DOCSLIB.ORG

Lecture Notes Number Theory and Cryptography Matt Kerr

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes Number Theory and Cryptography Matt Kerr Lecture notes Number Theory and Cryptography Matt Kerr Contents Introduction 5 Part 1. Primes and divisibility 9 Chapter 1. The Euclidean Algorithm 11 Chapter 2. Primes and factorization 21 Chapter 3. The distribution of primes 27 Chapter 4. The prime number theorem 35 Part 2. Congruences 43 Chapter 5. Modular arithmetic 45 Chapter 6. Consequences of Fermat’s theorem 53 Chapter 7. The Chinese Remainder Theorem 61 Chapter 8. Primality and compositeness testing 67 Chapter 9. Groups, rings, and fields 79 Chapter 10. Primitive roots 87 Chapter 11. Prime power moduli and power residues 93 Part 3. Introduction to cryptography 105 Chapter 12. Symmetric ciphers 107 Chapter 13. Public key cryptography 113 Chapter 14. Discrete log problem 117 3 4 CONTENTS Chapter 15. RSA Cryptosystem 125 Chapter 16. Introduction to PARI 131 Chapter 17. Breaking RSA 135 Part 4. Diophantine equations 141 Chapter 18. A first view of Diophantine equations 143 Chapter 19. Quadratic Diophantine equations 149 Chapter 20. Units in quadratic number rings 155 Chapter 21. Pell’s equation and related problems 163 Chapter 22. Unique factorization in number rings 171 Chapter 23. Elliptic curves 179 Chapter 24. Elliptic curves over Fp 189 Part 5. Elliptic cryptosystems 197 Chapter 25. Elliptic curve discrete log problem (ECDLP) 199 Chapter 26. Elliptic curve cryptography 207 Chapter 27. Lenstra’s factorization algorithm 211 Chapter 28. Pairing-based cryptography 215 Chapter 29. Divisors and the Weil pairing 221 Part 6. Algebraic numbers 231 Chapter 30. Algebraic number fields 233 Chapter 31. Discriminants and algebraic integers 239 Chapter 32. Ideals in number rings 247 Chapter 33. The ideal class group 253 Chapter 34. Fermat’s Last Theorem for regular exponents 259 Introduction Number theory has its roots in the study of the properties of the natural numbers N = f1, 2, 3, . .g and various “extensions” thereof, beginning with the integers Z = f..., −2, −1, 0, 1, 2, . .g and rationals a Q = b j a, b 2 Z, b 6= 0 . This leads directly to the first two parts of this course, of which the following may serve as a brief outline. ∗ ∗ ∗ I. Divisibility. • Euclidean algorithm and greatest common divisors. • Primes and the Fundamental Theorem of Algebra. • Results and conjectures concerning primes: Euclid’s theo- rem; the Riemann zeta function; arithmetic progressions. II. Congruences. • Modular (clock) arithmetic: ap−1 ≡ 1 (mod p) and its gener- alizations. • Chinese remainder theorem: given x ≡ a and x ≡ b, find x (p) (q) (mod pq). • A first view of primality testing and factorization. • Groups, rings and fields (especially finite abelian groups and finite fields). 5 6 INTRODUCTION • Primitive roots modulo a prime: e.g. mod 7, 3 · 3 ≡ 2, so 2 (7) has a square root! • Quadratic reciprocity: e.g., if 37 is a square modulo 11, this allows you to decide without computation whether 11 is a square modulo 37 (which it is). III. Cryptography (a first look). • Simple cryptosystems and symmetric ciphers. • Public key cryptography: answers the question “How can two parties communicate securely over an insecure channel without first privately exchanging some kind of ’key’ to each others’ messages?” They need a trapdoor function f that can be used to encode information easily but hard to invert with- out knowing “extra information”. • Diffie-Hellman key exchange (based on difficulty of solving ax ≡ b for x) and the discrete log problem. (p) • RSA cryptosystem: this is based on the difficulty of solving xe ≡ c when N = pq. (N) • Introduction to GP-PARI (computer package for number the- ory). • Pollard p − 1 factorization method: this helps us understand when RSA could be potentially broken. IV. Diophantine equations. • This is the part of number theory that studies polynomial equations in integers or rationals. A famous example is the insolubility of xm + ym = zm (apart from the “trivial” so- lution (0, 