Friendly Introduction to Number Theory

A Friendly Introduction to Number Theory, 4th Edition introduces students 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, students 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.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.

Table of Contents

Chapter 1 What Is Number Theory?

Chapter 2 Pythagorean Triples

Chapter 3 Pythagorean Triples and the Unit Circle

Chapter 4 Sums of Higher Powers and Fermat’s Last Theorem

Chapter 5 Divisibility and the Greatest Common Divisor

Chapter 6 Linear Equations and the Greatest Common Divisor

Chapter 7 Factorization and the Fundamental Theorem of Arithmetic

Chapter 8 Congruences

Chapter 9 Congruences, Powers, and Fermat’s Little Theorem

Chapter 10 Congruences, Powers, and Euler’s Formula

Chapter 11 Euler’s Phi Function and the Chinese Remainder Theorem

Chapter 12 Prime Numbers

Chapter 13 Counting Primes

Chapter 14 Mersenne Primes

Chapter 15 Mersenne Primes and Perfect Numbers

Chapter 16 Powers Modulo m and Successive Squaring

Chapter 17 Computing kth Roots Modulo m

Chapter 18 Powers, Roots, and “Unbreakable” Codes

Chapter 19 Primality Testing and Carmichael Numbers

Chapter 20 Squares Modulo p

Chapter 21 Is -1 a Square Modulo p? Is 2?

Chapter 22 Quadratic Reciprocity

Chapter 23 Proof of Quadratic Reciprocity

Chapter 24 Which Primes Are Sums of Two Squares?

Chapter 25 Which Numbers Are Sums of Two Squares?

Chapter 26 As Easy as One, Two, Three

Chapter 27 Euler’s Phi Function and Sums of Divisors

Chapter 28 Powers Modulo p and Primitive Roots

Chapter 29 Primitive Roots and Indices

Chapter 30 The Equation X4 + Y4 =Z4

Chapter 31 Square-Triangular Numbers Revisited

Chapter 32 Pell’s Equation

Chapter 33 Diophantine Approximation

Chapter 34 Diophantine Approximation and Pell’s Equation

Chapter 35 Number Theory and Imaginary Numbers

Chapter 36 The Gaussian Integers and Unique Factorization

Chapter 37 Irrational Numbers and Transcendental Numbers

Chapter 38 Binomial Coefficients and Pascal’s Triangle

Chapter 39 Fibonacci’s Rabbits and Linear Recurrence Sequences

Chapter 40 Oh, What a Beautiful Function

Chapter 41 Cubic CuNes and Elliptic CuNes

Chapter 42 Elliptic CuNes with Few Rational Points

Chapter 43 Points on Elliptic CuNes Modulo p

Chapter 44 Torsion Collections Modulo p and Bad Primes

Chapter 45 Defect Bounds and Modularity Patterns

Chapter 46 Elliptic CuNes and Fermat’s Last Theorem

About our author

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 7 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.