A course in computational number theory:
"The accompanying CD-Rom contains Mathematica files with all the commands and programs."--p. [4] of cover.
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Emeryville, CA
Key College Publ.
2000
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "The accompanying CD-Rom contains Mathematica files with all the commands and programs."--p. [4] of cover. |
| Physical Description: | XII, 367 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 1930190107 |
Staff View
MARC
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| 245 | 1 | 0 | |a A course in computational number theory |c David Bressoud ; Stan Wagon |
| 264 | 1 | |a Emeryville, CA |b Key College Publ. |c 2000 | |
| 300 | |a XII, 367 S. |b Ill., graph. Darst. |e 1 CD-ROM (12 cm) | ||
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| 520 | 3 | |a "The accompanying CD-Rom contains Mathematica files with all the commands and programs."--p. [4] of cover. | |
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| 650 | 4 | |a Nombres, Théorie des | |
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Record in the Search Index
| _version_ | 1804128013999144960 |
|---|---|
| adam_text | Contents
Preface v
Notation xi
Chapter 1. Fundamentals 1
1.0 Introduction 1
1.1 A Famous Sequence of Numbers 2
1.2 The Euclidean Algorithm 6
The Oldest Algorithm
Reversing the Euclidean Algorithm
The Extended GCD Algorithm
The Fundamental Theorem of Arithmetic
Two Applications
1.3 Modular Arithmetic 25
1.4 Fast Powers 30
A Fast Algorithm for Exponentiation
Powers of Matrices, Big O Notation
Chapter 2. Congruences, Equations, and Powers 41
2.0 Introduction 41
2.1 Solving Linear Congruences 41
Linear Diophantine Equations in Two Variables
Linear Equations in Several Variables
Linear Congruences
The Conductor
An Important Quadratic Congruence
2.2 The Chinese Remainder Theorem 49
viH CONTENTS
2.3 PowerMod Patterns 55
Fermat s Little Theorem
More Patterns in Powers
2.4 Pseudoprimes 59
Using the Pseudoprime Test
Chapter 3. Euler s f Function 65
3.0 Introduction 65
3.1 Euler s j Function 65
3.2 Perfect Numbers and Their Relatives 72
The Sum of Divisors Function
Perfect Numbers
Amicable, Abundant, and Deficient Numbers
3.3 Euler s Theorem 81
3.4 Primitive Roots for Primes 84
The Order of an Integer
Primes Have Primitive Roots
Repeating Decimals
3.5 Primitive Roots for Composites 90
3.6 The Universal Exponent 93
Universal Exponents
Power Towers
The Form of Carmichael Numbers
Chapter 4. Prime Numbers 99
4.0 Introduction 99
4.1 The Number of Primes 100
We ll Never Run Out of Primes
The Sieve of Eratosthenes
Chebyshev s Theorem and Bertrand s Postulate
4.2 Prime Testing and Certification 114
Strong Pseudoprimes
Industrial Grade Primes
Prime Certification Via Primitive Roots
An Improvement
Pratt Certificates
4.3 Refinements and Other Directions 131
Other Primality Tests
Strong Liars Are Scarce
Finding the nth Prime
4.4 A Dozen Prime Mysteries 141
CONTENTS IX
Chapter 5. Some Applications 145
5.0 Introduction 145
5.1 Coding Secrets 145
Tossing a Coin into a Well
The RSA Cryptosystem
Digital Signatures
5.2 The Yao Millionaire Problem 155
5.3 Check Digits 158
Basic Check Digit Schemes
A Perfect Check Digit Method
Beyond Perfection: Correcting Errors
5.4 Factoring Algorithms 167
Trial Division
Fermat s Algorithm
Pollard Rho
Pollard p 1
The Current Scene
Chapter 6. Quadratic Residues 179
6.0 Introduction 179
6.1 Pepin sTest 179
Quadratic Residues
Pepin s Test
Primes Congruent to 1 (Mod 4)
6.2 Proof of Quadratic Reciprocity 185
Gauss s Lemma
Proof of Quadratic Reciprocity
Jacobi s Extension
An Application to Factoring
6.3 Quadratic Equations 194
Chapter 7. Continued Fractions 201
7.0 Introduction 201