0, 0)) for m ≥ 3, known as Fermat’s last theorem (proved by Andrew Wiles). • Pythagoras’s theorem and Fibonacci numbers. • Pell’s equation (x2 − dy2 = ±1) and quadratic number fields. INTRODUCTION 7 • Cubic equations and the group law for elliptic curves.1 V. Elliptic curve cryptography. • The security of using elliptic curves for cryptography rests on the difficulty of solving an analogue of the discrete log problem. • We can also use the group law on an elliptic curve to factor large numbers (Lenstra’s algorithm). • A deeper, more flexible sort of cryptosystem can be obtained from the “Weil pairing” on m-torsion points of an elliptic curve. V. Algebraic numbers. • These appeared under the guise of “ideal numbers”p in the mid-19th century. Easy examples include a + b −1, where a, b 2 Z. • Cyclotomic fields and an “easy” case of Fermat’s last theo- rem. • Failure of unique factorization in general. • Irrationality and Galois groups. • Ideals and class groups. • Fermat’s last theorem (less easy case, still far from the whole thing). ∗ ∗ ∗ Now the natural numbers have a well-defined notion of order, which leads to the following property: 1Confusing terminology: these are not ellipses, which are defined by a quadratic 2 x2 + y = equation a2 b2 1, but rather are defined by cubic (and sometimes quartic) equations such as y2 = x3 + ax + b (or y2 = (1 − x2)(1 − k2x2)). They are called “elliptic” for the arcane historical reason that a related “elliptic integral” 2 2 1 p 1−k x dx arises in the course of determining the arclength of an ellipse. 0 ( − 2)( − 2 2) ´ 1 x 1 k x 8 INTRODUCTION THEOREM 1 (Principle of the least element). Let S ⊂ N be a nonempty subset. Then S has a least element, i.e. there exists s 2 S such that for every x 2 S, s ≤ x. (This also applies to N [ f0g.) Theorem 1 implies the well-known THEOREM 2 (Principle of mathematical induction). Let S(x) be a statement about any x 2 N. Suppose that (i) S(1) is true and (ii) S(x) true (8x < n) =) S(n) true. Them S(x) is true for all x 2 N. PROOF THAT THEOREM 1 =) THEOREM 2. Assume that (i) and (ii) hold, and suppose that F := fx 2 N j S(x) falseg is nonempty. Then F has a least element f , by Theorem 1. Hence, for any x < f , we have x 2/ F — i.e. S(x) is true. Now consider the following two cases: • f = 1: impossible, as it contradicts (i). • f > 1: by (ii), S( f ) is then true, contradicting f 2 F . Therefore our supposition was absurd, and F is empty. Part 1 Primes and divisibility CHAPTER 1 The Euclidean Algorithm We begin our discussion with the division algorithm: PROPOSITION 3. Given a, b 2 N, there exist unique q, r 2 Z such that a = b · q + r with 0 ≤ r < b. Of course, the “algorithm” isn’t in the formal statement, but in how we produce q and r. EXAMPLE 4. Suppose a = 313 and b = 9. In grade school, you learned to write 34 9 313 27 43 36 7 which yields 313 = 9 · 34 + 7 . |{z} |{z} q r The algorithm is simply long division with remainder. PROOF OF PROPOSITION 3. For the “existence” part, let S := fa − bk j k 2 Z, a − bk ≥ 0g ⊆ N [ f0g. Since a 2 S, S 6= Æ. Let r be the least element of S. Then r = a − bq ≥ 0 for some q 2 Z. If r ≥ b then S contains r − b = a − b(q + 1), contradicting minimality of r. So r < b. To see the uniqueness, write bq0 + r0 = a = bq + r, 11 12 1. THE EUCLIDEAN ALGORITHM with 0 ≤ r, r0 < b. This yields r = b(q0 − q) + r0, and if we had q0 > q, then q0 ≥ q + 1 would imply r ≥ b + r0 ≥ b + 0 = b, a contradiction. Symmetrically, one argues that q > q0 is 0 0 impossible. Therefore q = q and then also r = r . Next, we turn to divisibility and the GCD (= greatest common divisor). DEFINITION 5. Let a, b 2 Z, with b 6= 0. Then b j a () 9 c 2 Z such that a = bc. defn. (We say that “b divides a”.) Here are some basic examples: • everything divides 0; • 2ja () a is even; • bja () r = 0 in the division algorithm. and some basic properties: (i) ajb and bjc =) ajc (ii) ajb, c =) ajbx + cy for all x, y 2 Z (e.g. b + c, b − c) (iii) ajb and bja =) a = ±b. PROOF OF (III). Given b = ad, a = bc (and a, b 6= 0), we have a = adc =) dc = 1 =) d = ±1 = c. For any a, b 2 Z, not both 0, let S(a, b) := fd 2 N j dja, bg . DEFINITION 6. The GCD of a and b is (a, b) := the biggest element of S(a, b). (Of course, you need only check integers less than or equal to the smallest of jaj and jbj.) We say that a and b are relatively prime if (a, b) = 1. 