DOCSLIB.ORG

Computations on Normal Families of Primes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computations on Normal Families of Primes Computations on Normal Families of Primes bq' Erick Wong B-Sc., Simon Fraser University, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTEROF SCIENCE in the Department of k1athematic.s gd Statistics @ Erick Wong 1997 SIMON FRASER UNIVERSITY August 1997 iW rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. Acquisitions and Acquisitions et Bibliographic Se~*ces services bibliographiques 395 Wellington Street 395, rue Wellington ûttawa ON K1A ON4 ûttawa ON K1A ON4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, ioan, disbiiute or sell reproduire, prêter, distribuer ou copies of this thesis in microfom, vendre des copies de cette thèse sous paper or elecbonic formats. la fonne de rnicrofiche/~de reproducticn sur papier ou sur format électronique. The author retains ownership of the L'auteur conseme la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract We cal1 a farnily of primes P no.mnal if it contains no two primes p, q such that p divides q - 1. In this thesis we study two conjectures and their related variants. Giuga's conjecture is that C:!; kn-' =- n - 1 (mod n) implies n is prime. We study a group of eight variants of this equation and derive necessary and suficient conditions for which they hold. Lehmer's conjecture is that d(n) 1 n - 1 if and only if n is prime. This conjecture has been verified for up to 13 prime factors of n, and we extend this to 14 prime factors. We also examine the related condition d(n) 1 n + 1 which is known to have solutions with up to 6 prime factors and extend the search to 7 prime factors. For both of these conjectures the set of prime factors of any counterexample n is a normal family, and we exploit this property in our computations. Dedicat ion 1 dedicate this thesis in memory of Kaz Shinyashiki, my best friend of rnany years. To Kaz: you 're in the books for sure now.. Acknowledgement s First, 1 thank my supervisor Jon Borwein for his patience and constructive cornments throughout the production of this thesis. 1 would also like to thank rny friends in the Math Department at SFU, especially Kathy, Brian, Harvey and Luis, for their helpful advice and guidance over the past eight years. 1 thank rny family for their constant support throughout my studies, my rnother for fostering my early interest in rnathematics, and my father for driving me to school every day for so many years. Many thanks go to my good friend Jen Chang for her encouragement and assistance in writing Chapter 3. Finally, a very special thank-you goes to Janice Fung, who has been a dear friend through some hard times, and aiso helped to proofread the early drafts of Chapter 2. 1 would never have completed this work without her continuing support from beginning to end. Contents Approval ..................................... Abstract ..................................... Dedication .................................... Acknowledgements ............................... v ... List of Tables ............................... VIII 1 Introduction ................................ 1 1.1 Introduction ............................ 1 1.2 Normal Families .......................... 2 Giuga's Family ........................... .. 5 2.1 Giuga's Conjecture ........................ 5 2.2 The Eightfold Way ........................ 6 2.3 Evaluating sums ......................... 7 2.4 Classification ....................... 10 2.4.1 Sums of Type 1 ..................... 10 2.4.2 Sums of Type II .................... 12 2.5 Giuga numbers et d ....................... 14 2.5.1 Giuga numbers ..................... 14 2.5.2 Co-Giuga numbers ................... 15 2.5.3 Pseudo-Carmichael numbers .............. 17 2.6 Conclusion ............................. 18 3 Lehmer's Conjecture ........................... 20 3.1 Introduction ............................ 20 3.2 Computational Technique .................... 3-1 3.2.1 Obtaining bounds for p, ................ 22 3.2.2 Resolving prefixes of length 1 - 2 ........... 23 3.3 Results ............................... 25 3.3.1 Then-lcase ..................... 25 3.3.2 Then+lcase ..................... 26 3.4 Conclusion ............................. 27 Appendix: Source Code Listings ........................ 29 vii List of Tables 2.1 Conditions for CkEIkm r (mod n) . 18 ... Vlll Chapter 1 Introduction 1.1 Introduction The fields of number theory and computing science have enjoyed a fruitful partner- ship over the last few decades. Studies of such number-theoretic problems as primal- ity testing, factoring, and discrete logarithms have helped rnake practical public-key cryptography a reality. Conversely, the computer has become an invaluable tool for dealing with large numbers, making once impossibly long calculations feasible, and once tedious work trivial. There are manÿ conjectures which have simple and elegant statements, but for which there seems to be little hope of a complete resolution. This is particularly true in the field of number theory (for example, the twin prime conjecture, the existence of odd perfect numbers, and until quite recently Fermat's conjecture). In the absence of a proof or disproof, one method of