Computational Techniques in Quadratic Fields by Michael John Jacobson, Jr. A thesis presented to the University of Manitoba in partial fulfilment of the requirements for the degree of Master of Science in Computer Science Winnipeg, Manitoba, Canada, 1995 c Michael John Jacobson, Jr. 1995 ii I hereby declare that I am the sole author of this thesis. I authorize the University of Manitoba to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Manitoba to reproduce this thesis by photocopy- ing or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. iii The University of Manitoba requires the signatures of all persons using or photocopy- ing this thesis. Please sign below, and give address and date. iv Abstract Since Kummer's work on Fermat's Last Theorem, algebraic number theory has been a subject of interest for many mathematicians. In particular, a great amount of effort has been expended on the simplest algebraic extensions of the rationals, quadratic fields. These are intimately linked to binary quadratic forms and have proven to be a good test- ing ground for algebraic number theorists because, although computing with ideals and field elements is relatively easy, there are still many unsolved and difficult problems re- maining. For example, it is not known whether there exist infinitely many real quadratic fields with class number one, and the best unconditional algorithm known for computing the class number has complexity O D1=2+ : In fact, the apparent difficulty of com- puting class numbers has given rise to cryptographic algorithms based on arithmetic in quadratic fields. Factoring methods using quadratic fields have also been proposed which are dependent on being able to compute class numbers and regulators. The main goal of this thesis is to provide extensive numerical evidence in support of some unresolved conjectures related to quadratic fields. We first give an algorithm for computing class numbers and regulators of real quadratic fields, based on an algorithm due to Buchmann and Williams, that has complexity O D1=5+ and is conditional on the truth of the Extended Riemann Hypothesis. Our algorithm makes use of some im- provements, including a new method for estimating L (1; χ) due to Bach, and it performs about 1.5 times as quickly as similar algorithms which use truncated Euler products to estimate L (1; χ) : We then use this algorithm to compute class numbers of Q(pD) for all square-free D < 108 and Q(pp) for all prime p < 109 in order to test some heuristics due to Cohen and Lenstra on the distribution of real quadratic fields with certain class numbers, as well as a conjecture due to Hooley. Using a new sieving device, the MSSU, we examine the size of the regulator of Q(pD) by employing a strategy of Shanks to find fields with large L (1; χ) values. Our results lend support to a result of Littlewood that gives bounds on L (1; χ) assuming the truth of the Extended Riemann Hypothe- sis. Finally, we use MSSU to search for values of A such that the quadratic polynomial x2 + x + A has a high asymptotic density of prime values in order to test a conjecture of Hardy and Littlewood. v vi Acknowledgements I would like to take this opportunity to thank, first of all, my supervisor Dr. H.C. Williams. I first came into contact with him as a second year undergraduate student when he hired me as a summer research assistant, and I have been working with him ever since. It was he who introduced me to the fascinating subject of computational number theory. His generous financial support and constant encouragement have been immensely helpful to me throughout my studies, and are greatly appreciated. I also wish to thank the other members of my examining committee, Dr. M.W. Giesbrecht and Dr. J. Fabrykowski, for their careful reading of this thesis and their helpful suggestions for improving it. Special thanks go out to Richard Lukes for supplying me with all the numbers gener- ated by his sieve, the MSSU, which were used in Chapters 5 and 6. Finally, I wish to thank my family and friends, especially Barbara, for all the support and encouragement I received from them during my studies, and for their patience with my reclusiveness during the last few months while I finished this work. vii viii Contents 1 Introduction 1 1.1 Organization of the Thesis . 