1. THE EUCLIDEAN ALGORITHM 13 Again, here are some simple examples: • (4, −6) = 2 • (0, 7) = 7 • (12, 7) = 1 and some properties: (iv) (0, b) = b = (b, b) (v) (a, b) = (b, a) = (a, −b) (vi) (b, a − mb) = (a, b) for every m 2 Z. PROOF OF (VI). Let dja, b. Then dja − mb. Conversely, if djb, a − mb, then djmb + (a − mb) = a. So S(a, b) = S(b, a − mb) and they have identical largest elements. Property (vi) has the key consequence: LEMMA 7. Say a = bq + r in the division algorithm. Then (a, b) = (b, r).
Load more

Recommended publications
  • An Analysis of Primality Testing and Its Use in Cryptographic Applications
    An Analysis of Primality Testing and Its Use in Cryptographic Applications Jake Massimo Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Information Security Royal Holloway, University of London 2020 Declaration These doctoral studies were conducted under the supervision of Prof. Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathe- matics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Jake Massimo April, 2020 2 Abstract Due to their fundamental utility within cryptography, prime numbers must be easy to both recognise and generate. For this, we depend upon primality testing. Both used as a tool to validate prime parameters, or as part of the algorithm used to generate random prime numbers, primality tests are found near universally within a cryptographer's tool-kit. In this thesis, we study in depth primality tests and their use in cryptographic applications. We first provide a systematic analysis of the implementation landscape of primality testing within cryptographic libraries and mathematical software. We then demon- strate how these tests perform under adversarial conditions, where the numbers being tested are not generated randomly, but instead by a possibly malicious party. We show that many of the libraries studied provide primality tests that are not pre- pared for testing on adversarial input, and therefore can declare composite numbers as being prime with a high probability.
    [Show full text]
  • MATH 404: ARITHMETIC 1. Divisibility and Factorization After Learning To
    MATH 404: ARITHMETIC S. PAUL SMITH 1. Divisibility and Factorization After learning to count, add, and subtract, children learn to multiply and divide. The notion of division makes sense in any ring and much of the initial impetus for the development of abstract algebra arose from problems of division√ and factoriza- tion, especially in rings closely related to the integers such as Z[ d]. In the next several chapters we will follow this historical development by studying arithmetic in these and similar rings. √ To see that there are some serious issues to be addressed notice that in Z[ −5] the number 6 factorizes as a product of irreducibles in two different ways, namely √ √ 6 = 2.3 = (1 + −5)(1 − −5). 1 It isn’t hard to show these factors are irreducible but we postpone√ that for now . The key point for the moment is to note that this implies that Z[ −5] is not a unique factorization domain. This is in stark contrast to the integers. Since our teenage years we have taken for granted the fact that every integer can be written as a product of primes, which in Z are the same things as irreducibles, in a unique way. To emphasize the fact that there are questions to be understood, let us mention that Z[i] is a UFD. Even so, there is the question which primes in Z remain prime in Z[i]? This is a question with a long history: Fermat proved in ??? that a prime p in Z remains prime in Z[i] if and only if it is ≡ 3(mod 4).√ It is then natural to ask for each integer d, which primes in Z remain prime in Z[ d]? These and closely related questions lie behind much of the material we cover in the next 30 or so pages.
