Number theory: structures, examples, and problems
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston ; Basel ; Berlin
Birkhäuser
[2009]
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | xviii, 384 Seiten Diagramme |
| ISBN: | 9780817632458 |
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MARC
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| 264 | 4 | |c © 2009 | |
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Record in the Search Index
| _version_ | 1805089058532098048 |
|---|---|
| adam_text |
Contents
Preface
xiii
Acknowledgments
xv
Notation
xvii
I Fundamentals
1
1
Divisibility
3
1.1
Divisibility
. 3
1.2
Prime Numbers
. 9
1.3
The Greatest Common Divisor and Least Common Multiple
. 17
1.4
Odd and Even
. 27
1.5
Modular Arithmetic
. 29
1.6
Chinese Remainder Theorem
. 34
1.7
Numerical Systems
. 36
1.7.1
Representation of Integers in an Arbitrary Base
. 36
1.7.2
Divisibility Criteria in the Decimal System
. 38
2
Powers of Integers
47
2.1
Perfect Squares
. 47
2.2
Perfect Cubes
. 56
2.3
Mi Powers of Integers,
к
at least
4. 57
3
Floor Function and Fractional Part
61
3.1
General Problems
. 61
3.2
Floor Function and Integer Points
. 68
3.3
A Useful Result
. 73
viii Contents
4
Digits of Numbers 77
4.1
The Last Digits of a Number
. 77
4.2
The Sum of the Digits of a Number
. 79
4.3
Other Problems Involving Digits
. 85
5
Basic Principles in Number Theory
89
5.1
Two Simple Principles
. 89
5.1.1
Extremal Arguments
. 89
5.1.2
The Pigeonhole Principle
. 91
5.2
Mathematical Induction
. 93
5.3
Infinite Descent
. 98
5.4
Inclusion-Exclusion
. 99
6
Arithmetic Functions
105
6.1
Multiplicative Functions
.105
6.2
Number of Divisors
.112
6.3
Sum of Divisors
.115
6.4
Euler's Totient Function
.118
6.5
Exponent of a Prime and Legendre's Formula
.122
7
More on Divisibility
129
7.1
Congruences Modulo a Prime: Fermat's Little Theorem
.129
7.2
Euler's Theorem
.135
7.3
The Order of an Element
.138
7.4
Wilson's Theorem
.141
8
Diophantine Equations
145
8.1
Linear Diophantine Equations
.145
8.2
Quadratic Diophantine Equations
.148
8.2.1
The Pythagorean Equation
.148
8.2.2
Pell's Equation
.151
8.2.3
Other Quadratic Equations
.157
8.3
Nonstandard
Diophantine Equations
.159
8.3.1
Cubic Equations
.159
8.3.2
High-Order Polynomial Equations
.161
8.3.3
Exponential Diophantine Equations
.163
9
Some Special Problems in Number Theory
167
9.1
Quadratic Residues; the Legendre Symbol
.167
9.2
Special Numbers
.176
9.2.1
Fermat
Numbers
.176
9.2.2
Mersenne Numbers
.178
9.2.3
Perfect Numbers
.179
Contents ix
9.3
Sequences of Integers
. 180
9.3.1
Fibonacci and Lucas Sequences
. 180
9.3.2
Problems Involving Linear Recursive Relations
. 184
9.3.3
Nonstandard
Sequences of Integers
. 191
10
Problems Involving Binomial Coefficients
197
10.1
Binomial Coefficients
.197
10.2
Lucas's and Kummer's Theorems
.203
11
Miscellaneous Problems
207
II Solutions to Additional Problems
213
1
Divisibility
215
1.1
Divisibility
.215
1.2
Prime Numbers
.220
1.3
The Greatest Common Divisor and Least Common Multiple
. . . 227
1.4
Odd and Even
.231
1.5
Modular Arithmetic
.233
1.6
Chinese Remainder Theorem
.236
1.7
Numerical Systems
.238
2
Powers of Integers
245
2.1
Perfect Squares
.245
2.2
Perfect Cubes
.253
2.3
fohPowersof Integers ,k at least
4.256
3
Floor Function and Fractional Part
259
3.1
General Problems
.259
3.2
Floor Function and Integer Points
.263
3.3
A Useful Result
.264
4
Digits of Numbers
267
4.1
The Last Digits of a Number
.267
4.2
The Sum of the Digits of a Number
.268
4.3
Other Problems Involving Digits
.272
5
Basic Principles in Number Theory
275
5.1
Two Simple Principles
.275
5.2
Mathematical Induction
.278
5.3
Infinite Descent
.284
5.4
Inclusion-Exclusion
.284
x
Contents
6
Arithmetic Functions
287
6.1
Multiplicative Functions
.287
6.2
Number of Divisors
.289
6.3
Sum of Divisors
.291
6.4
Euler's Totient Function
.292
6.5
Exponent of a Prime and Legendre's Formula
.294
7
More on Divisibility
299
7.1
Congruences Modulo a Prime: Fermat's Little Theorem