7.1 Finite Continued Fractions 202
7.2 Infinite Continued Fractions 207
7.3 Periodic Continued Fractions 213
7.4 Pell s Equation 227
7.5 Archimedes and the Sun God s Cattle 232
Wurm s Version: Using Rectangular Bulls
The Real Cattle Problem
7.6 Factoring via Continued Fractions 238
X CONTENTS
Chapter 8. Prime Testing with Lucas Sequences 247
8.0 Introduction 247
8.1 Divisibility Properties of Lucas Sequences 248
8.2 Prime Tests Using Lucas Sequences 259
Lucas Certification
The Lucas Lehmer Algorithm Explained
Lucas Pseudoprimes
Strong Quadratic Pseudoprimes
Primality Testing s Holy Grail
Chapter 9. Prime Imaginaries and Imaginary Primes 279
9.0 Introduction 279
9.1 Sums of Two Squares 279
Primes
The General Problem
How Many Ways
Number Theory and Salt
9.2 The Gaussian Integers 302
Complex Number Theory
Gaussian Primes
The Moat Problem
The Gaussian Zoo
9.3 Higher Reciprocity 325
Appendix A. Mathematics Basics 333
A.O Introduction 333
A.1 Plotting 335
A.2 Typesetting 338
Sending Files By E Mail
A.3 Types of Functions 341
A.4 Lists 343
A.5 Programs 345
A.6 Solving Equations 347
A.7 Symbolic Algebra 349
Appendix B. Lucas Certificates Exist 351
References 355
Index of Mathematica Objects 359
Subject Index 363
|
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| discipline | Informatik Mathematik |
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| id | DE-604.BV013251605 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:42:30Z |
| institution | BVB |
| isbn | 1930190107 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009031976 |
| oclc_num | 41452561 |
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| owner_facet | DE-91G DE-BY-TUM DE-20 DE-739 DE-703 DE-634 DE-83 DE-11 DE-188 |
| physical | XII, 367 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Key College Publ. |
| record_format | marc |
| spelling | Bressoud, David M. 1950- Verfasser (DE-588)136747221 aut A course in computational number theory David Bressoud ; Stan Wagon Emeryville, CA Key College Publ. 2000 XII, 367 S. Ill., graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier "The accompanying CD-Rom contains Mathematica files with all the commands and programs."--p. [4] of cover. Algorithmes Nombres, Théorie des Algorithms Number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Computeralgebra (DE-588)4010449-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s Computeralgebra (DE-588)4010449-7 s DE-604 Wagon, Stan 1951- Verfasser (DE-588)172440114 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009031976&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bressoud, David M. 1950- Wagon, Stan 1951- A course in computational number theory Algorithmes Nombres, Théorie des Algorithms Number theory Zahlentheorie (DE-588)4067277-3 gnd Computeralgebra (DE-588)4010449-7 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4010449-7 |
| title | A course in computational number theory |
| title_auth | A course in computational number theory |
| title_exact_search | A course in computational number theory |
| title_full | A course in computational number theory David Bressoud ; Stan Wagon |
| title_fullStr | A course in computational number theory David Bressoud ; Stan Wagon |
| title_full_unstemmed | A course in computational number theory David Bressoud ; Stan Wagon |
| title_short | A course in computational number theory |
| title_sort | a course in computational number theory |
| topic | Algorithmes Nombres, Théorie des Algorithms Number theory Zahlentheorie (DE-588)4067277-3 gnd Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Algorithmes Nombres, Théorie des Algorithms Number theory Zahlentheorie Computeralgebra |
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