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Elementary Number Theory [Kõva köide]

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(Gilman School, Baltimore, Maryland, USA), (University of Maryland, College Park, USA)
  • Formaat: Hardback, 412 pages, kõrgus x laius: 234x156 mm, kaal: 920 g, 9 Illustrations, black and white
  • Ilmumisaeg: 24-Nov-2014
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498702686
  • ISBN-13: 9781498702683
Teised raamatud teemal:
Elementary Number TheorySuurem pilt
  • Formaat: Hardback, 412 pages, kõrgus x laius: 234x156 mm, kaal: 920 g, 9 Illustrations, black and white
  • Ilmumisaeg: 24-Nov-2014
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498702686
  • ISBN-13: 9781498702683
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781498702683.html
Märksõnad:

Elementary Number Theory takes an accessible approach to teaching students about the role of number theory in pure mathematics and its important applications to cryptography and other areas.

The first chapter of the book explains how to do proofs and includes a brief discussion of lemmas, propositions, theorems, and corollaries. The core of the text covers linear Diophantine equations; unique factorization; congruences; Fermat’s, Euler’s, and Wilson’s theorems; order and primitive roots; and quadratic reciprocity. The authors also discuss numerous cryptographic topics, such as RSA and discrete logarithms, along with recent developments.

The book offers many pedagogical features. The "check your understanding" problems scattered throughout the chapters assess whether students have learned essential information. At the end of every chapter, exercises reinforce an understanding of the material. Other exercises introduce new and interesting ideas while computer exercises reflect the kinds of explorations that number theorists often carry out in their research.

Arvustused

"This is a nice introduction to elementary number theory, designed for use in a basic undergraduate course. It can be used also for advanced high school students taking an accessible approach for an independent study. The book underlines the role of number theory in pure mathematics and its applications to cryptography and other areas." Zentralblatt MATH 1322