.299
7.2
Euler's Theorem
.305
7.3
The Order of an Element
.306
7.4
Wilson's Theorem
.309
8
Diophantine Equations
311
8.1
Linear Diophantine Equations
.311
8.2
Quadratic Diophantine Equations
.313
8.2.1
Pythagorean Equations
.313
8.2.2
Pell's Equation
.315
8.2.3
Other Quadratic Equations
.318
8.3
Nonstandard
Diophantine Equations
.320
8.3.1
Cubic Equations
.320
8.3.2
High-Order Polynomial Equations
.323
8.3.3
Exponential Diophantine Equations
.325
9
Some Special Problems in Number Theory
329
9.1
Quadratic Residues; the Legendre Symbol
.329
9.2
Special Numbers
.332
9.2.1
Fermat
Numbers
.332
9.2.2
Mersenne Numbers
.333
9.2.3
Perfect Numbers
.334
9.3
Sequences of Integers
.335
9.3.1
Fibonacci and Lucas Sequences
.335
9.3.2
Problems Involving Linear Recursive Relations
.338
9.3.3
Nonstandard
Sequences of Integers
.342
10
Problems Involving Binomial Coefficients
355
10.1
Binomial Coefficients
.355
10.2
Lucas's and Kummer's Theorems
.360
11
Miscellaneous Problems
363
Glossary
369
Contents xi
Bibliography
377
Index
of Authors
381
Subject Index
383 |
| adam_txt |
Contents
Preface
xiii
Acknowledgments
xv
Notation
xvii
I Fundamentals
1
1
Divisibility
3
1.1
Divisibility
. 3
1.2
Prime Numbers
. 9
1.3
The Greatest Common Divisor and Least Common Multiple
. 17
1.4
Odd and Even
. 27
1.5
Modular Arithmetic
. 29
1.6
Chinese Remainder Theorem
. 34
1.7
Numerical Systems
. 36
1.7.1
Representation of Integers in an Arbitrary Base
. 36
1.7.2
Divisibility Criteria in the Decimal System
. 38
2
Powers of Integers
47
2.1
Perfect Squares
. 47
2.2
Perfect Cubes
. 56
2.3
Mi Powers of Integers,
к
at least
4. 57
3
Floor Function and Fractional Part
61
3.1
General Problems
. 61
3.2
Floor Function and Integer Points
. 68
3.3
A Useful Result
. 73
viii Contents
4
Digits of Numbers 77
4.1
The Last Digits of a Number
. 77
4.2
The Sum of the Digits of a Number
. 79
4.3
Other Problems Involving Digits
. 85
5
Basic Principles in Number Theory
89
5.1
Two Simple Principles
. 89
5.1.1
Extremal Arguments
. 89
5.1.2
The Pigeonhole Principle
. 91
5.2
Mathematical Induction
. 93
5.3
Infinite Descent
. 98
5.4
Inclusion-Exclusion
. 99
6
Arithmetic Functions
105
6.1
Multiplicative Functions
.105
6.2
Number of Divisors
.112
6.3
Sum of Divisors
.115
6.4
Euler's Totient Function
.118
6.5
Exponent of a Prime and Legendre's Formula
.122
7
More on Divisibility
129
7.1
Congruences Modulo a Prime: Fermat's Little Theorem
.129
7.2
Euler's Theorem
.135
7.3
The Order of an Element
.138
7.4
Wilson's Theorem
.141
8
Diophantine Equations
145
8.1
Linear Diophantine Equations
.145
8.2
Quadratic Diophantine Equations
.148
8.2.1
The Pythagorean Equation
.148
8.2.2
Pell's Equation
.151
8.2.3
Other Quadratic Equations
.157
8.3
Nonstandard
Diophantine Equations
.159
8.3.1
Cubic Equations
.159
8.3.2
High-Order Polynomial Equations
.161
8.3.3
Exponential Diophantine Equations
.163
9
Some Special Problems in Number Theory
167
9.1
Quadratic Residues; the Legendre Symbol
.167
9.2
Special Numbers
.176
9.2.1
Fermat
Numbers
.176
9.2.2
Mersenne Numbers
.178
9.2.3
Perfect Numbers
.179
Contents ix
9.3
Sequences of Integers
. 180
9.3.1
Fibonacci and Lucas Sequences
. 180
9.3.2
Problems Involving Linear Recursive Relations
. 184
9.3.3
Nonstandard
Sequences of Integers
. 191
10
Problems Involving Binomial Coefficients
197
10.1
Binomial Coefficients
.197
10.2
Lucas's and Kummer's Theorems
.203
11
Miscellaneous Problems
207
II Solutions to Additional Problems
213
1
Divisibility
215
1.1
Divisibility
.215
1.2
Prime Numbers
.220
1.3
The Greatest Common Divisor and Least Common Multiple
. . . 227
1.4
Odd and Even
.231
1.5
Modular Arithmetic
.233
1.6
Chinese Remainder Theorem
.236
1.7
Numerical Systems
.238
2
Powers of Integers
245
2.1
Perfect Squares
.245
2.2
Perfect Cubes
.253
2.3
fohPowersof Integers ,k at least
4.256
3
Floor Function and Fractional Part
259
3.1
General Problems
.259
3.2
Floor Function and Integer Points
.263
3.3
A Useful Result
.264
4
Digits of Numbers
267
