Number theory: an introduction via the distribution of primes
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2007
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 342 S. Ill., graph. Darst. |
| ISBN: | 0817644725 9780817644727 |
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| 245 | 1 | 0 | |a Number theory |b an introduction via the distribution of primes |c Benjamin Fine ; Gerhard Rosenberger |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XIII, 342 S. |b Ill., graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804135356124102656 |
|---|---|
| adam_text | Contents
Preface
......................................................... xi
1
Introduction
and Historical Remarks
............................
I
2
Basic Number Theory
......................................... 7
2.1
The Ring of Integers
........................................ 7
2.2
Divisibility, Primes, and Composites
.......................... 11
2.3
The Fundamental Theorem of Arithmetic
....................... 16
2.4
Congruences and Modular Arithmetic
......................... 21
2.4.1
Basic Theory of Congruences
.......................... 22
2.4.2
The Ring of Integers Modulo«
......................... 23
2.4.3
Units and the
Euler
Phi Function
....................... 26
2.4.4
Fermat s Little Theorem and the Order of an Element
...... 31
2.4.5
On Cyclic Groups
.................................... 34
2.5
The Solution of Polynomial Congruences Modulo
m
............. 37
2.5.1
Linear Congruences and the Chinese Remainder Theorem
.. 37
2.5.2
Higher-Degree Congruences
........................... 42
2.6
Quadratic Reciprocity
....................................... 45
3
The Infinitude of Primes
...................................... 55
3.1
The Infinitude of Primes
..................................... 55
3.1.1
Some Direct Proofs and Variations
...................... 55
3.1.2
Some Analytic Proofs and Variations
.................... 58
3.1.3
The
Fermat
and Mersenne Numbers
.................... 61
3.1.4
The Fibonacci Numbers and the Golden Section
.......... 65
3.1.5
Some Simple Cases of Dirichlet s Theorem
.............. 78
3.1.6
A Topological Proof and a Proof Using Codes
............ 83
3.2
Sums of Squares
........................................... 86
3.2.1
Pythagorean Triples
.................................. 87
3.2.2
Fermat s Two-Square Theorem
......................... 89
3.2.3
The Modular Group
.................................. 94
Contents
100
3 24
Lagrange s Four-Square Theorem
......................
3.2.5
The Infinitude of Primes Through Continued Fractions
.....
HU
3.3
Dirichlcťs
Theorem
........................................
3.4
Twin Prime Conjecture and Related Ideas
......................
3.5
Primes Between
χ
and 2x
....................................
3.6
Arithmetic Functions and the
Möbius
Inversion Formula
.......... > **
The Density of Primes
........................................
4.1
The Prime Number Theorem: Estimates and History
............. ■>*
4.2
Chebychcv s Estimate and Some Consequences
................. 36
4.3
Equivalent Formulations of the Prime Number Theorem
.......... 149
4.4
The Riemann
Zeta
Function and the Riemann Hypothesis
......... 157
4.4.1
The Real
Zeta
Function of
Euler
........................ 158
4.4.2
Analytic Functions and Analytic Continuation
............ 163
4.4.3
The Riemann
Zeta
Function
........................... 166
4.5
The Prime Number Theorem
................................. 173
4.6
The Elementary Proof
....................................... 80
4.7
Some Extensions and Comments
..............................