    [Show full text]
  • GRACEFUL and PRIME LABELINGS -Algorithms, Embeddings and Conjectures
    GRACEFUL AND PRIME LABELINGS-Algorithms, Embeddings and Conjectures Group Discussion (KREC, Surathkal: Aug.16-25, 1999). Improved Version. GRACEFUL AND PRIME LABELINGS -Algorithms, Embeddings and Conjectures Suryaprakash Nagoji Rao* Exploration Business Group MRBC, ONGC, Mumbai-400 022 Abstract. Four algorithms giving rise to graceful graphs from a known (non-)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and highly graceful graphs. The Ringel-Rosa-Kotzig Conjecture is generalized to highly graceful graphs. Mayeda-Seshu Tree Generation Algorithm is modified to generate all possible graceful labelings of trees of order p. An alternative algorithm in terms of integers modulo p is described which includes all possible graceful labelings of trees of order p and some interesting properties are observed. Optimal and graceful graph embeddings (not necessarily connected) are given. Alternative proofs for embedding a graph into a graceful graph as a subgraph and as an induced subgraph are included. An algorithm to obtain an optimal graceful embedding is described. A necessary condition for a graph to be supergraceful is given. As a consequence some classes of non-supergraceful graphs are obtained. Embedding problems of a graph into a supergraceful graph are studied. A catalogue of super graceful graphs with at most five nodes is appended. Optimal graceful and supergraceful embeddings of a graph are given. Graph theoretical properties of prime and superprime graphs are listed. Good upper bound for minimum number of edges in a nonprime graph is given and some conjectures are proposed which in particular includes Entringer’s prime tree conjecture.
    [Show full text]
  • 13 Lectures on Fermat's Last Theorem
    I Paulo Ribenboim 13 Lectures on Fermat's Last Theorem Pierre de Fermat 1608-1665 Springer-Verlag New York Heidelberg Berlin Paulo Ribenboim Department of Mathematics and Statistics Jeffery Hall Queen's University Kingston Canada K7L 3N6 Hommage a AndrC Weil pour sa Leqon: goat, rigueur et pCnCtration. AMS Subiect Classifications (1980): 10-03, 12-03, 12Axx Library of Congress Cataloguing in Publication Data Ribenboim, Paulo. 13 lectures on Fermat's last theorem. Includes bibliographies and indexes. 1. Fermat's theorem. I. Title. QA244.R5 512'.74 79-14874 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. @ 1979 by Springer-Verlag New York Inc. Printed in the United States of America. 987654321 ISBN 0-387-90432-8 Springer-Verlag New York ISBN 3-540-90432-8 Springer-Verlag Berlin Heidelberg Preface Fermat's problem, also called Fermat's last theorem, has attracted the attention of mathematicians for more than three centuries. Many clever methods have been devised to attack the problem, and many beautiful theories have been created with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topic is presented in the form of lectures, where I survey the main lines of work on the problem. In the first two lectures, there is a very brief description of the early history, as well as a selection of a few of the more representative recent results. In the lectures which follow, I examine in suc- cession the main theories connected with the problem.
    [Show full text]
  • A Performant, Misuse-Resistant API for Primality Testing
    A Performant, Misuse-Resistant API for Primality Testing Jake Massimo1 and Kenneth G. Paterson2 1 Information Security Group, Royal Holloway, University of London [email protected] 2 Department of Computer Science, ETH Zurich [email protected] Abstract. Primality testing is a basic cryptographic task. But developers today are faced with complex APIs for primality testing, along with documentation that fails to clearly state the reliability of the tests being performed. This leads to the APIs being incorrectly used in practice, with potentially disastrous consequences. In an effort to overcome this, we present a primality test having a simplest-possible API: the test accepts a number to be tested and returns a Boolean indicating whether the input was composite or probably prime. For all inputs, the output is guaranteed to be correct with probability at least 1 − 2−128. The test is performant: on random, odd, 1024-bit inputs, it is faster than the default test used in OpenSSL by 17%. We investigate the impact of our new test on the cost of random prime generation, a key use case for primality testing. The OpenSSL developers have adopted our suggestions in full; our new API and primality test are scheduled for release in OpenSSL 3.0. 1 Introduction Primality testing, and closely related tasks like random prime generation and testing of Diffie-Hellman parameters, are core cryptographic tasks. Primality testing is by now very well understood mathematically; there is a clear distinction between accuracy and running time of different tests in settings that are malicious (i.e.
