Friendly Introduction to Number Theory, A

by Silverman, Joseph H.

• ISBN13: 9780131861374
• ISBN10: 0131861379
• Additional ISBN(s): 9780130309549, 9780321816191
• Edition: 3rd
• Format: Hardcover
• Copyright: 2006-01-01
• Publisher: Prentice Hall
• More Book Details
• Purchase Benefits

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