Elementary Number Theory with Programming

Preface xi Words xiii Notation in Mathematical Writing and in Programming xv 1 Special Numbers: Triangular, Oblong, Perfect, Deficient, and Abundant 1 The programs include one for factoring numbers and one to test a conjecture up to a fixed limit. Triangular Numbers 1 Oblong Numbers and Squares 3 Deficient, Abundant, and Perfect Numbers 4 Exercises 7 2 Fibonacci Sequence, Primes, and the Pell Equation 13 The programs include examples that count steps to compare two different approaches. Prime Numbers and Proof by Contradiction 13 Proof by Construction 17 Sums of Two Squares 18 Building a Proof on Prior Assertions 18 Sigma Notation 19 Some Sums 19 Finding Arithmetic Functions 20 Fibonacci Numbers 22 An Infinite Product 26 The Pell Equation 26 Goldbach's Conjecture 30 Exercises 31 3 Pascal's Triangle 44 The programs include examples that generate factorial using iteration and using recursion and thus demonstrate and compare important techniques in programming. Factorials 44 The Combinatorial Numbers n Choose k 46 Pascal's Triangle 48 Binomial Coefficients 50 Exercises 50 4 Divisors and Prime Decomposition 56 The programs include one that uses the algorithm to produce the GCD of a pair of numbers and a program to produce the prime decomposition of a number. Divisors 56 Greatest Common Divisor 58 Diophantine Equations 65 Least Common Multiple 67 Prime Decomposition 68 Semiprime Numbers 70 When Is a Number an mth Power? 71 Twin Primes 73 Fermat Primes 73 Odd Primes Are Differences of Squares 74 When Is n a Linear Combination of a and b? 75 Prime Decomposition of n! 76 No Nonconstant Polynomial with Integer Coefficients Assumes Only Prime Values 77 Exercises 78 5 Modular Arithmetic 85 One program checks if a mod equation is true, and another determines the solvability of a mod equation and then solves an equation that is solvable by a brute-force approach. Congruence Classes Mod k 85 Laws of Modular Arithmetic 87 Modular Equations 90 Fermat's Little Theorem 91 Fermat's Little Theorem 92 Multiplicative Inverses 92 Wilson's Theorem 93 Wilson's Theorem 95 Wilson's Theorem (2nd Version) 95 Squares and Quadratic Residues 96 Lagrange's Theorem 98 Lagrange's Theorem 99 Reduced Pythagorean Triples 100 Chinese Remainder Theorem 102 Chinese Remainder Theorem 103 Exercises 104 6 Number Theoretic Functions 111 The programs include two distinct approaches to calculating the tau function. The Tau Function 111 The Sigma Function 114 Multiplicative Functions 115 Perfect Numbers Revisited 115 Mersenne Primes 116 F(n) = Sigmaf(d) Where d is a Divisor of n 117 The Möbius Function 119 The Riemann Zeta Function 121 Exercises 124 7 The Euler Phi Function 134 The programs demonstrate two approaches to calculating the phi function. The Phi Function 134 Euler's Generalization of Fermat's Little Theorem 138 Phi of a Product of m and n When gcd(m,n) > 1 139 The Order of a (mod n) 139 Primitive Roots 140 The Index of m (mod p) Relative to a 141 To Be or Not to Be a Quadratic Residue 145 The Legendre Symbol 146 Quadratic Reciprocity 147 Law of Quadratic Reciprocity 148 When Does x2 = a (mod n) Have a Solution? 148 Exercises 150 8 Sums and Partitions 158 The exposition explains the central role of binary representation in computing and the programs produce the binary partition using a built-in function. An nth Power is the Sum of Two Squares 158 Solutions to the Diophantine Equation a2 + b2 + c2 = d2 159 Row Sums of a Triangular Array of Consecutive Odd Numbers 160 Partitions 160 When is a Number the Sum of Two Squares? 167 Sums of Four or Fewer Squares 170 Exercises 175 9 Cryptography 182 The programs include different ways to generate counts of letters and also Fermat factoring. Introduction and History 182 Public-Key Cryptography 187 Factoring Large Numbers 188 The Knapsack Problem 191 Superincreasing Sequences 192 Exercises 194 Sample programs at the end of each chapter. You can access working examples of the sample programs at the website: http://faculty.purchase.edu/jeanine.meyer/numbertheory/ Answers or Hints to Selected Exercises 203 Index 207