Catalan's conjecture: are 8 and 9 the only consecutive powers?
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston u.a.
Academic Press
1994
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 364 S. |
| ISBN: | 0125871708 |
Staff View
MARC
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| 245 | 1 | 0 | |a Catalan's conjecture |b are 8 and 9 the only consecutive powers? |c Paulo Ribenboim |
| 264 | 1 | |a Boston u.a. |b Academic Press |c 1994 | |
| 300 | |a XIV, 364 S. | ||
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| 650 | 7 | |a Nombres, Théorie des |2 ram | |
| 650 | 4 | |a Puissances consécutives (Algèbre) | |
| 650 | 4 | |a Consecutive powers (Algebra) | |
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Record in the Search Index
| _version_ | 1804124157525360640 |
|---|---|
| adam_text | Contents
Preface xi
Reader s Guide xiii
Introduction 1
History 5
Part P PRELIMINARIES 9
1. Binomials and Cyclotomic Polynomials 9
2. The Cyclotomic Field 27
3. The Pythagorean Equation, Special Cases of Fermat s
Last Theorem and Related Equations 31
4. Continued Fractions 55
5. The Equations EX2 DY2 = ±C 60
Part A SPECIAL CASES 67
1. Preliminary Lemmas 67
2. The Sequence of Squares or Cubes 69
3. The Equation Xm Y2 = 1 78
4. The Result of St0rmer on Fermat s Equation 80
5. The Attempts to Solve X2 Yn = 1 89
6. The Equation X2 Yn = I, n 3 92
vii
viii Contents
7. The Equations X3 Yn = 1 and Xm Y3 = 1, with
m, n 3 96
8. The Equation X ~ * = Fm 110
v — 1
9. The Sequence of Powers of 2 or 3 124
10. Interlude 127
11. The Equation 2Xn 1 = Z2 129
12. 7T and Grave s Problem 132
13. A Problem of Fermat on Pythagorean Triangles and the
Equation 2X4 Y4 = Z2 144
14. The Equations X4 ± 2mY4 = ±Z2 and X4 ± Y4 =
2mZ2 164
15. Representation of Integers by Binary Cubic Forms 177
16. Some Quartic Equations 192
Part B DIVISIBILITY CONDITIONS 201
1. Getting the Consecutive Powers 8 and 9 201
2. The Theorem of Cassels and First Consequences 204
3. Prime Factors of Solutions of Catalan s Equation 214
4. The Theorem of Hyyro 216
5. The Theorems of Inkeri 219
Part C ANALYTICAL METHODS 241
1. Some General Theorems for Diophantine Equations 241
I. The Equation Xm Yn = 1 247
2. Upper Bounds for the Number and Size of Solutions ... 247
3. Lower Bounds for Solutions 256
4. Algorithm to Determine the Eventual Solutions 266
II. The Equation au bv = 1 269
5. What Will Be Discussed 269
6. Finiteness of the Number of Solutions 271
7. Algorithm to Determine the Eventual Solutions 282
8. The Largest Prime Factor of Values of Quadratic
Polynomials 283
9. Effective Results 288
III. The Equation Xu Yv = 1 298
Contents ix
10. The Theorem of Tijdeman 298
11. A Density Result 308
Appendix 1. Catalan s Equation in Other Domains 313
(A) Catalan s Equation over Number Fields 313
(B) Catalan s Equation over Fields K(t) and
Domains K[t] 314
(C) Catalan s Equation Over Function Fields of
Projective Varieties 318
Appendix 2. Powerful Numbers 319
(A) Distribution of Powerful Numbers 320
(B) Additive Problems 322
(C) Difference Problems 323
Bibliography 331
Index of Names 359
Subject Index 363
|
| any_adam_object | 1 |
| author | Ribenboim, Paulo 1928- |
| author_GND | (DE-588)117719404 |
| author_facet | Ribenboim, Paulo 1928- |
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| bvnumber | BV009806361 |
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| callnumber-raw | QA161.E95 |
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| dewey-full | 512/.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.74 |
| dewey-search | 512/.74 |
| dewey-sort | 3512 274 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV009806361 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:41:12Z |
| institution | BVB |
| isbn | 0125871708 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006489597 |
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| owner_facet | DE-355 DE-BY-UBR DE-20 DE-634 DE-11 |
| physical | XIV, 364 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Academic Press |
| record_format | marc |
| spelling | Ribenboim, Paulo 1928- Verfasser (DE-588)117719404 aut Catalan's conjecture are 8 and 9 the only consecutive powers? Paulo Ribenboim Boston u.a. Academic Press 1994 XIV, 364 S. txt rdacontent n rdamedia nc rdacarrier Nombres, Théorie des ram Puissances consécutives (Algèbre) Consecutive powers (Algebra) Catalan-Vermutung (DE-588)4369060-9 gnd rswk-swf Catalan-Vermutung (DE-588)4369060-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006489597&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ribenboim, Paulo 1928- Catalan's conjecture are 8 and 9 the only consecutive powers? Nombres, Théorie des ram Puissances consécutives (Algèbre) Consecutive powers (Algebra) Catalan-Vermutung (DE-588)4369060-9 gnd |
| subject_GND | (DE-588)4369060-9 |
| title | Catalan's conjecture are 8 and 9 the only consecutive powers? |
| title_auth | Catalan's conjecture are 8 and 9 the only consecutive powers? |
| title_exact_search | Catalan's conjecture are 8 and 9 the only consecutive powers? |
| title_full | Catalan's conjecture are 8 and 9 the only consecutive powers? Paulo Ribenboim |
| title_fullStr | Catalan's conjecture are 8 and 9 the only consecutive powers? Paulo Ribenboim |
| title_full_unstemmed | Catalan's conjecture are 8 and 9 the only consecutive powers? Paulo Ribenboim |
| title_short | Catalan's conjecture |
| title_sort | catalan s conjecture are 8 and 9 the only consecutive powers |
| title_sub | are 8 and 9 the only consecutive powers? |
| topic | Nombres, Théorie des ram Puissances consécutives (Algèbre) Consecutive powers (Algebra) Catalan-Vermutung (DE-588)4369060-9 gnd |
| topic_facet | Nombres, Théorie des Puissances consécutives (Algèbre) Consecutive powers (Algebra) Catalan-Vermutung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006489597&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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