Solving the Pell equation:
This work discusses Pell's equation. It presents the historical development of the equation and features the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's equation.
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
2009
|
| Ausgabe: | 1. Ed. |
| Schriftenreihe: | CMS Books in Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | This work discusses Pell's equation. It presents the historical development of the equation and features the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's equation. |
| Beschreibung: | XXII, 497 S. 235 mm x 155 mm |
| ISBN: | 9780387849225 |
Internformat
MARC
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| 100 | 1 | |a Jacobson, Michael J. |d 1971- |e Verfasser |0 (DE-588)121455610 |4 aut | |
| 245 | 1 | 0 | |a Solving the Pell equation |c Michael Jacobson ; Hugh Williams |
| 250 | |a 1. Ed. | ||
| 264 | 1 | |a New York, NY |b Springer New York |c 2009 | |
| 300 | |a XXII, 497 S. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a CMS Books in Mathematics | |
| 520 | 3 | |a This work discusses Pell's equation. It presents the historical development of the equation and features the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's equation. | |
| 650 | 4 | |a Pell's equation | |
| 650 | 0 | 7 | |a Pell-Gleichung |0 (DE-588)4720859-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Pell-Gleichung |0 (DE-588)4720859-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Williams, Hugh C. |d 1943- |e Verfasser |0 (DE-588)124230415 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017168133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017168133 | ||
Datensatz im Suchindex
| _version_ | 1804138689605926912 |
|---|---|
| adam_text | Contents
Preface
Vil
1
Introduction
............................................... 1
1.1
Diophantine Equations
................................... 1
1.2
The
Peli
Equation
....................................... 3
1.3
Representation of
AU Solutions
............................ 8
1.4
The Lucas
Functions
..................................... 13
2
Early History of the
Pełł
Equation
......................... 19
2.1
The Cattle Problem of Archimedes
........................ 19
2.2
Further Contributions of the Greeks
....................... 24
2.3
The Indian Mathematicians
............................... 31
2.4
Fermat
and His Successors
................................ 36
3
Continued Fractions
....................................... 43
3.1
General Continued Fractions
.............................. 43
3.2
Simple Continued Fractions
............................... 47
3.3
Simple Continued Fractions of Quadratic Irrationals
......... 53
3.4
Some Special Results
.................................... 63
4
Quadratic Number Fields
.................................. 75
4.1
Algebraic Numbers
...................................... 75
4.2
Modules and Orders of
К
................................. 78
4.3
The Units of
С
.......................................... 81
4.4
The Ideals of
О
......................................... 83
4.5
Equivalence and Norms
.................................. 88
4.6
Divisibility
aud
Prime Ideals
.............................. 93
5
Ideals and Continued Fractions
............................ 97
5.1
Reduced Ideals of
С
..................................... 97
5.2
Reduction Algorithms
.................................... 104
xiv Contents
5.3
Reduced
Ideals
When
Δ
> 0..............................109
5.4
Ideal Products and NUCOMP
............................
П6
6
Some Special Pell Equations
...............................125
6.1
Introduction
............................................125
6.2
Continued Fractions
.....................................128
6.3
Schinzeľs
Families
.......................................134
6.4
Creepers and Kreepers
...................................140
6.5
Yamamoto s Results
.....................................145
7
The Ideal Class Group
.....................................153
7.1
Introduction
............................................153
7.2
The Cohen-Lenstra Heuristics
.............................157
7.2.1
Imaginary Quadratic Fields
.........................157
7.2.2
Real Quadratic Fields
..............................164
7.3
The 2-Sylow Subgroup
...................................169
7.4
Infrastructure
...........................................172
8
The Analytic Class Number Formula
......................185
8.1
Dirichlet Characters
.....................................185
8.2
Primitive Characters
.....................................191
8.3
The L-Function
.........................................194
8.4
Ideal Density
...........................................197
8.5
The Class Number Formula
...............................202
9
Some Additional Analytic Results
.........................209
9.1
More on Gauss Sums
....................................209
9.2
A Closed Formula for hK
.................................212
9.3
The Riemann Zeta-Function
..............................217
9.4
The
Euler
Product for L(l,
χ)
.............................222
9.5
Bounds on
1(1,
χ)
.......................................226
10
Some Computational Techniques
..........................237
10.1
Introduction
............................................237
10.2
Computing the Regulator
.................................238
10.3
Computing the Class Number
.............................245
10.4
Computing the Class Group
..............................253
10.5
Numerical Results
.......................................256
10.5.1
Imaginary Quadratic Fields
.........................257
10.5.2
Real Quadratic Fields
..............................260
11 (ƒ,
p) Representations of O-ideals
..........................265
11.1
Basic Concepts and Definitions
............................265
11.2
¡е
-Near Representations
..................................270
11.3
Exponentiation of Ideals and Computation of a[x]
...........275
Contents xv
12
Compact
Representations
..................................285
12.1
Compact Representation of 9j
.............................285
