Perfect and amicable numbers:
"Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This book gives a complete presentation of the theory of two classes of special numbers (perfect numbers and amicable numbers)...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2023]
|
| Series: | Selected chapters of number theory: special numbers
volume 2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This book gives a complete presentation of the theory of two classes of special numbers (perfect numbers and amicable numbers) and give much of their properties, facts and theorems with full proofs of them. In the book, a complete detailed description of two classes of special numbers, perfect and amicable numbers, as well as their numerous analogue and generalizations, is given. Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This is also an important part of the history of prime numbers, since the main formulas generated perfect and amicable pairs, depends on the good choice of one or several primes of special form. Nowadays, the theory of perfect and amicable numbers contains many interesting mathematical facts and theorems, as well as a lot of important computer algorithms needed for searching for new large elements of these two famous classes of special numbers. The mathematical part of this theory is closely connected with classical Arithmetics and Number Theory. It contains information about divisibility properties of perfect and amicable numbers, structure and properties of their generalizations and analogue (sociable numbers, multiperfect numbers, quasiperfect and quasiamicable numbers, etc.), their connections with other classes of special numbers, etc. Moreover, perfect and amicable numbers are involved in the search for new large primes, and have numerous connections with contemporary Cryptography. For these applications, one should study well-known deterministic and probabilistic primality tests, standard algorithms of integer factorization, the questions and open problems of Computational Complexity Theory"-- |
| Item Description: | Includes bibliographical references |
| Physical Description: | xxiv, 437 Seiten |
| ISBN: | 9789811259623 |
Staff View
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| 520 | 3 | |a "Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This book gives a complete presentation of the theory of two classes of special numbers (perfect numbers and amicable numbers) and give much of their properties, facts and theorems with full proofs of them. In the book, a complete detailed description of two classes of special numbers, perfect and amicable numbers, as well as their numerous analogue and generalizations, is given. Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This is also an important part of the history of prime numbers, since the main formulas generated perfect and amicable pairs, depends on the good choice of one or several primes of special form. Nowadays, the theory of perfect and amicable numbers contains many interesting mathematical facts and theorems, as well as a lot of important computer algorithms needed for searching for new large elements of these two famous classes of special numbers. The mathematical part of this theory is closely connected with classical Arithmetics and Number Theory. It contains information about divisibility properties of perfect and amicable numbers, structure and properties of their generalizations and analogue (sociable numbers, multiperfect numbers, quasiperfect and quasiamicable numbers, etc.), their connections with other classes of special numbers, etc. Moreover, perfect and amicable numbers are involved in the search for new large primes, and have numerous connections with contemporary Cryptography. For these applications, one should study well-known deterministic and probabilistic primality tests, standard algorithms of integer factorization, the questions and open problems of Computational Complexity Theory"-- | |
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Record in the Search Index
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| adam_text |
Contents Preface vii About the Author xv Notations xvii 1. Preliminaries 1 Divisibility of Integers. 1 Modular Arithmetic . 8 Solution to Congruences. 13 Quadratic Residues, Legendre Symbol, and Jacobi Symbol. 19 1.5 Multiplicative Orders, Primitive Roots, and Indexes. 22 1.6 Continued Fractions and Their Applications. 25 1.1 1.2 1.3 1.4 2. Arithmetic Functions 2.1 2.2 2.3 2.4 2.5 2.6 35 Additive and Multiplicative Functions. 35 Floor Function and Its Relatives . 42 Mobius Function. 46 Euler’s Totient Function. 50 Prime Counting Functions. 59 Divisor Functions. 86 xxiii
Perfect and Amicable Numbers xxiv 3. Perfect Numbers 107 History of the Question . 107 Divisor Function and PerfectNumbers. 119 Properties of Perfect Numbers. 132 Search for Perfect Numbers. 155 Perfect Numbers in the Family of Special Numbers. 176 3.6 Open Problems. 217 3.1 3.2 3.3 3.4 3.5 4. Amicable Numbers 4.1 4.2 4.3 4.4 4.5 4.6 5. Generalizations and Analogue 5.1 5.2 5.3 5.4 5.5 5.6 229 History of the Question . 229 Divisor Function and AmicableNumbers. 238 Properties of Amicable Numbers. 258 Search for Amicable Numbers. 268 Amicable Numbers in the Family of Special Numbers. 280 Open Problems. 289 299 History of the Question . 299 Relatives of Perfect Numbers. 305 Relatives of Amicable Numbers. 333 Sociable Numbers. 345 Search for Numbers Under Consideration .354 Open
Problems. 365 6. Zoo of Numbers 371 7. Mini Dictionary 383 8. Exercises 393 Bibliography 419 Index 433 |
| adam_txt |
Contents Preface vii About the Author xv Notations xvii 1. Preliminaries 1 Divisibility of Integers. 1 Modular Arithmetic . 8 Solution to Congruences. 13 Quadratic Residues, Legendre Symbol, and Jacobi Symbol. 19 1.5 Multiplicative Orders, Primitive Roots, and Indexes. 22 1.6 Continued Fractions and Their Applications. 25 1.1 1.2 1.3 1.4 2. Arithmetic Functions 2.1 2.2 2.3 2.4 2.5 2.6 35 Additive and Multiplicative Functions. 35 Floor Function and Its Relatives . 42 Mobius Function. 46 Euler’s Totient Function. 50 Prime Counting Functions. 59 Divisor Functions. 86 xxiii