investigation is to derive a lower bound on the size of a counterexample. G. Giuga and D. H. Lehmer each studied such problems and obtained computational bounds for them through hand calculations and tables, well before the advent of modern computing. As computer speeds increase, more problems become tractable, and better bounds can be obtained. In this thesis we look at both Giuga's and Lehmer's conjectures CHAPTER 1. INTRODUCTION 2 as well as some variants, and see what new results we cm derive with the level of computational power available today. In Chapter 3 we study Giuga's conjecture n- 1 / 2; "-' n - 1 (mod n) iff n is prime, together with severd related variants. The bulk of this work is contained in (BW). In Chapter 3 we examine Lehmer's conjecture and obtain a new bound on the number of prime factors of a counterexample. We also partially extend Lehmer's work of determinhg which n satisfy d(n) 1 n + 1. We will find that these problems a11 share a cornmon structure which can be exploited to make the computations somewhat easier. 1.2 Normal Farnilies We cal1 a finite family of distinct primes P = pz, p3'. -., pk) nomal if pi 1p, - 1 for al1 1 5 i, j 5 k. It is clear that if k > 1 then 2 $ P. CVe Say that a prime q is normal to an existing normal faïnilj- F if q > max P and P u {q} is also normal. Exarnple: (3,5,23,29) is a normal farnily of primes. The smallest prime normal to this family is 53. In general, we denote by [Pl the first prime normal to P, and by rP,ql the first prime greater than or equal to q which is normal to P. Except for the trivial case P = (21, Dirichlet's theorem (Levef guarantees that such a prime must exist. Similarly, [P,qJ denotes the largest prime less than or equal to q which is normal to P (provided q 1 [Pl). It turns out that for both Giuga's and Lehmer's conjectures as well as many of the variants we consider, a necessary condition for n to be a counterexarnple is that the prime factors of n form a normal family. By examining only normal families of a given cardinality, we reduce the search space required to verify these conjectures for a given number of prime factors. Incidentally, these n have an interesting group- theoretic property (thanks to Dr. Derek Holt at the University of Wârwick for this observation). Fact. Let n be a product of primes from a normal fâmily. Then there is a unique group of order n up to isomorphisrn (namely 2,). Conversely, if n is not squarefree or if n is divisible by primes p, q where p 1 q - 1, then there exists a group of order n which is not cyclic. Sketch. Let G be a group of order n. If P is a Sylow psubgroup of G, then IAut(P)I = p - 1 does not divide IGI. We then apply Burnside's Transfer Theorern (see Hallj to show that G has a normal subgroup N of order n/p such that G = NP. Choosing P to be a normal Sylow psubgoup of G, we see that G iV x P,and an induction on the number of prime factors of n completes the argument. To prove the converse, it is sufficient to simply find non-cyclic groups of orders p2 and pq. The former is easily given by Zp x Zp. For the latter, let 6 be a primitive pth root of unity modulo q. Then the group with presentation Before proceeding to study the two conjectures in detail, we introduce the following concepts which will be useful. A normal sequence is an infinite sequence P = {pl < pz < p3 < --0) of primes where pi 1pj - 1 for al1 i, j 2 1. It is easy to construct such a sequence inductively €rom any prescribed finite subset Pr by repeatedly adjoining [FI.We define a restriction P[k]of P to be the elements of P which are less thon or equal to k. We denote by Pc the simple restriction coosisting of {pl,p2, . ., pk). Given two families of primes A = {ai < a* < < ak} and B = {bl < 6* < < br) we say that A dominates B if k 2 I and ai 5 bi for each 1 5 i < 1.
Load more

Recommended publications
  • Note to Users
    NOTE TO USERS This reproduction is the best copy available. UMI A SURVEY OF RESULTS ON GIUGA'S CONJECTURE AND RELATED CONJECTURES by Joseph R. Hobart BSc., University of Northern British Columbia, 2004 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in MATHEMATICAL, COMPUTER AND PHYSICAL SCIENCES (MATHEMATICS) THE UNIVERSITY OF NORTHERN BRITISH COLUMBIA July 2005 © Joseph R. Hobart, 2005 Library and Bibliothèque et 1 ^ 1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-28392-9 Our file Notre référence ISBN: 978-0-494-28392-9 NOTICE: AVIS: The author has granted a non­ L'auteur a accordé une licence non exclusive exclusive license allowing Library permettant à la Bibliothèque et Archives and Archives Canada to reproduce,Canada de reproduire, publier, archiver, publish, archive, preserve, conserve,sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet,distribuer et vendre des thèses partout dans loan, distribute and sell theses le monde, à des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, électronique commercial purposes, in microform,et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriété du droit d'auteur ownership and moral rights in et des droits moraux qui protège cette thèse. this thesis. Neither the thesis Ni la thèse ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent être imprimés ou autrement may be printed or otherwise reproduits sans son autorisation.