12.2
Compact Representation of Quadratic Integers
..............290
12.3
The Arithmetic of Compact Representations
................297
13
The Subexponential Method
...............................307
13.1
Introduction
............................................307
13.2
Solving the Discrete Logarithm Problem in
Cl¿
.............308
13.3
Computing the Class Number and Class Group
.............316
13.4
Computing the Regulator
.................................322
13.5
Principality Testing
......................................331
13.6
Complexity
.............................................333
13.7
Practical Improvements
..................................337
13.7.1
Improvements to the Random Exponents Method
.....337
13.7.2
The Large Prime Variation
.........................338
13.7.3
Parallelism
.......................................340
13.7.4
Computing Relations Using Sieving
..................340
13.7.5
Self-initialization
..................................342
13.8
Computational Results
...................................345
13.9
Open Problems and Further Improvements
.................348
14
Applications to Cryptography
.............................353
14.1
Introduction
............................................353
14.2
The Pell Equation in a Public-Key
CryptoSystem
............355
14.3
Cryptography in Imaginary Quadratic Fields
................360
14.3.1
Cryptographic Protocols
...........................363
14.3.2
Efficiency
........................................363
14.4
Cryptography in Real Quadratic Fields
.....................364
14.4.1
Security
..........................................369
14.4.2
Efficiency
........................................373
14.4.3
Other Cryptosystems
..............................374
14.5
Cryptosystems in Non-Maximal Quadratic Orders
...........374
14.5.1
NICE
............................................376
14.5.2
REAL-NICE
......................................378
14.5.3
Trapdoor Discrete Logarithm Computation
...........380
15
Unconditional Verification of the Regulator and the Class
Number
...................................................387
15.1
Introduction
............................................387
15.2
Some Preliminary Results
................................388
15.3
The Algorithm and Some Implementation Issues
.............393
15.4
The Class Number
.......................................399
xii Contents
16 Principal Ideal
Testing
in
О
...............................405
16.1
Introduction............................................
405
16.2
Another Approach to Problem
Ρ
..........................410
16.3
The Equation X2
-
DY2 = N
............................415
17
Conclusion
................................................423
17.1
A More General Equation
................................423
17.2
Other Generalizations of the Pell Equation
.................426
17.3
Some Questions
.........................................432
Appendix
......................................................439
A.I NUCOMP
..............................................439
A.2 NUMULT
..............................................446
A.3 Theoretical Background for WNEAR
......................449
A.4 WNEAR
...............................................454
References
.....................................................461
Index
..........................................................489
|
| any_adam_object | 1 |
| author | Jacobson, Michael J. 1971- Williams, Hugh C. 1943- |
| author_GND | (DE-588)121455610 (DE-588)124230415 |
| author_facet | Jacobson, Michael J. 1971- Williams, Hugh C. 1943- |
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| ctrlnum | (OCoLC)245561348 (DE-599)DNB989571203 |
| dewey-full | 513.72 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 513 - Arithmetic |
| dewey-raw | 513.72 |
| dewey-search | 513.72 |
| dewey-sort | 3513.72 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. Ed. |
| format | Book |
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| id | DE-604.BV035364139 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T21:32:11Z |
| institution | BVB |
| isbn | 9780387849225 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017168133 |
| oclc_num | 245561348 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-20 DE-19 DE-BY-UBM |
| physical | XXII, 497 S. 235 mm x 155 mm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer New York |
| record_format | marc |
| series2 | CMS Books in Mathematics |
| spelling | Jacobson, Michael J. 1971- Verfasser (DE-588)121455610 aut Solving the Pell equation Michael Jacobson ; Hugh Williams 1. Ed. New York, NY Springer New York 2009 XXII, 497 S. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier CMS Books in Mathematics This work discusses Pell's equation. It presents the historical development of the equation and features the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's equation. Pell's equation Pell-Gleichung (DE-588)4720859-4 gnd rswk-swf Pell-Gleichung (DE-588)4720859-4 s DE-604 Williams, Hugh C. 1943- Verfasser (DE-588)124230415 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017168133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jacobson, Michael J. 1971- Williams, Hugh C. 1943- Solving the Pell equation Pell's equation Pell-Gleichung (DE-588)4720859-4 gnd |
| subject_GND | (DE-588)4720859-4 |
| title | Solving the Pell equation |
| title_auth | Solving the Pell equation |
| title_exact_search | Solving the Pell equation |
| title_full | Solving the Pell equation Michael Jacobson ; Hugh Williams |
| title_fullStr | Solving the Pell equation Michael Jacobson ; Hugh Williams |
| title_full_unstemmed | Solving the Pell equation Michael Jacobson ; Hugh Williams |
| title_short | Solving the Pell equation |
| title_sort | solving the pell equation |
| topic | Pell's equation Pell-Gleichung (DE-588)4720859-4 gnd |
| topic_facet | Pell's equation Pell-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017168133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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