A Friendly Introduction to Number Theory 4th Edition by Joseph H. Silverman, ISBN-13: 978-0321816191
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- Publisher: Pearson; 4th edition (January 28, 2012)
- Language: English
- 432 pages
- ISBN-10: 0321816196
- ISBN-13: 978-0321816191
A Friendly Introduction to Number Theory, Fourth Edition is designed to introduce readers to the overall themes and methodology of mathematics through the detailed study of one particular facet—number theory. Starting with nothing more than basic high school algebra, readers are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
Table of Contents:
Title Page
Copyright Page
Contents
Preface
Flowchart of Chapter Dependencies
Introduction
1 What Is Number Theory?
2 Pythagorean Triples
3 Pythagorean Triples and the Unit Circle
4 Sums of Higher Powers and Fermat’s Last Theorem
5 Divisibility and the Greatest Common Divisor
6 Linear Equations and the Greatest Common Divisor
7 Factorization and the Fundamental Theorem of Arithmetic
8 Congruences
9 Congruences, Powers, and Fermat’s Little Theorem
10 Congruences, Powers, and Euler’s Formula
11 Euler’s Phi Function and the Chinese Remainder Theorem
12 Prime Numbers
13 Counting Primes
14 Mersenne Primes
15 Mersenne Primes and Perfect Numbers
16 Powers Modulo m and Successive Squaring
17 Computing k th Roots Modulo m
18 Powers, Roots, and “Unbreakable” Codes
19 Primality Testing and Carmichael Numbers
20 Squares Modulo p
21 Is -1 a Square Modulo p? Is 2?
22 Quadratic Reciprocity
23 Proof of Quadratic Reciprocity
24 Which Primes Are Sums of Two Squares?
25 Which Numbers Are Sums of Two Squares?
26 As Easy as One, Two, Three
27 Euler’s Phi Function and Sums of Divisors
28 Powers Modulo p and Primitive Roots
29 Primitive Roots and Indices
30 The Equation X 4+Y 4=Z 4
31 Square–Triangular Numbers Revisited
32 Pell’s Equation
33 Diophantine Approximation
34 Diophantine Approximation and Pell’s Equation
35 Number Theory and Imaginary Numbers
36 The Gaussian Integers and Unique Factorization
37 Irrational Numbers and Transcendental Numbers
38 Binomial Coeffcients and Pascal’s Triangle
39 Fibonacci’s Rabbits and Linear Recurrence Sequences
40 Oh, What a Beautiful Function
41 Cubic Curves and Elliptic Curves
42 Elliptic Curves with Few Rational Points
43 Points on Elliptic Curves Modulo p
44 Torsion Collections Modulo p and Bad Primes
45 Defect Bounds and Modularity Patterns
46 Elliptic Curves and Fermat’s Last Theorem
Further Reading
Index
Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and seven books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography. He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.
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