    [Show full text]
  • 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
    [Show full text]
  • Giuga Numbers and the Arithmetic Derivative
    1 2 Journal of Integer Sequences, Vol. 15 (2012), 3 Article 12.4.1 47 6 23 11 Giuga Numbers and the Arithmetic Derivative Jos´eMar´ıa Grau Departamento de Matematicas Universidad de Oviedo Avda. Calvo Sotelo, s/n 33007 Oviedo Spain [email protected] Antonio M. Oller-Marc´en Centro Universitario de la Defensa Ctra. de Huesca, s/n 50090 Zaragoza Spain [email protected] Abstract We characterize Giuga numbers as solutions to the equation n′ = an + 1, with a ∈ N and n′ being the arithmetic derivative. Although this fact does not refute Lava’s conjecture, it does suggest doubts about its veracity. 1 Introduction 1.1 The Arithmetic Derivative The arithmetic derivative was introduced by Barbeau [3] (see A003415 in Sloane’s On-Line Encyclopedia of Integer Sequences). The derivative of an integer is defined to be the unique map sending every prime to 1 and satisfying the Leibnitz rule, i.e., p′ = 1, for any prime p, and (nm)′ = nm′ + mn′ for any n, m ∈ N. This map makes sense and is well-defined [7]. 1 Proposition 1. n k pri n If = Qi=1 i is the factorization of in prime powers, then the only way to define n′ satisfying the desired properties is k ′ ri n = n X . pi i=1 1.2 Giuga numbers In [4], Giuga numbers were introduced in the following way motivated by previous work by Giuga [6]. n Definition 2. A Giuga number is a composite number n such that p divides p − 1 for every prime divisor p of n.
    [Show full text]
  • Numbers 1 to 100
    Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)
    [Show full text]
  • Giuga's Primality Conjecture for Number Fields
    Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 5 Giuga's Primality Conjecture for Number Fields Jamaris Burns Johnson C. Smith University Katherine Casey Cornell University Duncan Gichimu Towson University Kerrek Stinson Colorado School of Mines Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Burns, Jamaris; Casey, Katherine; Gichimu, Duncan; and Stinson, Kerrek (2017) "Giuga's Primality Conjecture for Number Fields," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 5. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/5 Rose- Hulman Undergraduate Mathematics Journal Giuga's Primality Conjecture for Number Fields Jamaris Burnsa Katherine Caseyb Duncan Gichimuc Kerrek Stinsond Volume 18, No. 1, Spring 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics aJohnson C. Smith University Terre Haute, IN 47803 bCornell University c [email protected] Towson University d scholar.rose-hulman.edu/rhumj Colorado School of Mines Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 1, Spring 2017 Giuga's Primality Conjecture for Number Fields Jamaris Burns Katherine Casey Duncan Gichimu Kerrek Stinson Abstract. Giuseppe Giuga conjectured in 1950 that a natural number n is prime Pn−1 n−1 if and only if it satisfies the congruence k=1 k ≡ −1 (mod n). Progress in validating or disproving the conjecture has been minimal, with the most significant advance being the knowledge that a counter-example would need at least 19,907 digits. To gain new insights into Giuga's conjecture, we explore it in the broader context of number fields. We present a generalized version of the conjecture and prove generalizations of many of the major results related to the conjecture.
    [Show full text]
  • Number Theory and Combinatorics
    Number Theory and Combinatorics B SURY Stat-Math Unit, Indian Statistical Institute, Bengaluru, India. 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 otherwise, without prior permission of the publisher. c Indian Academy of Sciences 2017 Reproduced from Resonance{journal of science education Reformatted by TNQ Books and Journals Pvt Ltd, www.tnq.co.in Published by Indian Academy of Sciences Foreword The Masterclass series of eBooks brings together pedagogical articles on single broad topics taken from Resonance, the Journal of Science Educa- tion, that has been published monthly by the Indian Academy of Sciences since January 1996. Primarily directed at students and teachers at the un- dergraduate level, the journal has brought out a wide spectrum of articles in a range of scientific disciplines. Articles in the journal are written in a style that makes them accessible to readers from diverse backgrounds, and in addition, they provide a useful source of instruction that is not always available in textbooks. The third book in the series, `Number Theory and Combinatorics', is by Prof. B Sury. A celebrated mathematician, Prof. Sury's career has largely been at the Tata Institute of Fundamental Research, Mumbai', and the Indian Statistical Institute, Bengaluru, where he is presently professor. He has contributed pedagogical articles regularly to Resonance, and his arti- cles on Number Theory and Combinatorics comprise the present book. He has also served for many years on the editorial board of Resonance. Prof.
