My numbers, my friends: popular lectures on number theory
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2000
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 375 S. |
| ISBN: | 0387989110 |
Staff View
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV013261017 | ||
| 003 | DE-604 | ||
| 005 | 00000000000000.0 | ||
| 007 | t | ||
| 008 | 000720s2000 |||| 00||| eng d | ||
| 016 | 7 | |a 95957283X |2 DE-101 | |
| 020 | |a 0387989110 |9 0-387-98911-0 | ||
| 035 | |a (OCoLC)247760868 | ||
| 035 | |a (DE-599)BVBBV013261017 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-20 |a DE-824 |a DE-355 |a DE-19 |a DE-11 |a DE-188 | ||
| 050 | 0 | |a QA241 | |
| 082 | 0 | |a 512.7 | |
| 084 | |a SK 180 |0 (DE-625)143222: |2 rvk | ||
| 084 | |a 17,1 |2 ssgn | ||
| 100 | 1 | |a Ribenboim, Paulo |d 1928- |e Verfasser |0 (DE-588)117719404 |4 aut | |
| 245 | 1 | 0 | |a My numbers, my friends |b popular lectures on number theory |c Paulo Ribenboim |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2000 | |
| 300 | |a IX, 375 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Zahlentheorie | |
| 650 | 4 | |a Number theory | |
| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Zahlentheorie |0 (DE-588)4067277-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009039483&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-009039483 | ||
Record in the Search Index
| _version_ | 1804128025173819392 |
|---|---|
| adam_text | Contents
Preface xi
1 The Fibonacci Numbers and the Arctic Ocean 1
1 Basic definitions 2
A Lucas sequences 2
B Special Lucas sequences 3
C Generalizations 3
2 Basic properties 5
A Binet s formulas 5
B Degenerate Lucas sequences 5
C Growth and numerical calculations 6
D Algebraic relations 7
E Divisibility properties 9
3 Prime divisors of Lucas sequences 10
A The sets V(U), V{V), and the rank of
appearance 10
B Primitive factors of Lucas sequences .... 17
4 Primes in Lucas sequences 26
5 Powers and powerful numbers in Lucas sequences . 28
A General theorems for powers 29
B Explicit determination in special sequences . 30
vi Contents
C Uniform explicit determination of
multiples, squares, and square classes for
certain families of Lucas sequences 35
D Powerful numbers in Lucas sequences .... 41
2 Representation of Real Numbers by Means of
Fibonacci Numbers 51
3 Prime Number Records 62
4 Selling Primes 78
5 Euler s Famous Prime Generating Polynomial 91
1 Quadratic extensions 94
2 Rings of integers 94
3 Discriminant 95
4 Decomposition of primes 96
A Properties of the norm 96
5 Units 100
6 The class number 101
A Calculation of the class number 103
B Determination of all quadratic fields with
class number 1 106
7 The main theorem 108
6 Gauss and the Class Number Problem 112
1 Introduction 112
2 Highlights of Gauss life 112
3 Brief historical background 114
4 Binary quadratic forms 115
5 The fundamental problems • 118
6 Equivalence of forms 118
7 Conditional solution of the fundamental problems . 120
8 Proper equivalence classes of definite forms 122
A Another numerical example 126
9 Proper equivalence classes of indefinite forms ... 126
A Another numerical example 131
10 The automorph of a primitive form 131
11 Composition of proper equivalence classes of
primitive forms 135
Contents vii
12 The theory of genera 137
13 The group of proper equivalence classes of
primitive forms 143
14 Calculations and conjectures 145
15 The aftermath of Gauss (or the math after
Gauss) 146
16 Forms versus ideals in quadratic fields 146
17 Dirichlet s class number formula 153
18 Solution of the class number problem for definite
forms 157
19 The class number problem for indefinite forms . . . 161
20 More questions and conjectures 164
21 Many topics have not been discussed 168
7 Consecutive Powers 175
1 Introduction 175
2 History 177
3 Special cases 178
4 Divisibility properties 190
5 Estimates 195
A. The equation au bv = 1 196
B. The equation Xm Yn = 1 197
C. The equation Xu Yv = 1 201
6 Final comments and applications 204
8 1093 213
A. Determination of the residue of qp(a) ... . 216
B. Identities and congruences for the Fermat
quotient 217
9 Powerless Facing Powers 229
1 Powerful numbers 229
A. Distribution of powerful numbers 230
B. Additive problems 232
C. Difference problems 233
2 Powers 235
A. Pythagorean triples and Fermat s problem . 235
B. Variants of Fermat s problem 238
C. The conjecture of Euler 239
D. The equation AX1 + BYm = CZn 240
viii Contents
E. Powers as values of polynomials 245
3 Exponential congruences 246
A. The Wieferich congruence 246
B. Primitive factors 248
4 Dream mathematics 251
A. The statements 251
B. Statements 252
C. Binomials and Wieferich congruences .... 254
D. Erdos conjecture and Wieferich congruence . 257
E. The dream in the dream 257
10 What Kind of Number Is v^? 271
0 Introduction 271
1 Kinds of numbers 271
2 How numbers are given 276
3 Brief historical survey 284
4 Continued fractions 287
A. Generalities 288
B. Periodic continued fractions 289
C. Simple continued fractions of ir and e . . . . 291
5 Approximation by rational numbers 295
A. The order of approximation 295
B. The Markoff numbers 296
C. Measures of irrationality 298
D. Order of approximation of irrational