Perfect and Amicable Numbers xxiv 3. Perfect Numbers 107 History of the Question . 107 Divisor Function and PerfectNumbers. 119 Properties of Perfect Numbers. 132 Search for Perfect Numbers. 155 Perfect Numbers in the Family of Special Numbers. 176 3.6 Open Problems. 217 3.1 3.2 3.3 3.4 3.5 4. Amicable Numbers 4.1 4.2 4.3 4.4 4.5 4.6 5. Generalizations and Analogue 5.1 5.2 5.3 5.4 5.5 5.6 229 History of the Question . 229 Divisor Function and AmicableNumbers. 238 Properties of Amicable Numbers. 258 Search for Amicable Numbers. 268 Amicable Numbers in the Family of Special Numbers. 280 Open Problems. 289 299 History of the Question . 299 Relatives of Perfect Numbers. 305 Relatives of Amicable Numbers. 333 Sociable Numbers. 345 Search for Numbers Under Consideration .354 Open
Problems. 365 6. Zoo of Numbers 371 7. Mini Dictionary 383 8. Exercises 393 Bibliography 419 Index 433 |
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| author | Deza, Elena 1961- |
| author_GND | (DE-588)1027553648 |
| author_facet | Deza, Elena 1961- |
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| author_sort | Deza, Elena 1961- |
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| building | Verbundindex |
| bvnumber | BV048875002 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)1366290377 (DE-599)KXP181542785X |
| dewey-full | 512.7/4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7/4 |
| dewey-search | 512.7/4 |
| dewey-sort | 3512.7 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV048875002 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T21:44:47Z |
| indexdate | 2024-07-20T06:23:53Z |
| institution | BVB |
| isbn | 9789811259623 |
| language | English |
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| physical | xxiv, 437 Seiten |
| publishDate | 2023 |
| publishDateSearch | 2023 |
| publishDateSort | 2023 |
| publisher | World Scientific |
| record_format | marc |
| series | Selected chapters of number theory: special numbers |
| series2 | Selected chapters of number theory: special numbers |
| spelling | Deza, Elena 1961- Verfasser (DE-588)1027553648 aut Perfect and amicable numbers Elena Deza, Moscow State Pedagogical University, Russia New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2023] xxiv, 437 Seiten txt rdacontent n rdamedia nc rdacarrier Selected chapters of number theory: special numbers volume 2 Includes bibliographical references "Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This book gives a complete presentation of the theory of two classes of special numbers (perfect numbers and amicable numbers) and give much of their properties, facts and theorems with full proofs of them. In the book, a complete detailed description of two classes of special numbers, perfect and amicable numbers, as well as their numerous analogue and generalizations, is given. Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This is also an important part of the history of prime numbers, since the main formulas generated perfect and amicable pairs, depends on the good choice of one or several primes of special form. Nowadays, the theory of perfect and amicable numbers contains many interesting mathematical facts and theorems, as well as a lot of important computer algorithms needed for searching for new large elements of these two famous classes of special numbers. The mathematical part of this theory is closely connected with classical Arithmetics and Number Theory. It contains information about divisibility properties of perfect and amicable numbers, structure and properties of their generalizations and analogue (sociable numbers, multiperfect numbers, quasiperfect and quasiamicable numbers, etc.), their connections with other classes of special numbers, etc. Moreover, perfect and amicable numbers are involved in the search for new large primes, and have numerous connections with contemporary Cryptography. For these applications, one should study well-known deterministic and probabilistic primality tests, standard algorithms of integer factorization, the questions and open problems of Computational Complexity Theory"-- Vollkommene Zahl (DE-588)7683309-4 gnd rswk-swf Befreundete Zahl (DE-588)7684537-0 gnd rswk-swf Perfect numbers Amicable numbers Vollkommene Zahl (DE-588)7683309-4 s Befreundete Zahl (DE-588)7684537-0 s DE-604 Erscheint auch als Online-Ausgabe 978-981-125-963-0 Erscheint auch als Online-Ausgabe 978-981-125-964-7 Selected chapters of number theory: special numbers volume 2 (DE-604)BV047513069 2 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034139842&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Deza, Elena 1961- Perfect and amicable numbers Selected chapters of number theory: special numbers Vollkommene Zahl (DE-588)7683309-4 gnd Befreundete Zahl (DE-588)7684537-0 gnd |
| subject_GND | (DE-588)7683309-4 (DE-588)7684537-0 |
| title | Perfect and amicable numbers |
| title_auth | Perfect and amicable numbers |
| title_exact_search | Perfect and amicable numbers |
| title_exact_search_txtP | Perfect and amicable numbers |
| title_full | Perfect and amicable numbers Elena Deza, Moscow State Pedagogical University, Russia |
| title_fullStr | Perfect and amicable numbers Elena Deza, Moscow State Pedagogical University, Russia |
| title_full_unstemmed | Perfect and amicable numbers Elena Deza, Moscow State Pedagogical University, Russia |
| title_short | Perfect and amicable numbers |
| title_sort | perfect and amicable numbers |
| topic | Vollkommene Zahl (DE-588)7683309-4 gnd Befreundete Zahl (DE-588)7684537-0 gnd |
| topic_facet | Vollkommene Zahl Befreundete Zahl |
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