Solved and unsolved problems in number theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Chelsea Publ. Co.
1978
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 258 S. graph. Darst. |
| ISBN: | 0828402973 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804116737114767360 |
|---|---|
| adam_text | CONTENTS
Page
Preface vii
Chapter I
FROM PERFECT NUMBERS TO THE QUADRATIC
RECIPROCITY LAW
Section
1. Perfect Numbers 1
2. Euclid 4
3. Euler s Converse Proved 8
4. Euclid s Algorithm 8
5. Cataldi and Others 12
6. The Prime Number Theorem 15
7. Two Useful Theorems 17
8. Fermat and Others 19
9. Euler s Generalization Proved 23
10. Perfect Numbers, II 25
11. Euler and Msl 25
12. Many Conjectures and their Interrelations 29
13. Splitting the Primes into Equinumerous Classes 31
14. Euler s Criterion Formulated 33
15. Euler s Criterion Proved 35
16. Wilson s Theorem 37
17. Gauss s Criterion 38
18. The Original Legendre Symbol 40
19. The Reciprocity Law 42
20. The Prime Divisors of n2 + a 47
Chapter II
THE UNDERLYING STRUCTURE
21. The Residue Classes as an Invention 51
22. The Residue Classes as a Tool 55
23. The Residue Classes as a Group 59
24. Quadratic Residues 63
V
vi Solved and Unsolved Problems in Number Theory
Section Page
25. Is the Quadratic Reciprocity Law a Deep Theorem? 64
26. Congruential Equations with a Prime Modulus 66
27. Euler s 4 Function 68
28. Primitive Roots with a Prime Modulus 71
29. arc, as a Cyclic Group 73
30. The Circular Parity Switch 76
31. Primitive Roots and Fermat Numbers 78
32. Artin s Conjectures 80
33. Questions Concerning Cycle Graphs 83
34. Answers Concerning Cycle Graphs 92
35. Factor Generators of 3Rm 98
36. Primes in Some Arithmetic Progressions and a General Divisi¬
bility Theorem 104
37. Scalar and Vector Indices 109
38. The Other Residue Classes 113
39. The Converse of Fermat s Theorem 115
40. Sufficient Conditions for Primality 118
Chapter III
PYTHAGOREANISM AND ITS MANY CONSEQUENCES
41. The Pythagoreans 121
42. The Pythagorean Theorem 123
43. The y/2 and the Crisis 126
44. The Effect upon Geometry 127
45. The Case for Pythagoreanism 130
46. Three Greek Problems 138
47. Three Theorems of Fermat 142
48. Fermat s Last Theorem 144
49. The Easy Case and Infinite Descent 147
50. Gaussian Integers and Two Applications 149
51. Algebraic Integers and Kummer s Theorem 151
52. The Restricted Case, Sophie Germain, and Wieferich 154
53. Euler s Conjecture 157
54. Sum of Two Squares 159
55. A Generalization and Geometric Number Theory 161
56. A Generalization and Binary Quadratic Forms 165
57. Some Applications 168
58. The Significance of Fermat s Equation 171
59. The Main Theorem 174
60. An Algorithm 178
Contents vii
Section Page
61. Continued Fractions for /N 180
62. From Archimedes to Lucas 188
63. The Lucas Criterion 193
64. A Probability Argument 197
65. Fibonacci Numbers and the Original Lucas Test 198
Appendix to Chapters I—III
Supplementary Comments, Theorems, and Exercises 201
Chapter IV
PROGRESS
Section
66. Chapter I Fifteen Years Later 217
67. Artin s Conjectures, II 222
68. Cycle Graphs and Related Topics 225
69. Pseudoprimes and Primality. 226
70. Fermat s Last Theorem, II 231
71. Binary Quadratic Forms with Negative Discriminants 233
72. Binary Quadratic Forms with Positive Discriminants 235
73. Lucas and Pythagoras 237
74. The Progress Report Concluded 238
Appendix
Statement on Fundamentals 239
Table of Definitions 241
References 243
Index 255
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T15:43:15Z |
| institution | BVB |
| isbn | 0828402973 |
| language | English |
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| physical | XIII, 258 S. graph. Darst. |
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| spelling | Shanks, Daniel 1917-1996 Verfasser (DE-588)117725293 aut Solved and unsolved problems in number theory 2. ed. New York Chelsea Publ. Co. 1978 XIII, 258 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001497289&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shanks, Daniel 1917-1996 Solved and unsolved problems in number theory Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Solved and unsolved problems in number theory |
| title_auth | Solved and unsolved problems in number theory |
| title_exact_search | Solved and unsolved problems in number theory |
| title_full | Solved and unsolved problems in number theory |
| title_fullStr | Solved and unsolved problems in number theory |
| title_full_unstemmed | Solved and unsolved problems in number theory |
| title_short | Solved and unsolved problems in number theory |
| title_sort | solved and unsolved problems in number theory |
| topic | Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Number theory Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001497289&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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