algebraic numbers 299
6 Irrationality of special numbers 301
7 Transcendental numbers 309
A. Liouville numbers 310
B. Approximation by rational numbers:
sharper theorems 311
C. Hermite, Lindemann, and Weierstrass . . . 316
D. A result of Siegel on exponentials 318
E. Hilbert s 7th problem 320
F. The work of Baker 321
G. The conjecture of Schanuel 323
H. Transcendence measure and the
classification of Mahler 328
8 Final comments 331
Contents ix
11 Galimatias Arithmeticae 344
Index of Names 361
Index of Subjects 369
|
| any_adam_object | 1 |
| author | Ribenboim, Paulo 1928- |
| author_GND | (DE-588)117719404 |
| author_facet | Ribenboim, Paulo 1928- |
| author_role | aut |
| author_sort | Ribenboim, Paulo 1928- |
| author_variant | p r pr |
| building | Verbundindex |
| bvnumber | BV013261017 |
| callnumber-first | Q - Science |
| callnumber-label | QA241 |
| callnumber-raw | QA241 |
| callnumber-search | QA241 |
| callnumber-sort | QA 3241 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)247760868 (DE-599)BVBBV013261017 |
| 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 |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01373nam a2200385 c 4500</leader><controlfield tag="001">BV013261017</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">00000000000000.0</controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">000720s2000 |||| 00||| eng d</controlfield><datafield tag="016" ind1="7" ind2=" "><subfield code="a">95957283X</subfield><subfield code="2">DE-101</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0387989110</subfield><subfield code="9">0-387-98911-0</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)247760868</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV013261017</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-20</subfield><subfield code="a">DE-824</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-19</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA241</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">512.7</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 180</subfield><subfield code="0">(DE-625)143222:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">17,1</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Ribenboim, Paulo</subfield><subfield code="d">1928-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)117719404</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">My numbers, my friends</subfield><subfield code="b">popular lectures on number theory</subfield><subfield code="c">Paulo Ribenboim</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">New York [u.a.]</subfield><subfield code="b">Springer</subfield><subfield code="c">2000</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">IX, 375 S.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Zahlentheorie</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Number theory</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Zahlentheorie</subfield><subfield code="0">(DE-588)4067277-3</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Zahlentheorie</subfield><subfield code="0">(DE-588)4067277-3</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009039483&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-009039483</subfield></datafield></record></collection> |
| id | DE-604.BV013261017 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:42:41Z |
| institution | BVB |
| isbn | 0387989110 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009039483 |
| oclc_num | 247760868 |
| open_access_boolean | |
| owner | DE-20 DE-824 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-188 |
| owner_facet | DE-20 DE-824 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-188 |
| physical | IX, 375 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| spelling | Ribenboim, Paulo 1928- Verfasser (DE-588)117719404 aut My numbers, my friends popular lectures on number theory Paulo Ribenboim New York [u.a.] Springer 2000 IX, 375 S. txt rdacontent n rdamedia nc rdacarrier Zahlentheorie 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=009039483&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ribenboim, Paulo 1928- My numbers, my friends popular lectures on number theory Zahlentheorie Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | My numbers, my friends popular lectures on number theory |
| title_auth | My numbers, my friends popular lectures on number theory |
| title_exact_search | My numbers, my friends popular lectures on number theory |
| title_full | My numbers, my friends popular lectures on number theory Paulo Ribenboim |
| title_fullStr | My numbers, my friends popular lectures on number theory Paulo Ribenboim |
| title_full_unstemmed | My numbers, my friends popular lectures on number theory Paulo Ribenboim |
| title_short | My numbers, my friends |
| title_sort | my numbers my friends popular lectures on number theory |
| title_sub | popular lectures on number theory |
| topic | Zahlentheorie Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie Number theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009039483&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ribenboimpaulo mynumbersmyfriendspopularlecturesonnumbertheory |