Number theory through inquiry:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Washington, D.C. ; The Mathematical Association of America
2008
|
| Edition: | 1. printing |
| Series: | MAA textbooks
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 140 S. |
| ISBN: | 9780883857519 |
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MARC
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| 245 | 1 | 0 | |a Number theory through inquiry |c David C. Marshall ; Edward Odell ; Michael Starbird |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Introduction
1
Number Theory and Mathematical Thinking
........... 1
Note on the approach and organization
............. 2
Methods of thought
....................... 3
Acknowledgments
........................ 4
Divide and Conquer
7
Divisibility in the Natural Numbers
................ 7
Definitions and examples
.................... 7
Divisibility and congruence
................... 9
The Division Algorithm
..................... 14
Greatest common divisors and linear Diophantine equations
. 16
Linear Equations Through the Ages
................ 23
Prime Time
27
The Prime Numbers
........................ 27
Fundamental Theorem of Arithmetic
.............. 28
Applications of the Fundamental Theorem of Arithmetic
... 32
The infinitude of primes
..................... 35
Primes of special form
...................... 37
The distribution of primes
.................... 38
From Antiquity to the Internet
................... 41
A Modular World
43
Thinking Cyclically
........................ 43
Powers and polynomials modulo
η
............... 43
Linear congruences
....................... 48
vii
viii
Number Theory Through Inquiry
Systems of linear congruences: the Chinese
Remainder Theorem
................... 50
A Prince and a Master
....................... 51
4
Fermat s Little Theorem and Euler s Theorem
53
Abstracting the Ordinary
...................... 53
Orders of an integer modulo
η
................. 54
Fermat s Little Theorem
..................... 55
An alternative route to Fermat s Little Theorem
........ 58
Euler s Theorem and Wilson s Theorem
............ 59
Fermat,
Wilson and
...
Leibniz?
.................. 62
5
Public Key Cryptography
65
Public Key Codes and RSA
.................... 65
Public key codes
......................... 65
Overview of RSA
........................ 65
Let s decrypt
........................... 66
6
Polynomial Congruences and Primitive Roots
73
Higher Order Congruences
..................... 73
Lagrange s Theorem
....................... 73
Primitive roots
.......................... 74
Euler s ^»-function and sums of divisors
............ 77
Euler s (/»-function is multiplicative
............... 79
Roots modulo a number
..................... 81
Sophie Germain is Germane, Part I
................ 84
7
The Golden Rule: Quadratic Reciprocity
87
Quadratic Congruences
....................... 87
Quadratic residues
........................ 87
Gauss Lemma and quadratic reciprocity
............ 91
Sophie Germain is germane, Part II
............... 95
8
Pythagorean Triples, Sums of Squares,
and Fermat s Last Theorem
99
Congruences to Equations
..................... 99
Pythagorean triples
....................... 99
Sums of squares
......................... 102
Pythagorean triples revisited
................... 104
Fermat s Last Theorem
..................... 104
Who s Represented?
........................ 106
Sums of squares
......................... 106
Contents
¡χ
Sums of cubes, taxicabs, and
Fermat
s
Last Theorem
..... 107
9
Rationals Close to Irrationals and the Pell Equation
109
Diophantine Approximation and Pell Equations
......... 109
A plunge into rational approximation
.............. 110
Out with the trivial
....................... 114
New solutions from old
..................... 115
Securing the elusive solution
.................. 116
The structure of the solutions to the Pell equations
...... 117
Bovine Math
............................ 119
10
The Search for Primes
123
Primality Testing
.......................... 123
Is it prime?
............................ 123
Fermat s Little Theorem and probable primes
......... 124
AKS
primality
.......................... 126
Record Primes
........................... 127
A Mathematical Induction: The Domino Effect
129
The Infinitude Of Facts
...................... 129
Gauss formula
.......................... 129
Another formula
......................... 131
On your own
........................... 132
Strong induction
......................... 133
On your own
........................... 134
Index
135
About the Authors
139
|
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| author | Marshall, David C. Odell, Edward Starbird, Michael |
| author_facet | Marshall, David C. Odell, Edward Starbird, Michael |
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| author_sort | Marshall, David C. |
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| building | Verbundindex |
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| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)254668427 (DE-599)BVBBV025517933 |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7 |
| dewey-search | 512.7 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. printing |
| format | Book |
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| id | DE-604.BV025517933 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:35:49Z |
| institution | BVB |
| isbn | 9780883857519 |
| language | English |
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| physical | IX, 140 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| record_format | marc |
| series2 | MAA textbooks |
| spelling | Marshall, David C. Verfasser aut Number theory through inquiry David C. Marshall ; Edward Odell ; Michael Starbird 1. printing Washington, D.C. ; The Mathematical Association of America 2008 IX, 140 S. txt rdacontent n rdamedia nc rdacarrier MAA textbooks Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Odell, Edward Verfasser aut Starbird, Michael Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020126828&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Marshall, David C. Odell, Edward Starbird, Michael Number theory through inquiry Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Number theory through inquiry |
| title_auth | Number theory through inquiry |
| title_exact_search | Number theory through inquiry |
| title_full | Number theory through inquiry David C. Marshall ; Edward Odell ; Michael Starbird |
| title_fullStr | Number theory through inquiry David C. Marshall ; Edward Odell ; Michael Starbird |
| title_full_unstemmed | Number theory through inquiry David C. Marshall ; Edward Odell ; Michael Starbird |
| title_short | Number theory through inquiry |
| title_sort | number theory through inquiry |
| topic | Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020126828&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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