Perfect, amicable and sociable numbers: a computational approach
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
1996
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 338 S. |
| ISBN: | 9810228473 |
Staff View
MARC
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| 245 | 1 | 0 | |a Perfect, amicable and sociable numbers |b a computational approach |c Song Y. Yan |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 1996 | |
| 300 | |a XX, 338 S. | ||
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| 650 | 7 | |a Analyse diophantienne |2 ram | |
| 650 | 7 | |a Arithmétique - Histoire |2 ram | |
| 650 | 4 | |a Complexité algorithme | |
| 650 | 7 | |a Maple (logiciel) |2 ram | |
| 650 | 4 | |a Nombre amiable | |
| 650 | 4 | |a Nombre parfait | |
| 650 | 4 | |a Amicable numbers | |
| 650 | 4 | |a Perfect numbers | |
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Record in the Search Index
| _version_ | 1804125770578132992 |
|---|---|
| adam_text | Contents
1 Concepts and History 1
1.1 Basic Concepts 1
1.1.1 Prime Numbers 2
1.1.2 Amicable and Perfect Numbers 10
1.1.3 Amicable k Tuplets 19
1.1.4 Aliquot k Cycles 21
1.2 The History of Amicable Numbers 24
1.2.1 Ancient Times 25
1.2.2 Around Euler s Time 28
1.2.3 The Modern Computer Era 31
1.3 Brief History of Perfect and Sociable Numbers 34
2 Mathematical Tools 39
2.1 Computational Complexity Theory 39
2.2 Primality Testing 46
2.2.1 Basic Idea 46
2.2.2 Probable Primes and Strong Pseudoprimality Tests ... 48
2.2.3 Lucas Sequences and Lucas Tests 54
2.2.4 Primality Testing in Maple 63
2.2.5 Elliptic Curve Test 67
2.2.6 Computational Complexity of Primality Testing 71
2.3 Integer Factorization 74
2.3.1 Computational Complexity of Integer Factorization ... 74
2.3.2 Factoring by Trial Divisions 77
2.3.3 Continued FRACtion Method (CFRAC) 80
2.3.4 Pollard s rho and p 1 Algorithms 84
2.3.5 Lenstra s Elliptic Curve Algorithm 87
2.3.6 Progress on Factoring Fermat Numbers 89
xiii
xiv CONTENTS
2.3.7 Integer Factorization in Maple 92
2.4 Diophantine Equations 97
2.4.1 Continued Fraction Approach 98
2.4.2 Combinatorial Approach 100
3 Exhaustive Numerical Methods 103
3.1 Numerical Methods for Perfect Numbers 104
3.2 Computing Different Types of Perfect Numbers 107
3.3 Computing Odd Perfect Numbers Ill
3.4 Numerical Methods for Amicable Numbers 113
3.5 te Riele s Seminumerical Method for Amicable Pairs 118
3.6 Computing Reduced Amicable Pairs 121
3.7 Numerical Methods for Aliquot k Cycles 128
4 Algebraic Assumption Methods 137
4.1 Thabit s Algebraic Assumption Method 137
4.2 Euler s Version of Thabit s Method 140
4.3 Analogue of Thabit s Method 146
4.4 Some Conjectures on Amicable Pairs 151
4.5 Algebraic Rules for Aliquot k Cycles 153
4.5.1 Algebraic Rule for Aliquot 3 Cycles 154
4.5.2 Algebraic Rules for Aliquot 4 Cycles 163
5 Algebraic Constructive Methods 169
5.1 A Special Algebraic Constructive Method 169
5.2 A General Algebraic Constructive Method 180
5.3 Two Other More General Constructive Methods 189
5.3.1 Forward Methods 190
5.3.2 Backward Methods 199
5.4 Algebraic Methods for Amicable k Tuplets 201
5.4.1 Methods for Amicable Triplets 202
5.4.2 Methods for Amicable k Tuplets 203
5.5 A Practical Application in Cryptography 207
6 Conclusions and Open Problems 215
6.1 Summary 215
6.1.1 The History of Amicable Numbers 215
6.1.2 Mathematical Tools for Amicable Numbers 217
6.1.3 Exhaustive Numerical and Semi numerical Methods . . . 219
CONTENTS xv
6.1.4 Algebraic Assumption Methods 220
6.1.5 Algebraic Constructive Methods 221
6.1.6 Computer Aided Proof 221
6.2 Some Open Problems 222
6.2.1 Problems Related to Amicable Numbers 222
6.2.2 Problems in General Computational Number Theory . . 225
Bibliography 233
A Basic Number Theory 245
A.I What is Number Theory 245
A.2 Divisibility 248
A.3 Euclid s Algorithm 252
A.4 Arithmetic Functions 256
A.5 Congruences 259
B Introduction to the Maple numtheory/combinat Packages 267
B.I Basic Concepts 267
B.2 Number Theoretic Functions 268
B.2.1 Basic Arithmetic Functions 270
B.2.2 Modular Arithmetic Functions 272
B.2.3 Primality Testing Related Functions 273
B.2.4 Integer Factorization Related Functions 278
B.2.5 Continued Fractions 281
B.2.6 Legendre/Jacobi Symbols 287
B.3 Combinatorial Functions 289
B.4 Programming in Maple 293
C Selected Maple Programs 299
C.I Programs for Numerical Methods 299
C.2 Programs for Algebraic Assumption Methods 307
C.3 Programs for Algebraic Constructive Methods 316
Index 335
|
| any_adam_object | 1 |
| author | Yan, Song Y. 1954- |
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| dewey-tens | 510 - Mathematics |
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| id | DE-604.BV011267316 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:06:50Z |
| institution | BVB |
| isbn | 9810228473 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007565334 |
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| physical | XX, 338 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Yan, Song Y. 1954- Verfasser (DE-588)121556034 aut Perfect, amicable and sociable numbers a computational approach Song Y. Yan Singapore [u.a.] World Scientific 1996 XX, 338 S. txt rdacontent n rdamedia nc rdacarrier Analyse diophantienne ram Arithmétique - Histoire ram Complexité algorithme Maple (logiciel) ram Nombre amiable Nombre parfait Amicable numbers Perfect numbers HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007565334&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yan, Song Y. 1954- Perfect, amicable and sociable numbers a computational approach Analyse diophantienne ram Arithmétique - Histoire ram Complexité algorithme Maple (logiciel) ram Nombre amiable Nombre parfait Amicable numbers Perfect numbers |
| title | Perfect, amicable and sociable numbers a computational approach |
| title_auth | Perfect, amicable and sociable numbers a computational approach |
| title_exact_search | Perfect, amicable and sociable numbers a computational approach |
| title_full | Perfect, amicable and sociable numbers a computational approach Song Y. Yan |
| title_fullStr | Perfect, amicable and sociable numbers a computational approach Song Y. Yan |
| title_full_unstemmed | Perfect, amicable and sociable numbers a computational approach Song Y. Yan |
| title_short | Perfect, amicable and sociable numbers |
| title_sort | perfect amicable and sociable numbers a computational approach |
| title_sub | a computational approach |
| topic | Analyse diophantienne ram Arithmétique - Histoire ram Complexité algorithme Maple (logiciel) ram Nombre amiable Nombre parfait Amicable numbers Perfect numbers |
| topic_facet | Analyse diophantienne Arithmétique - Histoire Complexité algorithme Maple (logiciel) Nombre amiable Nombre parfait Amicable numbers Perfect numbers |
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