Linear Forms in Logarithms and Applications:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Zürich
European Mathematical Society Publishing House
2018
|
| Ausgabe: | 1. Auflage |
| Schriftenreihe: | IRMA Lectures in Mathematics and Theoretical Physics
28 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030675654&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XVI, 224 Seiten 24 cm x 17 cm |
| ISBN: | 9783037191835 303719183X |
Internformat
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| 245 | 1 | 0 | |a Linear Forms in Logarithms and Applications |c Yann Bugeaud |
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| 264 | 1 | |a Zürich |b European Mathematical Society Publishing House |c 2018 | |
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| 653 | |a Baker’s theory | ||
| 653 | |a Diophantine equation | ||
| 653 | |a Thue equation | ||
| 653 | |a abc-conjecture | ||
| 653 | |a irrationality measure | ||
| 653 | |a linear form in logarithms | ||
| 653 | |a p-adic analysis | ||
| 653 | |a primitive divisor | ||
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Datensatz im Suchindex
| _version_ | 1819275939140337664 |
|---|---|
| adam_text | Contents Preface ......................................... v Frequently used notation............................ xv 1. Brief introduction to linear forms in logarithms............................................ 1.1. Linear forms in complex logarithms............... 1.2. Linear forms in p-adic logarithms............................................................ . 1.3. Linear forms in elliptic logarithms............................................................... 1 1 6 6 2. Lower bounds for linear forms in complex and p-adic logarithms...................... 2.1. Lower bounds for linear forms in complex logarithms . ............................. 2.2. Multiplicative dependence relations between algebraic numbers................ 2.3. Lower bounds for linear forms in/7-adic logarithms.............. .................. 2.4. Notes........................................................................ 9 9 17 17 22 3. First applications..................................................................................................... 3.1. On the distance between powers of 2 and powers of 3 . ............................. 3.2. Effective irrationality measures for quotients of logarithms of integers . . 3.3. On the distance between two integral 5-units . . . ................................... 3.4. Effective irrationality measures for n-th roots of algebraic numbers .... 3.5. On the greatest prime factor of values of integer polynomials................... 3.6. On the greatest prime, factor of terms of linear recurrence sequences ... 3.7. Perfect powers in linear recurrence
sequences............................................ 3.8. Simultaneous Pelhan equations and Diophantine quadruples ................... 3.9. On the representation of integers in distinct bases...................................... 3.10. Further applications (without proofs) ............................................................ 3.11. Exercises ........................................................................................................ 3.12. Notes . . . .............................................................·............................ ... 23 23 25 25 26 27 30 32 35 39 43 44 44 4. Classical families of Diophantine equations............................... ......................... 4.1. Proof of Baker’s Theorem 1.10..................................................................... 4.2. The unit equation............................................................... -........................... 4.3. The Thue equation ........................................................................................ 4.4. Effective improvement of Liouville’s inequality......................................... 4.5. The superelliptic and hyperelliptic equations............................................... 4.6. The Diophantine equation x2 + C = yn...................................................... 4.7. Perfect powers at integer values of polynomials......................................... 47 47 50 52 55 55 57 60 «
xii Contents 4.8. The Catalan equation and the Pillai conjecture............................................. 4.9. Exercises ......................................................................................................... 4.10. Notes......................... 62 65 65 5. Further applications................... .............................................................................. 5.1. Effective irrationality measures for quotients of logarithms of rational numbers............................................................................................................ 5.2. Effective irrationality measures for n-th roots of rational numbers............. 5.3. The Thue equation axn — byn = c............................................................... 5.4. A generalization of Diophantine quadruples . . .......................................... 5.5. Exercises ............................... 5.6. Notes ........................................... 67 6. Applications of linear forms in p-adic logarithms................................................ 6.1. On the p-adic distance between two integral 5-units................................... 6.2. Waring’s problem........................................................... 6.3. On the ծ-ary expansion of an algebraic number.......................................... 6.4. Repunits and perfect powers .................................. 6.5. Perfect powers with few binary digits . .......................................................... 6.6. The 5-unit equation.................. 6.7. The Thue-Mahler equation and other
classical equations.......................... 6.8. Perfect powers in binary recurrence sequences............................................. 6.9. Perfect powers as sum of two integral 5-units............................................. 6.10. On the digital representation of integral 5-units......................................... 6.11. Exercises ......................................................................................................... 6.12. Notes ................................................................................................................ 75 75 76 77 80 81 83 85 86 88 89 90 92 67 68 69 71 72 73 7. Primitive divisors and the greatest prime factor of 2 — 1................................... 95 7.1. Primitive divisors............................................................................................ 95 7.2. Primitive divisors of Lucas-Lehmer sequences ......................................... 98 7.3. The Diophantine equation x2 + C = yn, continued...................................... 101 7.4. On the number of solutions to the Diophantine equation x2 + D — pn . . 102 7.5. On the greatest prime factor of 2 — 1............................................................ 103 7.6. Exercises ...................... 106 7.7. Notes..................................................................................................................106 8. The a¿c-conjecture.................................................................................................. 109 8.1. Effective results towards the üòc-
