Catalan's conjecture:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Springer
2008
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | IX, 124 S. 10 schw.-w. Fotos 235 mm x 155 mm |
| ISBN: | 9781848001848 9781848001855 |
Staff View
MARC
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| 245 | 1 | 0 | |a Catalan's conjecture |c René Schoof |
| 264 | 1 | |a London |b Springer |c 2008 | |
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| 600 | 1 | 4 | |a Catalan, Eugène Charles <1814-1894> |
| 600 | 1 | 4 | |a Mihăilescu, Preda |
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| 650 | 4 | |a Roots, Numerical | |
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Record in the Search Index
| _version_ | 1804138032291381248 |
|---|---|
| adam_text | Contents
1
Introduction
................................................... 1
2
The Case
q =
2 ............................................... 9
3
The Case
p
= 2 ............................................... 13
4
The
Nontrivial
Solution
......................................... 17
5
Runge s Method
................................................ 21
6
Casseis
theorem
............................................... 33
7
An Obstruction Group
.......................................... 41
8
Small
p
or
q
.................................................... 47
9
The Stickelberger Ideal
......................................... 55
10
The Double Wieferich Criterion
.................................. 65
11
The Minus Argument
........................................... 69
12
The Plus Argument I
........................................... 77
13 Semisimple
Group Rings
........................................ 85
14
The Plus Argument II
........................................... 91
15
The Density Theorem
........................................... 95
16
Thaine s Theorem
..............................................107
Appendix. Euler s Theorem
..........................................117
Index
.............................................................123
Catalan s Conjecture
Eugène
Charles Catalan made his famous conjecture
-
that
8
and
9
are the only two
consecutive perfect powers of natural numbers
-
in
1844
in a letter to the editor of
Crelle s mathematical journal. One hundred and fifty-eight years later,
Preda
Mihăilescu
proved it.
Catalan s Conjecture presents this spectacular result in a way that is accessible to the
advanced undergraduate. The first few sections of the book require little more than a
basic mathematical background and some knowledge of elementary number theory,
while later sections involve Galois theory, algebraic number theory and a small
amount of commutative algebra. The prerequisites, such as the basic facts from the
arithmetic of cyclotomic fields, are all discussed within the text.
The author dissects both
Mihăilescu s
proof and the earlier work it made use of,
taking great care to select streamlined and transparent versions of the arguments and
to keep the text self-contained. Only in the proof of Thaine s theorem is a little class
field theory used; it is hoped that this application will motivate the interested reader
to study the theory further.
Beautifully clear and concise, this book will appeal not only to specialists in number
theory but to anyone interested in seeing the application
of the ideas of algebraic number theory to a famous
mathematical problem.
|
| adam_txt |
Contents
1
Introduction
. 1
2
The Case
"q =
2". 9
3
The Case
"p
= 2". 13
4
The
Nontrivial
Solution
. 17
5
Runge's Method
. 21
6
Casseis'
theorem
. 33
7
An Obstruction Group
. 41
8
Small
p
or
q
. 47
9
The Stickelberger Ideal
. 55
10
The Double Wieferich Criterion
. 65
11
The Minus Argument
. 69
12
The Plus Argument I
. 77
13 Semisimple
Group Rings
. 85
14
The Plus Argument II
. 91
15
The Density Theorem
. 95
16
Thaine's Theorem
.107
Appendix. Euler's Theorem
.117
Index
.123
Catalan's Conjecture
Eugène
Charles Catalan made his famous conjecture
-
that
8
and
9
are the only two
consecutive perfect powers of natural numbers
-
in
1844
in a letter to the editor of
Crelle's mathematical journal. One hundred and fifty-eight years later,
Preda
Mihăilescu
proved it.
Catalan's Conjecture presents this spectacular result in a way that is accessible to the
advanced undergraduate. The first few sections of the book require little more than a
basic mathematical background and some knowledge of elementary number theory,
while later sections involve Galois theory, algebraic number theory and a small
amount of commutative algebra. The prerequisites, such as the basic facts from the
arithmetic of cyclotomic fields, are all discussed within the text.
The author dissects both
Mihăilescu's
proof and the earlier work it made use of,
taking great care to select streamlined and transparent versions of the arguments and
to keep the text self-contained. Only in the proof of Thaine's theorem is a little class
field theory used; it is hoped that this application will motivate the interested reader
to study the theory further.
Beautifully clear and concise, this book will appeal not only to specialists in number
theory but to anyone interested in seeing the application
of the ideas of algebraic number theory to a famous
mathematical problem. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Schoof, René |
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| dewey-ones | 512 - Algebra |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035080384 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:06:53Z |
| indexdate | 2024-07-09T21:21:44Z |
| institution | BVB |
| isbn | 9781848001848 9781848001855 |
| language | English |
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| physical | IX, 124 S. 10 schw.-w. Fotos 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Schoof, René Verfasser aut Catalan's conjecture René Schoof London Springer 2008 IX, 124 S. 10 schw.-w. Fotos 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Universitext Catalan, Eugène Charles <1814-1894> Mihăilescu, Preda Number theory Roots, Numerical Catalan-Vermutung (DE-588)4369060-9 gnd rswk-swf Catalan-Vermutung (DE-588)4369060-9 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016748634&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016748634&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Schoof, René Catalan's conjecture Catalan, Eugène Charles <1814-1894> Mihăilescu, Preda Number theory Roots, Numerical Catalan-Vermutung (DE-588)4369060-9 gnd |
| subject_GND | (DE-588)4369060-9 |
| title | Catalan's conjecture |
| title_auth | Catalan's conjecture |
| title_exact_search | Catalan's conjecture |
| title_exact_search_txtP | Catalan's conjecture |
| title_full | Catalan's conjecture René Schoof |
| title_fullStr | Catalan's conjecture René Schoof |
| title_full_unstemmed | Catalan's conjecture René Schoof |
| title_short | Catalan's conjecture |
| title_sort | catalan s conjecture |
| topic | Catalan, Eugène Charles <1814-1894> Mihăilescu, Preda Number theory Roots, Numerical Catalan-Vermutung (DE-588)4369060-9 gnd |
| topic_facet | Catalan, Eugène Charles <1814-1894> Mihăilescu, Preda Number theory Roots, Numerical Catalan-Vermutung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016748634&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016748634&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT schoofrene catalansconjecture |