Elementary number theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston
McGraw-Hill, Higher Education
2010
|
| Ausgabe: | 7th ed., [pbk] |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xii, 436 p. Illustrationen 24 cm |
| ISBN: | 9780071289191 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804176696230805504 |
|---|---|
| adam_text | CONTENTS
Preface viii
New to This Edition x
1 Preliminaries 01
1.1 Mathematical Induction 01
1.2 The Binomial Theorem 08
2 Divisibility Theory in the Integers 13
2.1 Early Number Theory 13
2.2 The Division Algorithm 17
2.3 The Greatest Common Divisor 19
2.4 The Euclidean Algorithm 26
2.5 The Diophantine Equation ax + by = c 32
3 Primes and Their Distribution 39
3.1 The Fundamental Theorem of Arithmetic 39
3.2 The Sieve of Eratosthenes 44
3.3 The Goldbach Conjecture 50
4 The Theory of Congruences 61
4.1 Carl Friedrich Gauss 61
4.2 Basic Properties of Congruence 63
4.3 Binary and Decimal Representations of Integers 69
4.4 Linear Congruences and the Chinese Remainder Theorem 76
5 Fermat’s Theorem 85
5.1 Pierre de Fermat 85
5.2 Fermat’s Little Theorem and Pseudoprimes 87
v
vi CONTENTS
5.3 Wilson’s Theorem 93
5.4 The Fermat-Kraitchik Factorization Method 97
6 Number-Theoretic Functions 103
6.1 The Sum and Number of Divisors 103
6.2 The Möbius Inversion Formula 112
6.3 The Greatest Integer Function 117
6.4 An Application to the Calendar 122
7 Euler’s Generalization of Fermat’s Theorem 129
7.1 Leonhard Euler 129
7.2 Euler’s Phi-Function 131
7.3 Euler’s Theorem 136
7.4 Some Properties of the Phi-Function 141
8 Primitive Roots and Indices 147
8.1 The Order of an Integer Modulo n 147
8.2 Primitive Roots for Primes 152
8.3 Composite Numbers Having Primitive Roots 158
8.4 The Theory of Indices 163
9 The Quadratic Reciprocity Law 169
9.1 Euler’s Criterion 169
9.2 The Legendre Symbol and Its Properties 175
9.3 Quadratic Reciprocity 185
9.4 Quadratic Congruences with Composite Moduli 192
10 Introduction to Cryptography 197
10.1 From Caesar Cipher to Public Key Cryptography 197
10.2 The Knapsack Cryptosystem 209
10.3 An Application of Primitive Roots to Cryptography 214
11 Numbers of Special Form 219
11.1 Marin Mersenne 219
11.2 Perfect Numbers 221
11.3 Mersenne Primes and Amicable Numbers 227
11.4 Fermat Numbers 237
12 Certain Nonlinear Diophantine Equations 245
12.1 The Equation x2 4- y2 = z2 245
12.2 Fermat’s Last Theorem 252
CONTENTS VÜ
13 Representation of Integers as Sums of Squares 261
13.1 Joseph Louis Lagrange 261
13.2 Sums of Two Squares 263
13.3 Sums of More Than Two Squares 272
14 Fibonacci Numbers 283
14.1 Fibonacci 283
14.2 The Fibonacci Sequence 285
14.3 Certain Identities Involving Fibonacci Numbers 292
15 Continued Fractions 303
15.1 Srinivasa Ramanujan 303
15.2 Finite Continued Fractions 306
15.3 Infinite Continued Fractions 319
15.4 Farey Fractions 334
15.5 Pell’s Equation 337
16 Some Modern Developments 353
16.1 Hardy, Dickson, and Erdos 353
16.2 Primality Testing and Factorization 358
16.3 An Application to Factoring: Remote Coin Flipping 371
16.4 The Prime Number Theorem and Zeta Function 375
Miscellaneous Problems 384
Appendixes 387
General References 387
Suggested Further Reading 390
Tables 393
Answers to Selected Problems 410
Index
421
|
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| dewey-hundreds | 500 - Natural sciences and mathematics |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 7th ed., [pbk] |
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| spelling | Burton, David M. 1930- Verfasser (DE-588)101498372X aut Elementary number theory David M. Burton 7th ed., [pbk] Boston McGraw-Hill, Higher Education 2010 xii, 436 p. Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Elementare Zahlentheorie (DE-588)4294368-1 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content 2\p (DE-588)4123623-3 Lehrbuch gnd-content Zahlentheorie (DE-588)4067277-3 s DE-604 Elementare Zahlentheorie (DE-588)4294368-1 s 3\p DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029243141&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Burton, David M. 1930- Elementary number theory Includes bibliographical references and index Number theory Zahlentheorie (DE-588)4067277-3 gnd Elementare Zahlentheorie (DE-588)4294368-1 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4294368-1 (DE-588)4151278-9 (DE-588)4123623-3 |
| title | Elementary number theory |
| title_auth | Elementary number theory |
| title_exact_search | Elementary number theory |
| title_full | Elementary number theory David M. Burton |
| title_fullStr | Elementary number theory David M. Burton |
| title_full_unstemmed | Elementary number theory David M. Burton |
| title_short | Elementary number theory |
| title_sort | elementary number theory |
| topic | Number theory Zahlentheorie (DE-588)4067277-3 gnd Elementare Zahlentheorie (DE-588)4294368-1 gnd |
| topic_facet | Number theory Zahlentheorie Elementare Zahlentheorie Einführung Lehrbuch |
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