1
85
Primality Testing: An Overview
................................ 197
5.1
Primality Testing and Factorization
............................ 197
5.2
Sieving Methods
........................................... 198
5.2.1
Brun s Sieve and Brun s Theorem
...................... 204
5.3
Primality Testing and Prime Records
.......................... 212
5.3.1
Pseudoprimes
and Probabilistic Testing
.................. 218
5.3.2
The Lucas-Lehmer Test and Prime Records
.............. 225
5.3.3
Some Additional Primality Tests
........................ 231
5.4
Cryptography and Primes
.................................... 234
5.4.1
Some Number-Theoretic Cryptosystems
................. 237
5.4.2
Public Key Cryptography and the RSA Algorithm
......... 240
5.5
The
AKS
Algorithm
........................................ 243
ι
Primes and Algebraic Number Theory
...........................253
6.1
Algebraic Number Theory
................................... 253
6.2
Unique Factorization Domains
............................... 255
6.2.1
Euclidean Domains and the Gaussian Integers
............ 261
6.2.2
Principal Ideal Domains
.............................. 268
6.2.3
Prime and Maximal Ideals
............................. 272
6.3
Algebraic Number Fields
................ 275
6.3.1
Algebraic Extensions of
Q
.......................282
63.2
Algebraic and Transcendental Numbers
.................. 284
6.3..3 Symmetric Polynomials
.................... 287
6.3.4
Discriminant and Norm
............. 290
6.4
Algebraic Integers
...................... .................. 294
6.4.1
The Ring of Algebraic Integers
................,. ..... 296
Contents
¡χ
6.4.2
Integrai Bases
....................................... 297
6.4.3
Quadratic Fields and Quadratic Integers
................. 300
6.4.4
The Transcendence of
e
and
π
......................... 303
6.4.5
The Geometry of Numbers: Minkowski Theory
........... 306
6.4.6
Dirichlet s Unit Theorem
.............................. 308
6.5
The Theory of Ideals
........................................ 311
6.5.1
Unique Factorization of Ideals
......................... 313
6.5.2
An Application of Unique Factorization
................. 319
6.5.3
The Ideal Class Group
................................ 321
6.5.4
Norms of Ideals
..................................... 323
6.5.5
Class Number
....................................... 326
Bibliography and Cited References
................................. 333
Index
........................................................... 337
|
| adam_txt |
Contents
Preface
. xi
1
Introduction
and Historical Remarks
.
I
2
Basic Number Theory
. 7
2.1
The Ring of Integers
. 7
2.2
Divisibility, Primes, and Composites
. 11
2.3
The Fundamental Theorem of Arithmetic
. 16
2.4
Congruences and Modular Arithmetic
. 21
2.4.1
Basic Theory of Congruences
. 22
2.4.2
The Ring of Integers Modulo«
. 23
2.4.3
Units and the
Euler
Phi Function
. 26
2.4.4
Fermat's Little Theorem and the Order of an Element
. 31
2.4.5
On Cyclic Groups
. 34
2.5
The Solution of Polynomial Congruences Modulo
m
. 37
2.5.1
Linear Congruences and the Chinese Remainder Theorem
. 37
2.5.2
Higher-Degree Congruences
. 42
2.6
Quadratic Reciprocity
. 45
3
The Infinitude of Primes
. 55
3.1
The Infinitude of Primes
. 55
3.1.1
Some Direct Proofs and Variations
. 55
3.1.2
Some Analytic Proofs and Variations
. 58
3.1.3
The
Fermat
and Mersenne Numbers
. 61
3.1.4
The Fibonacci Numbers and the Golden Section
. 65
3.1.5
Some Simple Cases of Dirichlet's Theorem
. 78
3.1.6
A Topological Proof and a Proof Using Codes
. 83
3.2
Sums of Squares
. 86
3.2.1
Pythagorean Triples
. 87
3.2.2
Fermat's Two-Square Theorem
. 89
3.2.3
The Modular Group
. 94
Contents
100
3 24
Lagrange's Four-Square Theorem
.
3.2.5
The Infinitude of Primes Through Continued Fractions
.
HU
3.3
Dirichlcťs
Theorem
.
3.4
Twin Prime Conjecture and Related Ideas
.
3.5
Primes Between
χ
and 2x
.
3.6
Arithmetic Functions and the
Möbius
Inversion Formula
. > **
The Density of Primes
.
4.1
The Prime Number Theorem: Estimates and History
. ' ■>*
4.2
Chebychcv's Estimate and Some Consequences
. '36
4.3
Equivalent Formulations of the Prime Number Theorem
. 149
4.4
The Riemann
Zeta
Function and the Riemann Hypothesis
. 157
4.4.1
The Real
Zeta
Function of
Euler
. 158
4.4.2
Analytic Functions and Analytic Continuation
. 163
4.4.3
The Riemann
Zeta
Function
. 166
4.5
The Prime Number Theorem
. 173
4.6
The Elementary Proof
. '80
4.7
Some Extensions and Comments
.
