A Friendly Introduction to Number Theory 4th Edition by Joseph H. Silverman, ISBN-13: 978-0321816191

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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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