On the infinitude of elliptic Carmichael numbers Item Type text; Dissertation-Reproduction (electronic) Authors Ekstrom, Aaron Todd Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 26/09/2021 05:20:16 Link to Item http://hdl.handle.net/10150/289053 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMi films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. 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Bell & Howell Infbmurtion and teaming 300 North Zaeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 ON THE INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS by Aaron Todd Ekstrom A Dissertation Submitted to the Faculty of the DEPARTMENT OF .MATHEMATICS In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 1999 UMI Number 9957966 UMI' UMI Microfomfi9957g66 Copyright 2000 by Bell & Howell Infomiation and Leaming Company. Ail rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Leaming Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Mi 48106-1346 2 THE UNIVERSITY OF ARIZONA « GRADUATE COLLEGE As members of the Final Examination Committee, we certify that we have read the dissertation prepared by AARON TODD EKSTRQM entitled ON THE INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of nn^^nr nf P^^^r.Qnphy^ Dirvcyk S* • TVNqkwy 3J Dine S/iThav,m.T Date John /Vrillhart Date' ' /n//y/f? Date Danie Date Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ifxfcrk S. "I V I ^ Octgbe< SJi Dissertation Director Dinesh S. Thakur Date 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation &om or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: 4 ACKNOWLEDGEMENTS I would like to thank my advisor, Dinesh Thakur, for guiding me through graduate school and demonstrating the nature of mathematics. I wish to express my gratitude to the following people: Dan Madden for posing the problem and for sharing his wisdom, Andrew Granville for his suggestion which improved the bound of the main theorem, and Bjom Poonen for his suggestion which helped prove the lifting lemma. In addition I wish to thank Hugh Williams and Susan Hammond Marshall for their helpful suggestions, Jennifer Christian Smith for many careful readings of this document and for all the time spent listening to me, and the Number Theory Group at the University of Arizona for their support. Finally, I will forever be indebted to Abbie Warrick for all her love and support. DEDICATION This work is dedicated to my mother. 6 TABLE OF CONTENTS LIST OF TABLES 8 LIST OF FIGURES 9 ABSTRACT 10 CHAPTER 1. INTRODUCTION 11 CHAPTER 2. COMPOSITENESS TESTS 16 2.1. The Lucas-Fennat Test and Carmichael Numbers 16 2.2. A Compositeness Test Using Elliptic Curves 19 2.2.1. Elliptic Curves 19 2.2.2. Elliptic Curve Compositeness Test 24 CHAPTER 3. THE INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS . 30 3.1. Preliminary Lemmas and Hypothesis 30 3.1.1. Sketch of the Proof of Theorem 3.7 30 3.1.2. Group theory lemma 31 3.1.3. Preliminaries regarding the distribution of primes 33 3.1.4. Prachar's theorem revisited 34 3.1.5. Least prime in an arithmetic progression 37 3.2. Lower Bound for Elliptic Carmichael Numbers 38 CHAPTER 4. CONSTRUCTING ELLIPTIC CARMICHAEL NUMBERS 43 4.1. iV = Pi • p2 • P3 43 4.2. N = pi---pk 47 CHAPTER 5. A LIFTING LEMMA AND IMPLICATIONS FOR THE ELLIPTIC CURVE COMPOSITENESS TEST 51 5.1. Lifting Lemma 51 5.2. Division Poljmomials 55 5.3. Computing [N + IJQ (mod N) 57 5.4. Elliptic Carmichael Numbers are Squarefiree 59 5.5. There are No Strong Elliptic Carmichael Numbers 61 5.6. Elliptic Carmichael Numbers with k Factors 66 CHAPTER 6. CONCLUSION 69 APPENDIX A. NOTATION 71 TABLE OF CONTENTS—Continued REFERENCES 8 LIST OF TABLES TABLE 1.1. Fifteen Carmichael Numbers 13 TABLE 2.1. Pseudoprimes vs. Primes: The entries in this table are from [PSW80] and unpublished work of Richard Pinch 18 TABLE 2.2. Sample Curves for Elliptic Curve Compositeness Test, and Elliptic Pseudoprimes foT Q eE 26 TABLE 2.3. Elliptic Carmichael Numbers with Prime Factors pi, pa, Pa • • 28 TABLE 4.1. Values of tsuch that PI, P2, and PA are all prime numbers (for Ti = 1, ra = 2, and ra = 3) 46 TABLE 4.2. Values of t such that PI, P2, and PA are all prime numbers (for Ti = 1, ^2 = 2, and ra = 5) 48 TABLE 4.3. Values of tsuch that pi, p2, pa, P4 and ps are all prime numbers (for ri = 1, r2 = 2,r3 = 3, r4= 5, and rs = 79) 50 TABLE A.l. Notation, part 1 72 TABLE A.2. Notation, part 2 73 LIST OF FIGURES FIGURE 2.1. Group Addition 10 ABSTRACT For any elliptic curve E with complex multiplication by an order in K = QiV^), a point Q of infinite order on E, and any prime p with gcd{A£,p} = 1, {-d\p) — —1, we have that [p+1] Q = 0 (mod p) where Ae is the discriminant oiE,Ovs the point at infinity and calculations are done using the addition law for E. Any composite number p which satisfies these conditions for any rational point on any CM elliptic curve over Q with discriminant prime to p is called an elliptic Carmichael number. For our main result, we modify the techniques of Alford, Granville and Pomerance to show there exist infinitely many elliptic Carmichael numbers under the assumption that the smallest prime congruent to —1 modulo q is at most gexp[(logg)^~®]. (Note that this assumption is much weaker than current conjectures about the smallest prime in an arithmetic progression.) We modify the construction of Chemick to provide many examples of elliptic Carmichael numbers. We also show that elliptic Carmichael numbers are square&ee, and modify the techniques of Pomerance, Selfridge, and Wagstaff to prove that the number of elliptic Carmichael numbers up to x with exactly k factors is at most for large enough z. Finally, we prove there are no strong elliptic Carmichael numbers, an analogue of Lehmer's result about strong Carmichael numbers. 11 Chapter 1 INTRODUCTION la this thesis, we examine the following problem: Given a large odd integer iV, can we quickly determine if AT is a prime number or a composite number? This problem has occupied mathematicians for centuries; in the words of Gauss[Gau66, page 396]: The problem of distinguishing prime numbers &om composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modem geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimable men, i.e., for numbers that do not yield to artificial methods, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. Even more attention has been focused on this problem since the introduction of a cryptography system by Rivest, Shamir and Adleman (called the RS A cryptosystem) in 1978 [RSA78].