4.1
The Last Digits of a Number
.267
4.2
The Sum of the Digits of a Number
.268
4.3
Other Problems Involving Digits
.272
5
Basic Principles in Number Theory
275
5.1
Two Simple Principles
.275
5.2
Mathematical Induction
.278
5.3
Infinite Descent
.284
5.4
Inclusion-Exclusion
.284
x
Contents
6
Arithmetic Functions
287
6.1
Multiplicative Functions
.287
6.2
Number of Divisors
.289
6.3
Sum of Divisors
.291
6.4
Euler's Totient Function
.292
6.5
Exponent of a Prime and Legendre's Formula
.294
7
More on Divisibility
299
7.1
Congruences Modulo a Prime: Fermat's Little Theorem
.299
7.2
Euler's Theorem
.305
7.3
The Order of an Element
.306
7.4
Wilson's Theorem
.309
8
Diophantine Equations
311
8.1
Linear Diophantine Equations
.311
8.2
Quadratic Diophantine Equations
.313
8.2.1
Pythagorean Equations
.313
8.2.2
Pell's Equation
.315
8.2.3
Other Quadratic Equations
.318
8.3
Nonstandard
Diophantine Equations
.320
8.3.1
Cubic Equations
.320
8.3.2
High-Order Polynomial Equations
.323
8.3.3
Exponential Diophantine Equations
.325
9
Some Special Problems in Number Theory
329
9.1
Quadratic Residues; the Legendre Symbol
.329
9.2
Special Numbers
.332
9.2.1
Fermat
Numbers
.332
9.2.2
Mersenne Numbers
.333
9.2.3
Perfect Numbers
.334
9.3
Sequences of Integers
.335
9.3.1
Fibonacci and Lucas Sequences
.335
9.3.2
Problems Involving Linear Recursive Relations
.338
9.3.3
Nonstandard
Sequences of Integers
.342
10
Problems Involving Binomial Coefficients
355
10.1
Binomial Coefficients
.355
10.2
Lucas's and Kummer's Theorems
.360
11
Miscellaneous Problems
363
Glossary
369
Contents xi
Bibliography
377
Index
of Authors
381
Subject Index
383 |
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| id | DE-604.BV022463898 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:41:32Z |
| indexdate | 2024-07-20T09:17:53Z |
| institution | BVB |
| isbn | 9780817632458 |
| language | English |
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| publisher | Birkhäuser |
| record_format | marc |
| spelling | Andreescu, Titu 1956- (DE-588)12217805X aut Number theory structures, examples, and problems Titu Andreescu ; Dorin Andrica Boston ; Basel ; Berlin Birkhäuser [2009] © 2009 xviii, 384 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Zahlentheorie Number theory Number theory Problems, exercises, etc Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Andrica, Dorin 1956- (DE-588)113881789 aut Erscheint auch als Online-Ausgabe 978-0-8176-4645-5 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2839888&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015671515&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Andreescu, Titu 1956- Andrica, Dorin 1956- Number theory structures, examples, and problems Zahlentheorie Number theory Number theory Problems, exercises, etc Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Number theory structures, examples, and problems |
| title_auth | Number theory structures, examples, and problems |
| title_exact_search | Number theory structures, examples, and problems |
| title_exact_search_txtP | Number theory structures, examples, and problems |
| title_full | Number theory structures, examples, and problems Titu Andreescu ; Dorin Andrica |
| title_fullStr | Number theory structures, examples, and problems Titu Andreescu ; Dorin Andrica |
| title_full_unstemmed | Number theory structures, examples, and problems Titu Andreescu ; Dorin Andrica |
| title_short | Number theory |
| title_sort | number theory structures examples and problems |
| title_sub | structures, examples, and problems |
| topic | Zahlentheorie Number theory Number theory Problems, exercises, etc Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie Number theory Number theory Problems, exercises, etc |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2839888&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015671515&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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