Elementary number theory:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1998
|
| Ausgabe: | Repr. with corr. |
| Schriftenreihe: | Springer undergraduate mathematics series
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 301 S. |
| ISBN: | 3540761977 |
Internformat
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Datensatz im Suchindex
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|---|---|
| adam_text | IMAGE 1
CONTENTS
NOTES TO THE READER XIII
1. DIVISIBILITY 1
1.1 DIVISORS 2
1.2 BEZOUT S IDENTITY 7
1.3 LEAST COMMON MULTIPLES 12
1.4 LINEAR DIOPHANTINE EQUATIONS 13
1.5 SUPPLEMENTARY EXERCISES 16
2. PRIME NUMBERS 19
2.1 PRIME NUMBERS AND PRIME-POWER FACTORISATIONS 19
2.2 DISTRIBUTION OF PRIMES 25
2.3 FERMAT AND MERSENNE PRIMES 30
2.4 PRIMALITY-TESTING AND FACTORISATION 32
2.5 SUPPLEMENTARY EXERCISES 35
3. CONGRUENCES 37
3.1 MODULAR ARITHMETIC 37
3.2 LINEAR CONGRUENCES 46
3.3 SIMULTANEOUS LINEAR CONGRUENCES 52
3.4 SIMULTANEOUS NON-LINEAR CONGRUENCES 57
3.5 AN EXTENSION OF THE CHINESE REMAINDER THEOREM 59 3.6 SUPPLEMENTARY
EXERCISES 62
4. CONGRUENCES WITH A PRIME-POWER MODULUS 65
4.1 THE ARITHMETIC OF Z P 65
4.2 PSEUDOPRIMES AND CARMICHAEL NUMBERS 72
IMAGE 2
ELEMENTARY NUMBER THEORY
4.3 SOLVING CONGRUENCES MOD (P E ) 78
4.4 SUPPLEMENTARY EXERCISES 82
5. EULER S FUNCTION 83
5.1 UNITS ; 83
5.2 EULER S FUNCTION 85
5.3 APPLICATIONS OF EULER S FUNCTION 92
5.4 SUPPLEMENTARY EXERCISES 96
6. THE GROUP OF UNITS 97
6.1 THE GROUP U N 97
6.2 PRIMITIVE ROOTS 99
6.3 THE GROUP U P E , WHERE P IS AN ODD PRIME 103
6.4 THE GROUP U 2 * 106
6.5 THE EXISTENCE OF PRIMITIVE ROOTS 108
6.6 APPLICATIONS OF PRIMITIVE ROOTS 110
6.7 THE ALGEBRAIC STRUCTURE OF U N 113
6.8 THE UNIVERSAL EXPONENT 116
6.9 SUPPLEMENTARY EXERCISES 117
7. QUADRATIC RESIDUES 119
7.1 QUADRATIC CONGRUENCES 119
7.2 THE GROUP OF QUADRATIC RESIDUES 120
7.3 THE LEGENDRE SYMBOL 123
7.4 QUADRATIC RECIPROCITY 130
7.5 QUADRATIC RESIDUES FOR PRIME-POWER MODULI 135
7.6 QUADRATIC RESIDUES FOR ARBITRARY MODULI 138
7.7 SUPPLEMENTARY EXERCISES 140
8. ARITHMETIC FUNCTIONS 143
8.1 DEFINITION AND EXAMPLES 143
8.2 PERFECT NUMBERS 146
8.3 THE MOBIUS INVERSION FORMULA 148
8.4 AN APPLICATION OF THE MOBIUS INVERSION FORMULA 152 8.5 PROPERTIES OF
THE MOBIUS FUNCTION 154
8.6 THE DIRICHLET PRODUCT 157
8.7 SUPPLEMENTARY EXERCISES 162
9. THE RIEMANN ZETA FUNCTION 163
9.1 HISTORICAL BACKGROUND 163
9.2 CONVERGENCE 165
9.3 APPLICATIONS TO PRIME NUMBERS 166
IMAGE 3
CONTENTS XI
9.4 RANDOM INTEGERS 170
9.5 EVALUATING ((2) 174
9.6 EVALUATING ((2K) 176
9.7 DIRICHLET SERIES 179
9.8 EULER PRODUCTS 182
9.9 COMPLEX VARIABLES 185
9.10 SUPPLEMENTARY EXERCISES 188
10. SUMS OF SQUARES 191
10.1 SUMS OF TWO SQUARES 191
10.2 THE GAUSSIAN INTEGERS 196
10.3 SUMS OF THREE SQUARES 201
10.4 SUMS OF FOUR SQUARES 202
10.5 DIGRESSION ON QUATERNIONS 205
10.6 MINKOWSKI S THEOREM 206
10.7 SUPPLEMENTARY EXERCISES 214
11. FERMAT S LAST THEOREM 217
11.1 THE PROBLEM 217
11.2 PYTHAGORAS S THEOREM 218
11.3 PYTHAGOREAN TRIPLES 219
11.4 ISOSCELES TRIANGLES AND IRRATIONALITY 221
11.5 THE CLASSIFICATION OF PYTHAGOREAN TRIPLES 223
11.6 FERMAT 226
11.7 THE CASE N = 4 227
11.8 ODD PRIME EXPONENTS 228
11.9 LAME AND KUMMER 233
11.10 MODERN DEVELOPMENTS 234
11.11 FURTHER READING 237
APPENDIX A. INDUCTION AND WELL-ORDERING 239
APPENDIX B. GROUPS, RINGS AND FIELDS 243
APPENDIX C. CONVERGENCE 247
APPENDIX D. TABLE OF PRIMES P 1000 249
SOLUTIONS TO EXERCISES 251
BIBLIOGRAPHY 289
INDEX OF SYMBOLS 291
IMAGE 4
ELEMENTARY NUMBER THEORY
INDEX OF NAMES 295
INDEX 297
|
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| author | Jones, Gareth A. Jones, J. Mary |
| author_facet | Jones, Gareth A. Jones, J. Mary |
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| author_sort | Jones, Gareth A. |
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| building | Verbundindex |
| bvnumber | BV025876827 |
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| ctrlnum | (OCoLC)246084974 (DE-599)BVBBV025876827 |
| dewey-full | 512.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7 |
| dewey-search | 512.7 |
| dewey-sort | 3512.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Repr. with corr. |
| format | Book |
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| institution | BVB |
| isbn | 3540761977 |
| language | English |
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| physical | XIV, 301 S. |
| publishDate | 1998 |
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| publisher | Springer |
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| series2 | Springer undergraduate mathematics series |
| spelling | Jones, Gareth A. Verfasser aut Elementary number theory Gareth A. Jones and J. Mary Jones Repr. with corr. Berlin [u.a.] Springer 1998 XIV, 301 S. txt rdacontent n rdamedia nc rdacarrier Springer undergraduate mathematics series Zahlentheorie (DE-588)4067277-3 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Zahlentheorie (DE-588)4067277-3 s DE-604 Jones, J. Mary Verfasser aut SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019123940&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jones, Gareth A. Jones, J. Mary Elementary number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4123623-3 |
| title | Elementary number theory |
| title_auth | Elementary number theory |
| title_exact_search | Elementary number theory |
| title_full | Elementary number theory Gareth A. Jones and J. Mary Jones |
| title_fullStr | Elementary number theory Gareth A. Jones and J. Mary Jones |
| title_full_unstemmed | Elementary number theory Gareth A. Jones and J. Mary Jones |
| title_short | Elementary number theory |
| title_sort | elementary number theory |
| topic | Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie Lehrbuch |
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