    [Show full text]
  • On Giuga Numbers
    International Journal of Modern Mathematics 4(1) (2009), 13–18 °c 2009 Dixie W Publishing Corporation, U. S. A. On Giuga Numbers Florian Luca, Carl Pomerance, and Igor Shparlinski Received November 18, 2007; Revised July 31, 2008 Abstract Pn¡1 n¡1 A Giuga number is a composite integer n satisfying the congruence j=1 j ´ ¡1 (mod n). We show that the counting function #G(x) of the Giuga numbers n · x satisfies the estimate #G(x) = o(x1=2) as x ! 1, improving upon a result of V. Tipu. Keywords: Giuga numbers, Carmichael numbers. 2000 Mathematics Subject Classification: 11A41, 11N25. 1 Introduction 1.1 Background Fermat’s Little Theorem immediately implies that for n prime, nX¡1 jn¡1 ´ ¡1 (mod n): (1.1) j=1 Giuga [3] has conjectured that there are no composite integers n fulfilling (1.1); a coun- terexample is called a Giuga number. With G the set of all Giuga numbers, it is known that n 2 G if and only if n is composite and p2(p¡1) j n¡p (1.2) for all prime factors p of n. In particular, n is squarefree. Furthermore, it is also a Carmichael number; that is, the congruence an ´ a (mod n) holds for all integers a. We refer the reader to [5, pp. 21-22] and the introduction to [7] for more properties of the Giuga numbers. In [1], the relation (1.2) is relaxed to p2 j n¡p, and it is shown that this property is equivalent to the sum of the reciprocals of the prime factors of n being 1=n mod 1.
    [Show full text]
  • Download PDF # Integer Sequences
    4DHTUDDXIK44 ^ eBook ^ Integer sequences Integer sequences Filesize: 8.43 MB Reviews Extensive information for ebook lovers. It typically is not going to expense too much. I discovered this book from my i and dad recommended this pdf to learn. (Prof. Gerardo Grimes III) DISCLAIMER | DMCA DS4ABSV1JQ8G > Kindle » 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 number, Multiplicative partition, Perrin number, Smooth number, Ulam number, Primorial, Lambek Moser theorem,
    [Show full text]
  • Dictionary of Mathematics
    Dictionary of Mathematics English – Spanish | Spanish – English Diccionario de Matemáticas Inglés – Castellano | Castellano – Inglés Kenneth Allen Hornak Lexicographer © 2008 Editorial Castilla La Vieja Copyright 2012 by Kenneth Allen Hornak Editorial Castilla La Vieja, c/o P.O. Box 1356, Lansdowne, Penna. 19050 United States of America PH: (908) 399-6273 e-mail: [email protected] All dictionaries may be seen at: http://www.EditorialCastilla.com Sello: Fachada de la Universidad de Salamanca (ESPAÑA) ISBN: 978-0-9860058-0-0 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage or retrieval system without permission in writing from the author Kenneth Allen Hornak. Reservados todos los derechos. Quedan rigurosamente prohibidos la reproducción de este libro, el tratamiento informático, la transmisión de alguna forma o por cualquier medio, ya sea electrónico, mecánico, por fotocopia, por registro u otros medios, sin el permiso previo y por escrito del autor Kenneth Allen Hornak. ACKNOWLEDGEMENTS Among those who have favoured the author with their selfless assistance throughout the extended period of compilation of this dictionary are Andrew Hornak, Norma Hornak, Edward Hornak, Daniel Pritchard and T.S. Gallione. Without their assistance the completion of this work would have been greatly delayed. AGRADECIMIENTOS Entre los que han favorecido al autor con su desinteresada colaboración a lo largo del dilatado período de acopio del material para el presente diccionario figuran Andrew Hornak, Norma Hornak, Edward Hornak, Daniel Pritchard y T.S. Gallione. Sin su ayuda la terminación de esta obra se hubiera demorado grandemente.