    [Show full text]
  • Fermat's Last Theorem for Regular Primes: Dinesh S. Thakur
    Fermat's last theorem for regular primes Dinesh S. Thakur The so-called Fermat's last theorem (FLT for short) that `there are no integral solutions to xn + yn = zn, with n > 2 and xyz 6= 0 (a non-triviality condition)' is now proved by Wiles. The proof is outside the scope of this summer school, though we will sketch some ideas in the later lectures. Now we will prove the so-called `regular prime case' of FLT, following Kummer in essence, but connecting with later ideas and simplifications by other mathematicians. It may be advisable to first learn the proof for n = 3 from Hardy and Wright or from Ireland-Rosen pp. 285-286, which prove even stronger result, using only the basics of quadratic fields that we have seen. Since FLT for the exponent n implies FLT for multiples of n and since we proved it for n = 4, it is enough to prove FLT for n = p an odd prime. The basic strategy is the same as before: try to get an infinite descent and thus a contradiction starting with an assumed non-trivial solution. We have p Q i a factorization z = (x + ζpy) and we try to conclude that each factor (apart from GCD) is a p-th power. This conclusion is justified in the case of unique factorization domains. But since we only have unique factorization in ideals in general, we can conclude that each factor is a p-th power of an ideal, which can be further assumed to be principal, if p does not divide h, the class number of Q(ζp).
    [Show full text]
  • Kummer, Regular Primes, and Fermat's Last Theorem 1
    KUMMER, REGULAR PRIMES, AND FERMAT'S LAST THEOREM ILA VARMA CALIFORNIA INSTITUTE OF TECHNOLOGY Abstract. This paper rephrases Kummer's proof of many cases of Fermat's last theorem in contemporary notation that was in fact derived from his work. Additionally, it develops a reformulation of the proof using class field theory from a modern perspective in a manner similar to the tactics used for the complete proof, and describes how Kummer's proof strategy can generalize to solve the theorem for a broader set of primes. 1. Introduction Ernst Kummer was a 19th century mathematician who came across Fermat's last theorem in attempts to generalize the law of quadratic reciprocity and study higher reciprocity laws. While he described those as \the principal subject and the pinnacle of contemporary number theory," he considered Fermat's last theorem a \curiosity of number theory rather than a major item" [1]. A priori, this was not an unreasonable opinion of a problem that could be understood by a 12-year-old. We state this mere curiosity below. Theorem. For any integer n > 2, the equation xn + yn = zn has no non-trivial solutions in the integers, i.e. if x; y; z 2 Z satisfy this equation, then xyz = 0. Despite his disinterest, Kummer made the first substantial step in proving a part of Fermat's last theorem for many cases. This came only a few weeks after Gabriel Lam´eincorrectly announced that he had found a complete proof [1]. Lam´edid make the breakthrough in attempting to decompose xn + yn into linear factors by introducing the complex numbers satisfying ζn = 1, known today as roots of unity.
    [Show full text]
  • A Note on the Voronoi Congruences and the Residue of the Fermat Quotient
    A note on the Voronoi congruences and the residue of the Fermat quotient Claire Levaillant August 31, 2021 Abstract We prove a congruence on the residue of the Fermat quotient in base a which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the generalized harmonic number H p−1 , a generalization originally due to Sun of a classical 2, 2 congruence known since long for the harmonic number H p−1 as a special case 2 of the Lerch formula. We find a sharpening of the Voronoi congruences that is different from the one of Johnson and more computationally efficient. We prove an additional related congruence, which specialized to base 2, allows to retrieve several congruences that were originally due to Lehmer. 1 Past interest in computing the residue of the Fermat quotient in base a Computing the residue of the Fermat quotient qa in general base a has become important to mathematicians since the beginning of the twentieth century ever since young Arthur Wieferich showed in 1909 that if the first case of Fermat’s last theorem (FLT) is false for a prime p ≥ 5 then this prime is such that q2 = 0 mod p [26]. Such a prime got later called a Wieferich prime in his honor. arXiv:2108.12733v1 [math.NT] 29 Aug 2021 FLT. For n ≥ 3, the equation Xn + Y n = Zn has no solutions in integers X,Y,Z with XYZ 6= 0.