Preface xiii
Introduction 1(8)
1 Divisibility 9(46)
1.1 What Is a Proof?
9(8)
1.1.1 Proof by Contradiction
15(2)
1.2 Divisibility
17(2)
1.3 Euclid's Theorem
19(2)
1.4 Euclid's Original Proof
21(2)
1.5 The Sieve of Eratosthenes
23(2)
1.6 The Division Algorithm
25(2)
1.6.1 A Cryptographic Application
26(1)
1.7 The Greatest Common Divisor
27(3)
1.8 The Euclidean Algorithm
30(7)
1.8.1 The Extended Euclidean Algorithm
32(5)
1.9 Other Bases
37(3)
1.10 Fermat and Mersenne Numbers
40(3)
1.11
Chapter Highlights
43(1)
1.12 Problems
44(11)
1.12.1 Exercises
44(6)
1.12.2 Computer Explorations
50(1)
1.12.3 Answers to "Check Your Understanding"
51(4)
2 Linear Diophantine Equations 55(12)
2.1 ax + by = c
55(5)
2.2 The Postage Stamp Problem
60(4)
2.3
Chapter Highlights
64(1)
2.4 Problems
64(3)
2.4.1 Exercises
64(2)
2.4.2 Answers to "Check Your Understanding"
66(1)
3 Unique Factorization 67(10)
3.1 Preliminary Results
67(2)
3.2 The Fundamental Theorem of Arithmetic
69(5)
3.3 Euclid and the Fundamental Theorem of Arithmetic
74(1)
3.4
Chapter Highlights
75(1)
3.5 Problems
75(2)
3.5.1 Exercises
75(1)
3.5.2 Answers to "Check Your Understanding"
76(1)
4 Applications of Unique Factorization 77(18)
4.1 A Puzzle
77(2)
4.2 Irrationality Proofs
79(2)
4.3 The Rational Root Theorem
81(2)
4.4 Pythagorean Triples
83(4)
4.5 Differences of Squares
87(2)
4.6
Chapter Highlights
89(1)
4.7 Problems
90(5)
4.7.1 Exercises
90(3)
4.7.2 Computer Explorations
93(1)
4.7.3 Answers to "Check Your Understanding"
93(2)
5 Congruences 95(38)
5.1 Definitions and Examples
95(7)
5.2 Modular Exponentiation
102(2)
5.3 Divisibility Tests
104(4)
5.4 Linear Congruences
108(7)
5.5 The Chinese Remainder Theorem
115(4)
5.6 Fractions Mod m
119(2)
5.7 Queens on a Chessboard
121(2)
5.8
Chapter Highlights
123(1)
5.9 Problems
124(9)
5.9.1 Exercises
124(6)
5.9.2 Computer Explorations
130(1)
5.9.3 Answers to "Check Your Understanding"
130(3)
6 Fermat, Euler, Wilson 133(20)
6.1 Fermat's Theorem
133(5)
6.2 Euler's Theorem
138(7)
6.3 Wilson's Theorem
145(2)
6.4
Chapter Highlights
147(1)
6.5 Problems
147(6)
6.5.1 Exercises
147(3)
6.5.2 Computer Explorations
150(1)
6.5.3 Answers to "Check Your Understanding"
151(2)
7 Cryptographic Applications 153(46)
7.1 Introduction
153(3)
7.2 Shift and Affine Ciphers
156(4)
7.3 Vigenere Ciphers
160(6)
7.4 Transposition Ciphers
166(3)
7.5 RSA
169(7)
7.6 Stream Ciphers
176(5)
7.7 Block Ciphers
181(3)
7.8 Secret Sharing
184(3)
7.9
Chapter Highlights
187(1)
7.10 Problems
187(12)
7.10.1 Exercises
187(8)
7.10.2 Computer Explorations
195(1)
7.10.3 Answers to "Check Your Understanding"
196(3)
8 Order and Primitive Roots 199(30)
8.1 Orders of Elements
199(4)
8.1.1 Fermat Numbers
201(2)
8.1.2 Mersenne Numbers
203(1)
8.2 Primitive Roots
203(6)
8.3 Decimals
209(5)
8.3.1 Midy's Theorem
212(2)
8.4 Card Shuffling
214(2)
8.5 The Discrete Log Problem
216(2)
8.6 Existence of Primitive Roots
218(5)
8.7
Chapter Highlights
223(1)
8.8 Problems
223(6)
8.8.1 Exercises
223(4)
8.8.2 Computer Explorations
227(1)
8.8.3 Answers to "Check Your Understanding"
227(2)
9 More Cryptographic Applications 229(16)
9.1 Diffie-Hellman Key Exchange
229(2)
9.2 Coin Flipping over the Telephone
231(2)
9.3 Mental Poker
233(5)
9.4 Digital Signatures
238(2)
9.5
Chapter Highlights
240(1)
9.6 Problems
240(5)
9.6.1 Exercises
240(3)
9.6.2 Computer Explorations
243(1)
9.6.3 Answers to "Check Your Understanding"
243(2)
10 Quadratic Reciprocity 245(30)
10.1 Squares and Square Roots Mod Primes
245(8)
10.2 Computing Square Roots Mod p
253(2)
10.3 Quadratic Equations
255(1)
10.4 The Jacobi Symbol
256(5)
10.5 Proof of Quadratic Reciprocity
261(7)
10.6
Chapter Highlights
268(1)
10.7 Problems
268(7)
10.7.1 Exercises
268(5)
10.7.2 Answers to "Check Your Understanding"
273(2)
11 Primality and Factorization 275(22)
11.1 Trial Division and Fermat Factorization
275(4)
11.2 Primality Testing
279(6)
11.2.1 Pseudoprimes
279(5)
11.2.2 Mersenne Numbers
284(1)
11.3 Factorization
285(6)
11.3.1 x2 = y2
285(4)
11.3.2 Factoring Pseudoprimes and Factoring Using RSA Exponents
289(2)
11.4 Coin Flipping over the Telephone
291(2)
11.5
Chapter Highlights
293(1)
11.6 Problems
293(4)
11.6.1 Exercises
293(2)
11.6.2 Computer Explorations
295(1)
11.6.3 Answers to "Check Your Understanding"
296(1)
12 Sums of Squares 297(14)
12.1 Sums of Two Squares
297(4)
12.1.1 Algorithm for Writing p = 1 (mod 4) as a Sum of Two Squares
300(1)
12.2 Sums of Four Squares
301(5)
12.3 Other Sums of Powers
306(1)
12.4
Chapter Highlights
307(1)
12.5 Problems
307(4)
12.5.1 Exercises
307(4)
13 Arithmetic Functions 311(16)
13.1 Perfect Numbers
311(4)
13.2 Multiplicative Functions
315(6)
13.3
Chapter Highlights
321(1)
13.4 Problems
321(6)
13.4.1 Exercises
321(3)
13.4.2 Computer Explorations
324(1)
13.4.3 Answers to "Check Your Understanding"
325(2)
14 Continued Fractions 327(14)
14.1 Rational Approximations
328(4)
14.2 Evaluating Continued Fractions
332(2)
14.3 Pell's Equation
334(2)
14.4
Chapter Highlights
336(1)
14.5 Problems
337(4)
14.5.1 Exercises
337(2)
14.5.2 Computer Explorations
339(1)
14.5.3 Answers to "Check Your Understanding"
339(2)
15 Recent Developments 341(10)
15.1 Goldbach's Conjecture and the Twin Prime Problem
341(1)
15.2 Fermat's Last Theorem
342(3)
15.3 The Riemann Hypothesis
345(6)
A Supplementary Topics 351(26)
A.1 Geometric Series
351(2)
A.2 Mathematical Induction
353(5)
A.3 Pascal's Triangle and the Binomial Theorem
358(6)
A.4 Fibonacci Numbers
364(3)
A.5 Matrices
367(4)
A.6 Problems
371(6)
A.6.1 Exercises
371(3)
A.6.2 Answers to "Check Your Understanding"
374(3)
B Answers and Hints for Odd-Numbered Exercises 377(12)
Index 389
James S. Kraft teaches mathematics at the Gilman School. He has previously taught at the University of Rochester, St. Marys College of California, and Ithaca College. He has also worked in communications security. Dr. Kraft has published several research papers in algebraic number theory. He received his Ph.D. from the University of Maryland.

Lawrence C. Washington is a professor of mathematics and Distinguished Scholar-Teacher at the University of Maryland. Dr. Washington has published extensively in number theory, including books on cryptography (with Wade Trappe), cyclotomic fields, and elliptic curves. He received his Ph.D. from Princeton University.