Math 706, Theory of Numbers Kansas State University Spring 2019 Todd Cochrane Department of Mathematics Kansas State University Contents Notation3 Chapter 1. Axioms for the set of Integers Z 5 1.1. Ring Properties of Z 5 1.2. Order Properties of Z 5 1.3. Discreteness Axioms.6 1.4. Additional Properties of Z.7 Chapter 2. Divisibility and Unique Factorization9 2.1. Divisibility and Greatest Common Divisors9 2.2. Division Algorithms 10 2.3. Euclidean Algorithm 10 2.4. Euclidean Domains 11 2.5. Linear Combinations and GCDLC Theorem 13 2.6. Solving the equation ax + by = d, with d = (a; b) 13 2.7. The linear equation ax + by = c 15 2.8. Primes and Euclid's Lemma 16 2.9. Unique Factorization in Z 16 2.10. Properties of GCDs and LCMs 17 2.11. Units, Primes and Irreducibles 18 2.12. UFDs, PIDs and Euclidean Domains 19 2.13. Gaussian Integers 20 2.14. The Set of Primes 21 Chapter 3. Modular Arithmetic 25 3.1. Basic properties of congruences 25 3.2. The ring of integers (mod m), Zm 26 3.3. Congruences in general rings 27 3.4. Multiplicative inverses and Cancelation Laws 27 3.5. The Group of units (mod m) and the Euler phi-function 28 3.6. A few results from Group Theory 29 3.7. Fermat's Little Theorem, Euler's Theorem and Wilson's Theorem 29 3.8. Chinese Remainder Theorem 30 3.9. Group of units modulo a prime, G(p) 32 3.10. Group of units G(pe) 34 3.11. Group of units G(m) for arbitrary m 35 Chapter 4. Polynomial Congruences 37 4.1. Linear Congruences 37 4.2. Power Congruences, xn ≡ a (mod m) 37 3 4 CONTENTS 4.3. A general quadratic congruence 39 4.4. General Polynomial Congruences: Lifting Solutions 39 4.5. Counting Solutions of Polynomial Congruences 42 Chapter 5. Quadratic Residues and Quadratic Reciprocity 43 5.1. Introduction 43 5.2. Properties of the Legendre Symbol 43 5.3. Proof of the Law of Quadratic Reciprocity 45 5.4. The Jacobi Symbol 47 5.5. Local solvability implies global solvability 50 5.6. Sums of two Squares 51 Chapter 6. Primality Testing, Mersenne Primes and Fermat Primes 55 6.1. Basic Primality Test 55 6.2. Pseudoprimes and Carmichael Numbers 55 6.3. Mersenne Primes and Fermat Primes 57 Chapter 7. Arithmetic Functions 59 7.1. Properties of Greatest Integer Function and Binomial Coefficients 59 7.2. The Divisor function and Sigma function 60 7.3. Multiplicative Function 60 7.4. Perfect Numbers 62 7.5. The M¨obiusFunction 63 7.6. Estimating Arithmetic Sums 63 7.7. M¨obiusInversion Formula 65 7.8. Estimates for τ(n), σ(n) and φ(n) 66 Chapter 8. Recurrence Sequences 69 8.1. The Fibonacci Sequence 69 8.2. Second order linear recurrences 70 8.3. A Matrix view of the Fibonacci Sequence 70 8.4. Congruence and Divisibility Properties of the Fibonacci Sequence 71 8.5. Periodicity of the Fibonacci sequence (mod m) 72 8.6. Further Properties of the Fibonacci Sequence 74 Chapter 9. Diophantine Equations 77 9.1. Preliminaries 77 9.2. Systems of Linear Equations 77 9.3. Pythagorean Triples 81 9.4. Rational Points on Conics 82 9.5. The Equations x4 + y4 = z2 and x2 + 4y4 = z4 83 9.6. Cubic Curves 84 Chapter 10. Elliptic Curves 87 10.1. Definition of an Elliptic Curve 87 10.2. Addition of Points on an Elliptic Curve 87 10.3. The Projective Plane 89 10.4. Elliptic curves in the projective plane 90 10.5. The Elliptic Curve as an abelian Group 90 10.6. The Pollard (p − 1)-method of Factorization 93 CONTENTS 5 10.7. Elliptic Curve Method of Factorization 94 Chapter 11. Prime Number Theory 97 11.1. Euler-Maclaurin Summation Formula and Estimating Factorials 97 11.2. Chebyshev Estimate for π(x) 98 11.3. Bertrand's Postulate 100 11.4. The von Mangoldt function and the function 101 11.5. The sum of reciprical primes 103 Chapter 12. Binary Quadratic Forms 105 12.1. Matrix representation of quadratic form 105 12.2. Equivalent Forms and Reduced Forms 106 12.3. Representation by Positive Definite Binary Quadratic Forms 108 12.4. Class Number 109 12.5. Congruence test for Representation 109 12.6. Tree diagram of Values Represented by a Binary Quadratic Form 111 Chapter 13. Geometry of Numbers 113 13.1. Lattices and Bases 113 13.2. Discrete Subgroups of Rn 113 13.3. Minkowski's Fundamental Theorem 114 13.4. Canonical Basis Theorem and Sublattices 115 13.5. Lagrange's 4-squares Theorem 116 13.6. Sums of Three