    [Show full text]
  • History of Fermat's Last Theorem a Thesis Submitted to the Faculty of Atlanta University in Partial Fulfillment of the Requireme
    HISTORY OF FERMAT'S LAST THEOREM A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY BILL KALTSOUNIS DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA AUGUST 1968 TABLE OF CONTENTS Page INTRODUCTION K K K ..>... h K K K K K ... 1 Chapter I. FERMAT'S LAST THEOREM 2 II. EARLY ATTEMPTS AT PROOF 8 III. RECENT WORK ON FERMAT'S LAST THEOREM .... 17 IV. SUMMARY 22 LIST OF REFERENCES 23 iii INTRODUCTION The famed "Fermat's last theorem" has been chosen as a topic for rather detailed study in order to provide a relatively brief summary of the history and the research that has been done on the theorem as a possible stimulation and an aid to those who may be interested in broadening the horizon of existing proof. Further research on the proof of the theorem or any of its innumer¬ able restricted cases is clearly dependent upon an almost complete know¬ ledge of existing research. Therefore a brief study of this sort should serve well as an introduction to the necessary knowledge and might possibly also be a link in a chain of work resulting in some new ideas in the mathematical world. Since three centuries of research using known mathematical methods have resulted in no general proof, new methods or ideas must in all probability be developed before a general proof can evolve. In order to provide an adequate introduction to the knowledge necessary for further research, this study should include a brief his¬ tory of the birth of the theorem along with Fermat's actual and prob¬ able work on it, the plausibility of its validity, the minimum restricted cases necessary for general proof, and lastly, a summary of the actual valid work that has been and is being done with a consideration of mo¬ tives for continued interest such as prizes for complete proof.
    [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]
  • Chains of Semiprime and Prime Ideals in Leavitt Path Algebras 3
    CHAINS OF SEMIPRIME AND PRIME IDEALS IN LEAVITT PATH ALGEBRAS G. ABRAMS, B. GREENFELD, Z. MESYAN, AND K. M. RANGASWAMY Abstract. Semiprime ideals of an arbitrary Leavitt path algebra L are described in terms of their generators. This description is then used to show that the semiprime ideals form a complete sublattice of the lattice of ideals of L, and they enjoy a certain gap property identified by Kaplansky in prime spectra of commutative rings. It is also shown that the totally ordered sets that can be realized as prime spectra of Leavitt path algebras are precisely those that have greatest lower bounds on subchains and enjoy the aforementioned gap property. Finally, it is shown that a conjecture of Kaplansky regarding identifying von Neumann regular rings via their prime factors holds for Leavitt path algebras. 1. Introduction Kaplansky included the following results in two of his monographs from the 1970s, the first of which he attributed to R. Hamsher. (See [11, page 1] and [10, Theorem 11], respectively.) Theorem (K-1): A commutative ring R is von Neumann regular if and only if R is semiprime and every prime factor ring of R is von Neumann regular. Theorem (K-2): If P ( Q are prime ideals of a commutative unital ring R, then there are prime ideals P ′, Q′ of R such that P ⊆ P ′ ( Q′ ⊆ Q and Q′ covers P ′, that is, there are no prime ideals of R strictly between P ′ and Q′. Kaplansky took an interest in extending both theorems to noncommutative rings. In each case, his questions inspired substantial new results in the literature.
    [Show full text]
  • Number Theoretic Aspects of a Combinatorial Function
    Number theoretic aspects of a combinatorial function Lorenz Halbeisen1 and Norbert Hungerbuhler¨ Abstract We investigate number theoretic aspects of the integer sequence seq1 1(n) with identification number A000522 in Sloane’s On-Line Encyclopedia of Integer Sequences: seq1 1(n) counts the number of sequences without repetition one can build with n distinct objects. By introducing the the notion of the “shadow” of an integer function, we examine divisibility properties of the combinatorial function seq1 1(n): We show that seq1 1(n) has the reduction property and its shadow d therefore is multiplicative. As a consequence, the shadow d of seq1 1(n) is determined by its values at powers of primes. It turns out that there is a simple characterization of regular prime numbers, i.e. prime numbers p for which the shadow d of seq1 1 has the socket property d(pk) = d(p) for all integers k. Although a stochastic argument supports the conjecture that infinitely many irregular primes exist, there density is so thin that there is only one irregular prime number less than 2.5 · 106, namely 383. 1 Introduction The sequence we are interested in has the ID number A000522 in Sloane’s On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/ njas/sequences). Former identification numbers of this sequence were M1497 in [SP]e and N0589 in [Sl]. The sequence A000522 has many faces (see, e.g., [Ga], [Si] or [Ri]). The most accessible one is its combinatorial interpretation: Definition 1 For n ∈ N = {0, 1, 2,... } let seq1 1(n) denote the number of one-to- one sequences – these are sequences without repetitions – we can build with n distinct objects.
    [Show full text]