A course in computational number theory:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley
[2008]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 367 S. Ill., graph. Darst. |
| ISBN: | 9780470412152 |
Internformat
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| 100 | 1 | |a Bressoud, David M. |d 1950- |e Verfasser |0 (DE-588)136747221 |4 aut | |
| 245 | 1 | 0 | |a A course in computational number theory |c David Bressoud ; Stan Wagon |
| 264 | 1 | |a Hoboken, NJ |b Wiley |c [2008] | |
| 300 | |a XII, 367 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Number theory |x Data processing | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016745129 | ||
Datensatz im Suchindex
| _version_ | 1804138027305402369 |
|---|---|
| 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
-О
Notation
Chapter
2.
Congruences, Equations, and Powers
41
2.0
Introduction
............................................................................................ 41
2.1
Solving Linear Congruences
................................................................... 41
Linear Diophantine Equations in Two Varmbies
Linear Equations in Several Variables
Linear Congruences
The Conductor
An Important Quadratic Congruence
2.2
The Chinese Remainder Theorem
.......................................................... 49
VIU
CONTENTS
2.3
PowerMod Patterns
................................................................................ 55
Fermaťs
Little Theorem
More Patterns in Powers
2.4
Pseudoprimes
......................................................................................... 59
Using the
Pseudoprime Test
Chapter
3.
Euler s
φ
Function
65
3.0
Introduction
............................................................................................ 65
3.1
Euler s
φ
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
Pri
ma lity
Tests
Strong Liars Are Scarce
Finding the nth Prime
4.4
A Dozen Prime Mysteries
........................................................................ 141
CONTENTS
ЇХ
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
Pépin sTest
............................................................................................ 179
Quadratic Residues
Pépin 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
Imaginarles
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. Mathematica
Basics
333
A.0 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 Mathomatica Objects
359
Subject Index
363
|
| adam_txt |
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
-О
Notation
Chapter
2.
Congruences, Equations, and Powers
41
2.0
Introduction
. 41
2.1
Solving Linear Congruences
. 41
Linear Diophantine Equations in Two Varmbies
Linear Equations in Several Variables
Linear Congruences
The Conductor
An Important Quadratic Congruence
2.2
The Chinese Remainder Theorem
. 49
VIU
CONTENTS
2.3
PowerMod Patterns
. 55
Fermaťs
Little Theorem
More Patterns in Powers
2.4
Pseudoprimes
. 59
Using the
Pseudoprime Test
Chapter
3.
Euler's
φ
Function
65
3.0
Introduction
. 65
3.1
Euler's
φ
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
Pri
ma lity
Tests
Strong Liars Are Scarce
Finding the nth Prime
4.4
A Dozen Prime Mysteries
. 141
CONTENTS
ЇХ
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
Pépin'sTest
. 179
Quadratic Residues
Pépin'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
Imaginarles
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. Mathematica
Basics
333
A.0 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 Mathomatica Objects
359
Subject Index
363 |
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| id | DE-604.BV035076823 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:05:52Z |
| indexdate | 2024-07-09T21:21:39Z |
| institution | BVB |
| isbn | 9780470412152 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016745129 |
| oclc_num | 261197757 |
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| physical | XII, 367 S. Ill., graph. Darst. |
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| spelling | Bressoud, David M. 1950- Verfasser (DE-588)136747221 aut A course in computational number theory David Bressoud ; Stan Wagon Hoboken, NJ Wiley [2008] XII, 367 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Datenverarbeitung Number theory Data processing Computeralgebra (DE-588)4010449-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s Computeralgebra (DE-588)4010449-7 s DE-604 Wagon, Stan Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016745129&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bressoud, David M. 1950- Wagon, Stan A course in computational number theory Datenverarbeitung Number theory Data processing Computeralgebra (DE-588)4010449-7 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4010449-7 (DE-588)4067277-3 |
| 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_exact_search_txtP | 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 | Datenverarbeitung Number theory Data processing Computeralgebra (DE-588)4010449-7 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Datenverarbeitung Number theory Data processing Computeralgebra Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016745129&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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