1
85
Primality Testing: An Overview
. 197
5.1
Primality Testing and Factorization
. 197
5.2
Sieving Methods
. 198
5.2.1
Brun's Sieve and Brun's Theorem
. 204
5.3
Primality Testing and Prime Records
. 212
5.3.1
Pseudoprimes
and Probabilistic Testing
. 218
5.3.2
The Lucas-Lehmer Test and Prime Records
. 225
5.3.3
Some Additional Primality Tests
. 231
5.4
Cryptography and Primes
. 234
5.4.1
Some Number-Theoretic Cryptosystems
. 237
5.4.2
Public Key Cryptography and the RSA Algorithm
. 240
5.5
The
AKS
Algorithm
. 243
ι
Primes and Algebraic Number Theory
.253
6.1
Algebraic Number Theory
. 253
6.2
Unique Factorization Domains
. 255
6.2.1
Euclidean Domains and the Gaussian Integers
. 261
6.2.2
Principal Ideal Domains
. 268
6.2.3
Prime and Maximal Ideals
. 272
6.3
Algebraic Number Fields
. 275
6.3.1
Algebraic Extensions of
Q
.282
63.2
Algebraic and Transcendental Numbers
. 284
6.3.3 Symmetric Polynomials
. 287
6.3.4
Discriminant and Norm
. 290
6.4
Algebraic Integers
. . 294
6.4.1
The Ring of Algebraic Integers
.,. ' '. 296
Contents
¡χ
6.4.2
Integrai Bases
. 297
6.4.3
Quadratic Fields and Quadratic Integers
. 300
6.4.4
The Transcendence of
e
and
π
. 303
6.4.5
The Geometry of Numbers: Minkowski Theory
. 306
6.4.6
Dirichlet's Unit Theorem
. 308
6.5
The Theory of Ideals
. 311
6.5.1
Unique Factorization of Ideals
. 313
6.5.2
An Application of Unique Factorization
. 319
6.5.3
The Ideal Class Group
. 321
6.5.4
Norms of Ideals
. 323
6.5.5
Class Number
. 326
Bibliography and Cited References
. 333
Index
. 337 |
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| id | DE-604.BV021583101 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:42:09Z |
| indexdate | 2024-07-09T20:39:12Z |
| institution | BVB |
| isbn | 0817644725 9780817644727 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014798718 |
| oclc_num | 68804135 |
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| physical | XIII, 342 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Fine, Benjamin 1948- Verfasser (DE-588)132345048 aut Number theory an introduction via the distribution of primes Benjamin Fine ; Gerhard Rosenberger Boston [u.a.] Birkhäuser 2007 XIII, 342 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Teoria dos números larpcal Number theory Numbers, Prime Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Primzahlverteilung (DE-588)4175716-6 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Primzahlverteilung (DE-588)4175716-6 s 1\p DE-604 Rosenberger, Gerhard 1944- Verfasser (DE-588)13155221X aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014798718&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Fine, Benjamin 1948- Rosenberger, Gerhard 1944- Number theory an introduction via the distribution of primes Teoria dos números larpcal Number theory Numbers, Prime Zahlentheorie (DE-588)4067277-3 gnd Primzahlverteilung (DE-588)4175716-6 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4175716-6 |
| title | Number theory an introduction via the distribution of primes |
| title_auth | Number theory an introduction via the distribution of primes |
| title_exact_search | Number theory an introduction via the distribution of primes |
| title_exact_search_txtP | Number theory an introduction via the distribution of primes |
| title_full | Number theory an introduction via the distribution of primes Benjamin Fine ; Gerhard Rosenberger |
| title_fullStr | Number theory an introduction via the distribution of primes Benjamin Fine ; Gerhard Rosenberger |
| title_full_unstemmed | Number theory an introduction via the distribution of primes Benjamin Fine ; Gerhard Rosenberger |
| title_short | Number theory |
| title_sort | number theory an introduction via the distribution of primes |
| title_sub | an introduction via the distribution of primes |
| topic | Teoria dos números larpcal Number theory Numbers, Prime Zahlentheorie (DE-588)4067277-3 gnd Primzahlverteilung (DE-588)4175716-6 gnd |
| topic_facet | Teoria dos números Number theory Numbers, Prime Zahlentheorie Primzahlverteilung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014798718&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT finebenjamin numbertheoryanintroductionviathedistributionofprimes AT rosenbergergerhard numbertheoryanintroductionviathedistributionofprimes |