Friendly Introduction to Number Theory, A,9780131861374

Friendly Introduction to Number Theory, A

by
Edition: 3rd
ISBN13: 9780131861374
ISBN10: 0131861379
Format: Hardcover
Pub. Date: 2006-01-01
Publisher(s): Prentice Hall
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Summary

Starting with nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers.Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh notation and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Provides a new chapter that introduces the theory of continued fractions. Includes a new chapter on "Continued Fractions, Square Roots and Pellrs"s Equation." Contains additional historical material, including material on Pellrs"s equation and the Chinese Remainder Theorem.A useful reference for mathematics teachers.

Table of Contents

Preface v
Introduction 1(65)
1 What Is Number Theory?
6(7)
2 Pythagorean Triples
13(7)
3 Pythagorean Triples and the Unit Circle
20(4)
4 Sums of Higher Powers and Fermat's Last Theorem
24(4)
5 Divisibility and the Greatest Common Divisor
28(7)
6 Linear Equations and the Greatest Common Divisor
35(9)
7 Factorization and the Fundamental Theorem of Arithmetic
44(9)
8 Congruences
53(7)
9 Congruences, Powers, and Fermat's Little Theorem
60(6)
10 Congruences, Powers, and Euler's Formula 66(4)
11 Euler's Phi Function and the Chinese Remainder Theorem 70(8)
12 Prime Numbers 78(7)
13 Counting Primes 85(6)
14 Mersenne Primes 91(4)
15 Mersenne Primes and Perfect Numbers 95(10)
16 Powers Modulo m and Successive Squaring 105(7)
17 Computing kth Roots Modulo m 112(5)
18 Powers, Roots, and "Unbreakable" Codes 117(6)
19 Primality Testing and Carmichael Numbers 123(11)
20 Euler's Phi Function and Sums of Divisors 134(5)
21 Powers Modulo p and Primitive Roots 139(10)
22 Primitive Roots and Indices 149(7)
23 Squares Modulo p 156(8)
24 Is -1 a Square Modulo p? Is 2? 164(11)
25 Quadratic Reciprocity 175(11)
26 Which Primes Are Sums of Two Squares? 186(12)
27 Which Numbers Are Sums of Two Squares? 198(6)
28 The Equation X4 + Y4 = Z4 204(3)
29 Square-Triangular Numbers Revisited 207(9)
30 Pell's Equation 216(6)
31 Diophantine Approximation 222(10)
32 Diophantine Approximation and Pell's Equation 232(7)
33 Number Theory and Imaginary Numbers 239(14)
34 The Gaussian Integers and Unique Factorization 253(17)
35 Irrational Numbers and Transcendental Numbers 270(16)
36 Binomial Coefficients and Pascal's Triangle 286(11)
37 Fibonacci's Rabbits and Linear Recurrence Sequences 297(13)
38 Oh, What a Beautiful Function 310(14)
39 The Topsy-Turvy World of Continued Fractions 324(16)
40 Continued Fractions, Square Roots, and Pell's Equation 340(15)
41 Generating Functions 355(10)
42 Sums of Powers 365(11)
43 Cubic Curves and Elliptic Curves 376(12)
44 Elliptic Curves with Few Rational Points 388(7)
45 Points on Elliptic Curves Modulo p 395(11)
46 Torsion Collections Modulo p and Bad Primes 406(4)
47 Defect Bounds and Modularity Patterns 410(6)
48 Elliptic Curves and Fermat's Last Theorem 416(2)
Further Reading 418(1)
A Factorization of Small Composite Integers 419(2)
B A List of Primes 421(2)
Index 423

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