A Friendly Introduction to Number Theory

Cover......Page 1
Preface......Page 4
Introduction......Page 10
Chapter 1 What Is Number Theory?......Page 15
Chapter 2 Pythagorean Triples......Page 22
Chapter 3 Pythagorean Triplesand the Unit Circle......Page 30
Chapter 4 Sums of Higher Powers and Fermat's Last Theorem......Page 35
Chapter 5 Divisibility and the Greatest Common Divisor......Page 39
Chapter 6 Linear Equations and the Greatest Common Divisor......Page 46
Chapter 7 Factorization and the Fundamental Theoremof Arithmetic......Page 55
Chapter 8 Congruences......Page 64
Chapter 9 Congruences, Powers, and Fermat's Little Theorem......Page 74
Chapter 10 Congruences, Powers, and Euler's Formula......Page 80
Chapter 11 Euler's Phi Function and the Chinese Remainder Theorem......Page 84
Chapter 12 Prime Numbers......Page 92
Chapter 13 Counting Primes......Page 99
Chapter 14 Mersenne Primes......Page 105
Chapter 15 Mersenne Primes and Perfect Numbers......Page 110
Chapter 16 Powers Modulo m and Successive Squaring......Page 120
Chapter 17 Computing kth Roots Modulo m......Page 127
Chapter 18 Powers, Roots, and ''Unbreakable'' Codes......Page 132
Chapter 19 Primality Testing and Carmichael Numbers......Page 138
Chapter 20 Squares Modulo p......Page 150
Chapter 21 Is -1 a Square Modulo p? Is 2?......Page 157
Chapter 22 Quadratic Reciprocity......Page 168
Chapter 23 Proof of Quadratic Reciprocity......Page 180
Chapter 24 Which Primes Are Sums of Two Squares?......Page 190
Chapter 25 Which Numbers Are Sums of Two Squares?......Page 202
Chapter 26 As Easy as One, Two, Three......Page 208
Chapter 27 Euler's Phi Function and Sums of Divisors......Page 215
Chapter 28 Powers Modulo pand Primitive Roots......Page 220
Chapter 29 Primitive Roots and Indices......Page 233
Chapter 30 The Equation X4 + Y4 =Z4......Page 240
Chapter 31 Square-Triangular Numbers Revisited......Page 245
Chapter 32 Pell's Equation......Page 254
Chapter 33 Diophantine Approximation......Page 260
Chapter 34 Diophantine Approximation and Pell's Equation......Page 269
Chapter 35 Number Theory and Imaginary Numbers......Page 276
Chapter 36 The Gaussian Integers and Unique Factorization......Page 290
Chapter 37 Irrational Numbers and Transcendental Numbers......Page 306
Chapter 38 Binomial Coefficients and Pascal's Triangle......Page 322
Chapter 39 Fibonacci's Rabbits and Linear Recurrence Sequences......Page 333
Chapter 40 Oh, What a Beautiful Function......Page 348
Chapter 41 Cubic Curves and Elliptic Curves......Page 362
Chapter 42 Elliptic Curves with Few Rational Points......Page 375
Chapter 43 Points on Elliptic Curves Modulo p......Page 382
Chapter 44 Torsion Collections Modulo pand Bad Primes......Page 393
Chapter 45 Defect Bounds and Modularity Patterns......Page 397
Chapter 46 Elliptic Curves and Fermat's Last Theorem......Page 403
Further Reading......Page 405
Index......Page 406