Math 706, Theory of Numbers, Kansas State University, Spring 2019

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Notation......Page 7 1.2.1. Trichotomy Principle.......Page 9 1.3. Discreteness Axioms.......Page 10 1.4. Additional Properties of Z.......Page 11 2.1. Divisibility and Greatest Common Divisors......Page 13 2.3. Euclidean Algorithm......Page 14 2.4. Euclidean Domains......Page 15 2.6. Solving the equation ax+by=d, with d=(a,b)......Page 17 2.7. The linear equation ax+by=c......Page 19 2.9. Unique Factorization in Z......Page 20 2.10. Properties of GCDs and LCMs......Page 21 2.11. Units, Primes and Irreducibles......Page 22 2.12.1. Principal Ideal Domains......Page 23 2.13. Gaussian Integers......Page 24 2.14. The Set of Primes......Page 25 2.14.2. Twin Primes......Page 26 2.14.3. Number of primes up to x......Page 27 2.14.5. Goldbach Conjecture......Page 28 3.1. Basic properties of congruences......Page 29 3.2. The ring of integers 8mu(mod6mum), Zm......Page 30 3.4. Multiplicative inverses and Cancelation Laws......Page 31 3.5. The Group of units 8mu(mod6mum) and the Euler phi-function......Page 32 3.7. Fermat's Little Theorem, Euler's Theorem and Wilson's Theorem......Page 33 3.8. Chinese Remainder Theorem......Page 34 3.9. Group of units modulo a prime, G(p)......Page 36 3.10. Group of units G(pe)......Page 38 3.11. Group of units G(m) for arbitrary m......Page 39 4.2. Power Congruences, xn a 8mu(mod6mum)......Page 41 4.4. General Polynomial Congruences: Lifting Solutions......Page 43 4.5. Counting Solutions of Polynomial Congruences......Page 46 5.2. Properties of the Legendre Symbol......Page 47 5.3. Proof of the Law of Quadratic Reciprocity......Page 49 5.4. The Jacobi Symbol......Page 51 5.4.1. Euclidean-type algorithm for evaluating a Jacobi symbol......Page 53 5.5. Local solvability implies global solvability......Page 54 5.6. Sums of two Squares......Page 55 5.6.2. Sums of three squares and sums of four squares......Page 58 6.2. Pseudoprimes and Carmichael Numbers......Page 59 6.3. Mersenne Primes and Fermat Primes......Page 61 7.1. Properties of Greatest Integer Function and Binomial Coefficients......Page 63 7.3. Multiplicative Function......Page 64 7.4. Perfect Numbers......Page 66 7.6. Estimating Arithmetic Sums......Page 67 7.7. Möbius Inversion Formula......Page 69 7.8. Estimates for (n), (n) and (n)......Page 70 7.8.1. Estimates for (n)......Page 71 8.1. The Fibonacci Sequence......Page 73 8.3. A Matrix view of the Fibonacci Sequence......Page 74 8.4. Congruence and Divisibility Properties of the Fibonacci Sequence......Page 75 8.5. Periodicity of the Fibonacci sequence 8mu(mod6mum)......Page 76 8.6. Further Properties of the Fibonacci Sequence......Page 78 9.2. Systems of Linear Equations......Page 81 9.3. Pythagorean Triples......Page 85 9.4. Rational Points on Conics......Page 86 9.5. The Equations x4+y4=z2 and x2+4y4=z4......Page 87 9.6. Cubic Curves......Page 88 9.6.2. Method of Tangent Lines......Page 89 10.2. Addition of Points on an Elliptic Curve......Page 91 10.3. The Projective Plane......Page 93 10.5. The Elliptic Curve as an abelian Group......Page 94 10.6. The Pollard (p-1)-method of Factorization......Page 97 10.7. Elliptic Curve Method of Factorization......Page 98 11.1. Euler-Maclaurin Summation Formula and Estimating Factorials......Page 101 11.2. Chebyshev Estimate for (x)......Page 102 11.3. Bertrand's Postulate......Page 104 11.4. The von Mangoldt function and the function......Page 105 11.5. The sum of reciprical primes......Page 107 12.1. Matrix representation of quadratic form......Page 109 12.2. Equivalent Forms and Reduced Forms......Page 110 12.3. Representation by Positive Definite Binary Quadratic Forms......Page 112 12.5. Congruence test for Representation......Page 113 12.5.1. Ideal Class Number......Page 114 12.6. Tree diagram of Values Represented by a Binary Quadratic Form......Page 115 13.2. Discrete Subgroups of Rn......Page 117 13.3. Minkowski's Fundamental Theorem......Page 118 13.4. Canonical Basis Theorem and Sublattices......Page 119 13.5. Lagrange's 4-squares Theorem......Page 120 13.7. The Legendre Equation......Page 122 13.8. The Catalan Equation......Page 123 14.1. Approximating real numbers by rationals......Page 125 14.2.1. Simple Finite Continued Fractions......Page 126 14.3. Convergents to Continued Fractions......Page 127 14.4. Infinite Continued Fraction Expansions......Page 129 14.5. Best Rational Approximations to Irrationals......Page 130 14.6. Hurwitz's Theorem......Page 132 14.7. The set of all best rational approximations......Page 133 14.8. Quadratic Irrationals and Periodic Continued Fractions......Page 134 14.9. Pell Equations......Page 137 14.10. Liouville's Theorem......Page 140 15.1. Definition and Convergence of a Dirichlet series......Page 143 15.2. Important examples of Dirchlet Series......Page 144 15.3. Another Proof of the Möbius Inversion Formula......Page 145 15.5. Analytic properties of Dirichlet series......Page 146 15.6. The Riemann Zeta Function and the Riemann Hypothesis......Page 149 15.7. More on the zeta function......Page 150 Appendix A. Preliminaries......Page 153 B.0.2. Cancelation Law for Addition......Page 157 B.0.6. Basic consequence of Trichotomy.......Page 158 B.0.10. General Associative-Commutative Law.......Page 159 B.0.12. Binomial Square Formula......Page 160 C.1. Equivalence of the Discreteness Axioms......Page 161 C.3. Proof by Induction......Page 162 C.3.2. Examples of Induction Proofs......Page 163 C.3.3. Property 11. Binomial Expansion Formula......Page 164 D.1. Definition of a Ring......Page 167 D.3. Units and Zero Divisors......Page 168 D.5. Polynomial Rings......Page 169 D.6. Ring homomorphisms and Ideals......Page 172 D.7. Group Theory......Page 174 D.8. Lagrange's Theorem......Page 176 D.9. Normal Subgroups and Group Homomorphisms......Page 177 Bibliography......Page 179