Squares 118 13.7. The Legendre Equation 118 13.8. The Catalan Equation 119 Chapter 14. Best Rational Approximations and Continued Fractions 121 14.1. Approximating real numbers by rationals 121 14.2. Continued Fractions 122 14.3. Convergents to Continued Fractions 123 14.4. Infinite Continued Fraction Expansions 125 14.5. Best Rational Approximations to Irrationals 126 14.6. Hurwitz's Theorem 128 14.7. The set of all best rational approximations 129 14.8. Quadratic Irrationals and Periodic Continued Fractions 130 14.9. Pell Equations 133 14.10. Liouville's Theorem 136 Chapter 15. Dirichlet Series 139 15.1. Definition and Convergence of a Dirichlet series 139 15.2. Important examples of Dirchlet Series 140 15.3. Another Proof of the M¨obiusInversion Formula 141 15.4. Product Formula for Dirichlet Series 142 15.5. Analytic properties of Dirichlet series 142 15.6. The Riemann Zeta Function and the Riemann Hypothesis 145 15.7. More on the zeta function 146 Appendix A. Preliminaries 149 Appendix B. Proof of Additional Properties of Z 153 6 CONTENTS Appendix C. Discreteness Axioms for Z 157 C.1. Equivalence of the Discreteness Axioms 157 C.2. Proof of Additional Discreteness Properties 158 C.3. Proof by Induction 158 Appendix D. Review of Groups, Rings and Fields 163 D.1. Definition of a Ring 163 D.2. Basic properties of Rings 164 D.3. Units and Zero Divisors 164 D.4. Integral Domains and Fields 165 D.5. Polynomial Rings 165 D.6. Ring homomorphisms and Ideals 168 D.7. Group Theory 170 D.8. Lagrange's Theorem 172 D.9. Normal Subgroups and Group Homomorphisms 173 Appendix. Bibliography 175 Notation N = f1; 2; 3; 4; 5;::: g = Natural numbers Z = f0; ±1; ±; 2; ±3;::: g = Integers E = f0; ±2; ±4; ±6;::: g = Even integers O = {±1; ±3; ±5;::: g = Odd integers Q = fa=b : a; b 2 Z; b 6= 0g = Rational numbers R = Real numbers C = Complex numbers Z[i] = fa + bi : a; b 2 Zg = Gaussian Integers Zm = Ring of integers mod m [a]m = fa + mx : x 2 Zg = Residue class of a mod m Um = Multiplicative group of units mod m a−1 (mod m) = \multiplicative inverse of a (mod m)" φ(m) = Euler phi-function (a; b) = gcd(a; b) = greatest common divisor of a and b [a; b] = lcm[a; b] = least common multiple of a and b ajb = \a divides b" M2;2(R) = Ring of 2 × 2 matrices over a given ring R R[x] = Ring of polynomials over R jSj = order or cardinality of a set S Sn = n-th symmetric group log(x) = natural logarithm of x \ intersection [ union ; empty set ⊆ subset 9 there exists 9! there exists a unique 8 for all ) implies , equivalent to iff if and only if 2 element of ≡ congruent to 7 CHAPTER 1 Axioms for the set of Integers Z 1.1. Ring Properties of Z We shall assume the following properties as axioms for the set of integers. 1.1.1. Addition Properties. There is a binary operation + on Z, called addition, satisfying a) Addition is well defined, that is, given any two integers a; b, a+b is a uniquely defined integer. b) Substitution Law for addition: If a = b and c = d then a + c = b + d. c) The set of integers is closed under addition. For any a; b 2 Z, a + b 2 Z. d) Addition is commutative. For any a; b 2 Z, a + b = b + a. e) Addition is associative. For any a; b; c 2 Z,(a + b) + c = a + (b + c). f) There is a zero element 0 2 Z (also called the additive identity), satisfying 0 + a = a = a + 0 for any a 2 Z. g) For any a 2 Z, there exists an additive inverse −a 2 Z satisfying a + (−a) = 0 = (−a) + a. Note 1.1.1. Properties a),b), and c) above are implicit in the definition of a binary operation on Z. Definition 1.1.1. Subtraction is defined by a − b = a + (−b) for a; b 2 Z. 1.1.2. Multiplication Properties. There is an operation · (or ×) on Z called multiplication, satisfying, a) Multiplication is well defined, that is, given any two integers a; b, a · b is a uniquely defined integer. b) Substitution Law for multiplication: If a = b and c = d then ac = bd. c) Z is closed under multiplication. For any a; b 2 Z, a · b 2 Z. d) Multiplication is commutative. For any a; b 2 Z, ab = ba. e) Multiplication is associative. For any a; b; c 2 Z,(ab)c = a(bc). f) There is an identity element 1 2 Z satisfying 1 · a = a = a · 1 for any a 2 Z. 1.1.3. Distributive law. This is the one property that combines both ad- dition and multiplication.