    [Show full text]
  • On a Variant of Giuga Numbers
    On a variant of Giuga numbers Jose´ Mar´ıa Grau Departamento de Matem´aticas Universidad de Oviedo Avda. Calvo Sotelo, s/n, 33007 Oviedo, Spain [email protected] Florian Luca Instituto de Matem´aticas Universidad Nacional Autonoma de M´exico C.P. 58089, Morelia, Michoac´an, M´exico fl[email protected] and The John Knopfmacher Centre for Applicable Analysis and Number Theory University of the Witwatersrand, P.O. Wits 2050, South Africa Antonio M. Oller-Marcen´ Departamento de Matem´aticas arXiv:1103.3428v1 [math.NT] 17 Mar 2011 Universidad de Zaragoza C/Pedro Cerbuna 12, 50009 Zaragoza, Spain [email protected] November 7, 2018 1 Abstract In this paper, we characterize the odd positive integers n satisfying n−1 n−1 2 the congruence j=1 j ≡ 0 (mod n). We show that the set of such positive integers has an asymptotic density which turns out to P be slightly larger than 3/8. 1 Introduction Given any property P satisfied by the primes, it is natural to consider the set CP := {n composite : n satisfies P}. Elements of CP can be thought of as pseudoprimes with respect to the property P. Such sets of pseudoprimes have been of interest to number theorists. Putting aside practical primality tests such as Fermat, Euler, Euler–Jacobi, Miller–Rabin, Solovay–Strassen, and others, let us have a look at some in- teresting, although not very efficient, primality tests as summarized in the table below. Test Pseudoprimes Infinitely many 1 (n − 1)! ≡ −1 (mod n) None No 2 an ≡ a (mod n) for all a Carmichael numbers Yes n−1 φ(n) 3 j=1 j ≡ −1 (mod n) Giuga numbers Unknown 4 φ(n)|(n − 1) Lehmer numbers No example known Pn−1 n−1 5 j=1 j ≡ −1 (mod n) No example known P In the above table, φ(n) is the Euler function of n.
    [Show full text]
  • Tutorme Subjects Covered.Xlsx
    Subject Group Subject Topic Computer Science Android Programming Computer Science Arduino Programming Computer Science Artificial Intelligence Computer Science Assembly Language Computer Science Computer Certification and Training Computer Science Computer Graphics Computer Science Computer Networking Computer Science Computer Science Address Spaces Computer Science Computer Science Ajax Computer Science Computer Science Algorithms Computer Science Computer Science Algorithms for Searching the Web Computer Science Computer Science Allocators Computer Science Computer Science AP Computer Science A Computer Science Computer Science Application Development Computer Science Computer Science Applied Computer Science Computer Science Computer Science Array Algorithms Computer Science Computer Science ArrayLists Computer Science Computer Science Arrays Computer Science Computer Science Artificial Neural Networks Computer Science Computer Science Assembly Code Computer Science Computer Science Balanced Trees Computer Science Computer Science Binary Search Trees Computer Science Computer Science Breakout Computer Science Computer Science BufferedReader Computer Science Computer Science Caches Computer Science Computer Science C Generics Computer Science Computer Science Character Methods Computer Science Computer Science Code Optimization Computer Science Computer Science Computer Architecture Computer Science Computer Science Computer Engineering Computer Science Computer Science Computer Systems Computer Science Computer Science Congestion Control
    [Show full text]
  • Computations on Normal Families of Primes
    ~orn~utati6nson Normal Families of Pri-mes Erick Wong 1 B.Sc., Simon Fraser University, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIRE5lENTS FOR THE DEGREE OF - MASTEROF SCIENCE in the Department >lathematics & Statistics @ Erick Wong 1'997 SIMON FRASER UNIVERSITY August 1997 A11 rights,&served. This work may not be reproduced in whole or in part, by photocopy * or other means, without the permission of the author. National Library Bibliotheqve nationale of Canada du Canada 1 Acquisitions and Acquisitions et Bibliographic Services services bibliographiqqs 395 WeUington Street 395, rue Wellington Ottawa ON .K1 A ON4 Ottawa ON KIA ON4 * Canada Canada i Our file Notre ;'4 4r& 1). 4 The author has granted a non- L'auteur a accorde une licence non exclusive licence allowing the exclusive permettant a la National Libfary of Canada to Bibliotheque nationale du Canada de reproduce, loan, dstribute or sell reproduire, preter, distribuer ou copies of th~sthesis in microform, vendre des copies de cette these sous paper or electronic formats. la forme de rnicroficheifih, de reproduction sur papier ou sur format electronique. '. The author retains ownershp of the L'auteur conserve la propriete du copy@~t in th~sthesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts from it Ni la these ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent Ctre imprimes reproduced without the author's ou autrement reproduits sans son pemssion. autorisation. Name: Erick Wong Degree: vaster of Science '. & ' t Title of thesis: Computations on Normal ~akiliesof Primes 0 Examining Committee: Dr: C.
    [Show full text]