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.