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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. reproduced without the author's permission. In compliance with the Canadian Conformément à la loi canadienne Privacy Act some supporting sur la protection de la vie privée, forms may have been removed quelques formulaires secondaires from this thesis. ont été enlevés de cette thèse. While these forms may be includedBien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada APPROVAL Name; Joseph hobait Degree: Master of Science Thesis Title: CONJECTURAL PRIME TESTS USING BERNOULLI NUMBERS AND THE RIEMANN ZETA FUNCTION Examining Committee: Chair^%Dr. Kenneth Prkachin, Professor Psychology Program University of Northern British Columbia Co-Supervisor: Dr. Lee Keener, Professor Mathematical, Computer, and Physical Sciences Program University of Northern British Columbia Co-S^d^isor: Dr. PaüT^^ôntgomery, Associate Professor Mamematical, Computer, and Physical Sciences Program University of Northern British Columbia Committee Member: Dr. Charles Brown, Associate Professor Mathematical, Computer, and Physical Sciences Program University of Northern British Columbia External Examiner: Dr. Karl Dilcher, Professor Department of Mathematics and Statistics Dalhousie University Date Approved: Rl Q .005 A b stract In 1950, G. Giuga developed a method of determining if a natural number n is prime [20]; n—1 n is prime ^ = — 1 (mod n). k~l Giuga proved his conjecture up to 1000 digits. However, he was unable to extend his proof to all natural numbers. Thirty-five years later, in 1985, E. Bedocchi extended Giuga’s result to 1700 digits [6]. More recently, Borwein et al extended Giuga’s result to 13,800 digits [8]; however, they were unable to prove the conjecture. Since all previous attempts to prove the conjecture failed, researchers at­ tempted to reformulate Giuga’s conjecture. The first published attempt was by Agoh in 1991. Agoh’s reformulation n is prime 4=^ nBn~i = — 1 (mod n), became known as Agoh’s conjecture [2]. Concurrently, a 1994 thesis by Wong showed 8 variations on Giuga’s conjecture using different combinations of (p{n) in place of n — 1 [57]. This thesis attempts to detail every known result on Giuga’s conjecture. By consolidating all published results along with some unpublished results we hope to aid those who wish to study Giuga’s conjecture in the future. 11 C ontents Abstract ii Contents iii List of Figures v List of Tables v Acknowledgements vi 1 Introduction 1 2 Background Material 3 2.1 Number T h e o ry .................................................................................................. 3 2.2 Abstract Algebra .............................................................................................. 6 2.3 Quadratic R esid u e s........................................................................................... 9 2.4 Algorithms ................................................................................................. 12 2.5 Bernoulli N u m b e r s ........................................................................................... 15 2.6 Bernoulli Numbers Modulo p ......................................................................... 18 2.7 Euler Numbers .................................................................................................. 20 2.8 The Riemann Zeta F u n c tio n .......................................................................... 22 2.9 Prime Tests ........................................................................................................ 24 2.10 Factoring Algorithms ........................................................................................ 28 2.11 Computing Bn and ( ( n ) ................................................................................. 32 3 Giuga’s Conjecture 37 3.1 Giuga Numbers and Giuga Sequences ........................................................... 41 111 3.2 "The Eightfold W a y "....................................................................................... 48 3.3 Normal Families, Co-Giuga, and Pseudo-Carmichael Numbers . 55 3.3.1 Normal Families of Prim es ................................................................ 56 3.3.2 Pseudo-Carmichael Numbers ............................................................. 57 3.3.3 Co-Ciuga Numbers .............................................................................. 59 3.4 Computational Results ................................................................................... 61 3.4.1 Counterexamples ................................................................................. 61 3.4.2 Ciuga Sequences ................................................................................. 65 3.5 Agoh’s Conjecture and Related Results ..................................................... 69 3.5.1 Ciuga Quotients ................................................................................. 72 3.6 Open Problems ................................................................................................ 76 4 Conclusion 79 Appendix 1: Giuga’s Conjecture Results 81 Appendix 2: Modern Primality Tests 86 Appendix 3: Number Theoretic Congruences 95 Appendix 4: Algorithms 97 Notational Conventions 99 References 100 IV List of Figures 1 Contour for Riemann’s Zeta function [53] ..................................................... 24 List of Tables 1 The First 15 Bernoulli Numbers .................................................................... 16 2 Inequalities Related to Bernoulli Numbers [46] . ................................... 17 3 Congruences Involving Bernoulli N um bers ................................................ 19 4 The First 13 Euler Numbers ........................................................................... 21 5 Values of the Riemann Zeta function .......................................................... 23 6 Relations Between the Riemann Zeta Function and Bernoulli Numbers [1)............................................................................................................................. 25 7 Number of Giuga Sequences ........................................................................... 43 8 Proper Giuga Sequences ................................................................................. 44 9 The “Eightfold Way” ........................................................................................ 55 10 Conditions for Yhk&i ^(mod n) [57].................................................... 62 11 Giuga’s Conjecture Results .............................................................................. 81 12 Modern Prime Tests . ............................................... 86 13 Modern Pseudo-prime T ests ........................................................................... 89 14 Prime Number Conjectures .............................................................................. 89 15 Modern Sieving Methods ................................................................................. 90 16 Modern Factoring Techniques ....................................................................... 92 17 Number Theoretic Congruences .................................................................... 95 18 Notational Conventions ..................................................................................... 99 Acknowledgements First, I would like to thank Dr. Lee Keener for taking the time to teach so many extra courses on top of his paid workload. If not for the course MATH 499 - Special Topics in Number Theory, I would never have had an opportunity to first learn about Giuga’s Conjecture. I
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