conjecture...................................................109 8.2. Proofs of Theorems 8.2 and 8.3................................... ............................ ... . 110 8.3. Exercise...............................................................................................................115 8.4. Notes................ ............................................................................................... 115 9. Simultaneous linear forms in logarithms and applications................................... 117 9.1. A theorem of Loxton.........................................................................................117 9.2. Perfect powers in short intervals . . ................................................................118 9.3. Simultaneous Pelhan equations with at most one solution.............................121 9.4. Exercise...............................................................................................................123 9.5. Notes..................................................................................................................123
Contents xiii 10. Multiplicative dependence relations between algebraic numbers ..........................125 10.1. Lower bound for the height of an algebraic number...................................125 10.2. Existence of small multiplicative dependence relations ............................ 127 11. Lower bounds for linear forms in two complex logarithms: proofs................... .131 11.1. Three estimates for linear forms in two complex logarithms...................... 131 11.2. An auxiliary inequality involving several parameters...................................132 11.3. Proof of Theorem 11.4..................................................................................... 133 11.4. Deduction of Theorem 11.1 from Theorem 11.4............................................ 138 11.5. Deduction of Theorem 11.2 from Theorem 11.4............................................ 140 12. Lower bounds for linear forms in two /badie logarithms: proofs......................... 143 12.1. Three estimates for linear forms in two p-adic logarithms......................... 143 12.2. Two auxiliary inequalities involving several parameters............................... 145 12.3. Proofs of Theorems 12.4 and 12.5................................... ..........................147 12.4. Deduction of Theorem 12.1 from Theorem 12.4.................................. 153 12.5. Deduction of Theorem 12.2 from Theorem 12.5............................................155 12.6. Deduction of Theorem 12.3 from Theorem 12.1 ...................... 156 13. Open problems...................... 157 13.1.
Classical conjectures in transcendence theory.................................. 157 13.2. Diophantine equations ............................................................... 158 13.3. Miscellaneous.....................................................................................................161 Appendices....................................................................................................................... 165 A. Approximation by rational numbers ......................... 167 B. Heights ........................................................................................................................171 B.l. Definitions........................................................ 171 B.2. The Liouville inequality .................................................................................. 176 B.3. Linear forms in one /badie logarithm..................................... 177 C. Auxiliary results on algebraic number fields............................................................179 C.l. Real quadratic fields and Pellian equations................... . . . ................... 179 C.2. Algebraic number fields..................................................................................... 180 D. Classical results on prime numbers............................................... 183 E. A zero lemma..................................................................... 187 F. Tools from complex and p-adic analysis.................................................................. 191 F.l. The Schwarz lemma in complex
analysis.........................................................191 F.2. Auxiliary results on/badie fields............................................................ . . 191 F.3. The Schwarz lemma in /badie analysis...................... 193 Bibliography .................................. 195 Index.................................................................................................................................... 223
|
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| id | DE-604.BV045288223 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T18:23:07Z |
| institution | BVB |
| institution_GND | (DE-588)1066118477 |
| isbn | 9783037191835 303719183X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030675654 |
| oclc_num | 1025337058 |
| open_access_boolean | |
| owner | DE-20 DE-739 DE-355 DE-BY-UBR |
| owner_facet | DE-20 DE-739 DE-355 DE-BY-UBR |
| physical | XVI, 224 Seiten 24 cm x 17 cm |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | European Mathematical Society Publishing House |
| record_format | marc |
| series | IRMA Lectures in Mathematics and Theoretical Physics |
| series2 | IRMA Lectures in Mathematics and Theoretical Physics |
| spellingShingle | Bugeaud, Yann 1971- Linear Forms in Logarithms and Applications IRMA Lectures in Mathematics and Theoretical Physics Diophantische Gleichung (DE-588)4012386-8 gnd Zahlentheorie (DE-588)4067277-3 gnd Diophantische Ungleichung (DE-588)4200628-4 gnd |
| subject_GND | (DE-588)4012386-8 (DE-588)4067277-3 (DE-588)4200628-4 |
| title | Linear Forms in Logarithms and Applications |
| title_auth | Linear Forms in Logarithms and Applications |
| title_exact_search | Linear Forms in Logarithms and Applications |
| title_full | Linear Forms in Logarithms and Applications Yann Bugeaud |
| title_fullStr | Linear Forms in Logarithms and Applications Yann Bugeaud |
| title_full_unstemmed | Linear Forms in Logarithms and Applications Yann Bugeaud |
| title_short | Linear Forms in Logarithms and Applications |
| title_sort | linear forms in logarithms and applications |
| topic | Diophantische Gleichung (DE-588)4012386-8 gnd Zahlentheorie (DE-588)4067277-3 gnd Diophantische Ungleichung (DE-588)4200628-4 gnd |
| topic_facet | Diophantische Gleichung Zahlentheorie Diophantische Ungleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030675654&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV014300756 |
| work_keys_str_mv | AT bugeaudyann linearformsinlogarithmsandapplications AT europeanmathematicalsocietypublishinghouseethzentrumsewa27 linearformsinlogarithmsandapplications |