A Friendly Introduction to Number Theory, 4e  (English, Paperback, Joseph Silverman)

A Friendly Introduction to Number Theory, 4th Edition is designed to introduce 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. 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 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 Little Theorem Chapter 10: Congruences, Powers, and Euler Formula Chapter 11: Euler 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 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 Equation Chapter 33: Diophantine Approximation Chapter 34: Diophantine Approximation and Pell 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 Triangle Chapter 39: Fibonacci Rabbits and Linear Recurrence Sequences Chapter 40: Oh, What a Beautiful Function Chapter 41: Cubic Curves and Elliptic Curves Chapter 42: Elliptic Curves with Few Rational Points Chapter 43: Points on Elliptic Curves Modulo p Chapter 44: Torsion Collections Modulo p and Bad Primes Chapter 45: Defect Bounds and Modularity Patterns Chapter 46: Elliptic Curves and Fermat Last Theorem Chapter 47: The Topsy-Turvey World of Continued Fractions [online] Chapter 48: Continued Fractions, Square Roots, and Pell Equation [online] Chapter 49: Generating Functions [online] Chapter 50: Sums of Powers [online]