roots), and the divisibility properties of norms in Z will provide important information about divisibility properties in Z[i].

  • Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases

    INTERSECTIONS OF DELETED DIGITS CANTOR SETS WITH GAUSSIAN INTEGER BASES Athesissubmittedinpartialfulfillment of the requirements for the degree of Master of Science By VINCENT T. SHAW B.S., Wright State University, 2017 2020 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES May 1, 2020 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPER- VISION BY Vincent T. Shaw ENTITLED Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases BE ACCEPTED IN PARTIAL FULFILLMENT OF THE RE- QUIREMENTS FOR THE DEGREE OF Master of Science. ____________________ Steen Pedersen, Ph.D. Thesis Director ____________________ Ayse Sahin, Ph.D. Department Chair Committee on Final Examination ____________________ Steen Pedersen, Ph.D. ____________________ Qingbo Huang, Ph.D. ____________________ Anthony Evans, Ph.D. ____________________ Barry Milligan, Ph.D. Interim Dean, School of Graduate Studies Abstract Shaw, Vincent T. M.S., Department of Mathematics and Statistics, Wright State University, 2020. Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases. In this paper, the intersections of deleted digits Cantor sets and their fractal dimensions were analyzed. Previously, it had been shown that for any dimension between 0 and the dimension of the given deleted digits Cantor set of the real number line, a translate of the set could be constructed such that the intersection of the set with the translate would have this dimension. Here, we consider deleted digits Cantor sets of the complex plane with Gaussian integer bases and show that the result still holds. iii Contents 1 Introduction 1 1.1 WhystudyintersectionsofCantorsets? . 1 1.2 Prior work on this problem . 1 2 Negative Base Representations 5 2.1 IntegerRepresentations .............................