3 1.2 Frequently Used Notation . 6 2 Introduction to Quadratic Fields 9 2.1 Units and Prime Factorization in . 13 OK 2.2 Ideals and the Class Group . 14 2.3 Reduced Ideals in Imaginary Quadratic Fields . 19 2.4 Reduced Ideals in Real Quadratic Fields . 22 2.5 Infrastructure of the Principal Class . 26 2.6 L (1; χ) and the Analytic Class Number Formula . 30 3 Computing R and h in Real Quadratic Fields 35 3.1 Estimating L (1; χ) . 36 3.2 Evaluation of R . 39 3.3 Finding a divisor of h . 46 3.4 Evaluation of h . 49 4 The Cohen-Lenstra Heuristics 55 4.1 Heuristic Results . 56 4.2 Numerical Experiments . 65 4.3 Conclusion . 84 ix 5 The Size of R 85 5.1 Littlewood's Bounds on L (1; χ) . 86 5.2 Numerical Experiments . 90 5.3 Conclusion . 108 6 Polynomials With High Densities of Prime Values 111 6.1 The Conjecture of Hardy and Littlewood . 112 6.2 Evaluation of C(D) . 114 6.3 Computing h in Imaginary Quadratic Fields . 117 6.4 Numerical Experiments . 120 6.5 Conclusion . 139 7 Conclusion 143 x List of Tables 3.1 A and B values for A(Q; ∆) . 37 3.2 Growth of L . 40 3.3 Times for computing R using various algorithms . 46 3.4 Number of ideals used to compute h1 . 48 3.5 How often h1 was calculated . 49 3.6 How often Q > 18000 was required (truncated product method) . 51 3.7 How often Q > 5000 was required (Bach method) . 52 3.8 Times for computing h (truncated products Q=18000) . 53 3.9 Times for computing h (Bach's method Q=5000) . 54 4.1 q (x) for ∆ 1 (mod 4) . 67 i ≡ 4.2 q (x) for ∆ 0 (mod 4) . 68 i ≡ 4.3 q (x) for p 1 (mod 4) . 68 i ≡ 4.4 q (x) for p 3 (mod 4) . 69 i ≡ 4.5 t (x) for D 1 (mod 4) . 69 i ≡ 4.6 t (x) for D 0 (mod 4) . 70 i ≡ 4.7 t (x) for p 1 (mod 4) . 70 i ≡ 4.8 t (x) for p 3 (mod 4) . 71 i ≡ 4.9 H (x) for p 1 (mod 4) . 72 ∗ ≡ 4.10 H (x) for p 3 (mod 4) . 73 ∗ ≡ 5.1 D 5 (mod 8) | L (1; χ)-lochamps . 91 ≡ 5.2 D 5 (mod 8) | LLI-lochamps . 91 ≡ xi 5.3 D 1 (mod 8) | L (1; χ)-hichamps . 92 ≡ 5.4 D 1 (mod 8) | ULI-hichamps . 92 ≡ 5.5 D 6 (mod 8) | L (1; χ)-hichamps . 93 ≡ 5.6 D 6 (mod 8) | ULI-hichamps . 93 ≡ 5.7 D 1 (mod 4) | L (1; χ)-hichamps . 94 ≡ − 5.8 D 1 (mod 4) | ULI-hichamps . 94 ≡ − 5.9 5Np | Least Solutions . 96 5.10 5Np | Least Prime Solutions . 97 5.11 1Rp | Least Solutions . 98 5.12 1Rp | Least Prime Solutions . 99 5.13 6Rp | Least Solutions . 100 5.14 6R | Least 2 Prime Solutions . 101 p × 5.15 3Rp and 7Rp | Least Solutions . 102 5.16 3Rp and 7Rp | Least Prime Solutions . 103 6 6 6.1 PA(10 )=LA(10 ) for some values of D . 114 6.2 C(D)-hichamps (D < 0) . 122 6.3 C(D)-hichamps (D > 0) . 122 6.4 L (1; χ)-lochamps (D < 0) . 123 6.5 LLI-lochamps (D < 0) . 124 6.6 Np | Least Solutions . 127 6.7 Np | Least Prime Solutions . 128 6.8 Mp | Least Solutions . 129 6.9 Mp | Least Prime Solutions . 130 6.10 Np | Least Solutions (LLI) . 131 6.11 Np | Least Prime Solutions (LLI) . 132 γ 6.12 Np | Least Solutions (Z(Np) = C=e log log Np) . 133 γ 6.13 Np | Least Prime Solutions (Z(Np) = C=e log log Np) . 134 γ 6.14 Mp | Least Solutions (Z(Mp) = C=e log log Mp) . 135 γ 6.15 Mp | Least Prime Solutions (Z(Mp) = C=e log log Mp) . 136 xii List of Figures 4.1 x vs. q (x) for ∆ 1 (mod 4) . 74 1 ≡ 4.2 x vs. q (x) for ∆ 0 (mod 4) . 75 1 ≡ 4.3 x vs. q (x) for p 1 (mod 4) . 76 1 ≡ 4.4 x vs. q (x) for p 3 (mod 4) . 77 1 ≡ 4.5 x vs. t (x) for ∆ 1 (mod 4) . 78 1 ≡ 4.6 x vs. t (x) for ∆ 0 (mod 4) . 79 1 ≡ 4.7 x vs. t (x) for p 1 (mod 4) . 80 1 ≡ 4.8 x vs. t (x) for p 3 (mod 4) . 81 1 ≡ 4.9 x vs. 8H (x)=x for p 1 (mod 4) . 82 ∗ ≡ 4.10 x vs. 8H (x)=x for p 3 (mod 4) . 83 ∗ ≡ 5.1 Frequency values of Z for ∆ = p (prime), 8 108 < p < 109; p 1 (mod 8) 89 × ≡ 5.2 log 5Np vs. LLI . ..