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Aba Mbirika and Rita Post \Mathematics is the Queen of the Sciences, and Number Theory is the Queen of Mathematics." { Carl Friedrich Gauss (1777-1855) aBa Mbirika and Rita Post Spring 2019 LICENSE Creative Commons License (CC BY-NC-SA): This text, including the art and illustrations, are available under the Creative Commons license (CC BY-NC- SA), allowing anyone to reuse, revise, remix and redistribute the text. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/4.0/ Section CONTENTS Page 3 Contents 0 Preliminaries9 0.1 Math Symbols Used In This Book . .9 0.2 What Is Number Theory and What Is Its Purpose? . 12 0.2.1 Some Beautiful Conjectures Awaiting Proofs . 22 0.2.2 Some Beautiful Results That Have Been Proven . 23 0.2.3 Number Theory Flourishes In Hollywood Films . 27 0.3 Examples of the Beauty and Power of Number Theory . 29 1 Proofs { An Introduction 37 1.1 Statements and Truth Tables . 38 1.2 Implications . 42 1.3 Predicates and Quantifiers . 46 1.4 Writing Formal Proofs . 56 1.5 Quick Review of Set Theory and Some Set Theory Proofs . 63 1.6 Butternut Squash . 69 2 Mathematical Induction 71 2.1 Motivation . 71 2.2 The Principle and Method of Mathematical Induction . 74 2.3 Exercises . 77 3 Pascal's Triangle and Binomial Coefficients 81 3.1 Motivation . 81 3.2 Constructing Pascal's Triangle . 84 3.3 16 Rows of Pascal's Triangle . 86 3.4 Binomial Coefficients . 87 3.5 Exercises . 88 4 Primes, Prime Factorization, and Some Consequences 93 4.1 Motivation . 93 4.2 Density of Primes and the Celebrated Prime Number Theorem . 94 4.3 A Sm¨org˚asbord of Types of Primes . 99 4.4 The Fundamental Theorem of Arithmetic and Some Consequences . 106 4.5 The Goldbach Conjecture . 112 4.6 Exercises . 114 Section CONTENTS Page 4 5 Divisibility Theory 119 5.1 Motivation . 119 5.2 Divisibility . 120 5.3 The Division Algorithm . 126 5.4 GCD and LCM . 127 5.5 The Euclidean Algorithm and B´ezout'sIdentity . 129 5.6 The Diophantine Equation . 134 5.7 Exercises . 135 6 Congruence Theory (a Number Theory Perspective) 141 6.1 Motivation . 141 6.2 Gauss and Congruence Properties . 142 6.3 Solving Linear Congruences . 148 6.4 Chinese Remainder Theorem (CRT) . 153 6.5 Exercises . 159 7 Arithmetic Functions 167 7.1 Motivation . 167 7.2 Sigma and Tau Functions . 171 7.3 Perfect Numbers . 178 7.4 The Omega Functions ! and Ω . 182 7.5 Euler's Phi Function . 183 7.6 M¨obiusInversion Formula . 188 7.7 Exercises . 195 8 Euler, Fermat, and Wilson 205 8.1 Motivation . 205 8.2 Fermat's Little Theorem and Pseudoprimes . 206 8.3 Wilson's Theorem . 212 8.4 Euler's Theorem . 214 8.5 Exercises . 215 9 Congruence Theory (a Group Theory Perspective) 223 9.1 Motivation . 223 9.2 The Groups (Zn; ⊕) and (U(n); ).................. 226 9.3 Group Homomorphisms and Isomorphisms . 235 9.4 The Order of an Element in a Group and Primitive Roots . 239 Section CONTENTS Page 5 10 Quadratic Reciprocity 251 10.1 Motivation . 251 10.2 The Many Masks of Quadratic Reciprocity (QR) . 255 10.3 Solving Quadratic Congruences . 256 10.4 Euler's Criterion . 256 10.5 The Legendre Symbol and Its Properties . 256 Index 258 Section CONTENTS Page 6 Preface: This is an introductory course to both proofs and number theory. No mathematics beyond Calculus I is assumed. As such, Chapters 0 and 1 are dedi- cated to introducing the beauty and power of number theory proofs and teaching the foundations of logic and set theory which will be useful for the rest of the book contents. If you are holding the student version, then there are multiple blanks throughout the book. Together in class we will be creating the missing contents of this book. In the words of mathematician Paul Halmos (1916{2006): \The only way to learn mathematics is to do mathematics." To that end, we will also have one homework assignment (with proofs and/or com- putations and/or conceptual questions) for each section in addition to occasional computational WeBWorK homeworks. Each written homework will be done in LATEX and a template will be provided for each of those homeworks. Acknowledgments: Co-authors aBa and Post would like to express their sincere ap- preciation to the following people1 for their useful comments and suggestion in earlier drafts of this book: (1) Niels Abel (13) Carl Friedrich Gauss (26) Ada Lovelace (2) Claude Gaspar Bachet de (14) Sophie Germain (27) Marin Mersenne Merziriac (15) Christian Goldbach (28) Hermann Minkowski (3) Etienne´ B´ezout (16) G.H. Hardy (29) Sir Isaac Newton (4) Richard Dedekind (17) David Hilbert (30) Emmy Noether (5) Diophantus of Alexandria (18) Hypatia of Alexandria (31) Blaise Pascal (6) Lejeune Dirichlet (19) Carl Jacobi (32) Pythagoras of Samos (7) Paul Erd}os (20) Sofia Kovalevskaya (33) Srinivasa Ramanujan (8) Eratosthenes (21) Leopold Kronecker (34) Bernhard Riemann (9) Euclid of Alexandria (22) Ernst Kummer (35) Pafnuty Lvovich (10) Leonhard Euler (23) Joseph-Louis Lagrange Chebyshev. (11) Pierre de Fermat (24) Adrien-Marie Legendre (12) Leonardo Fibonacci (25) Joseph Liousville 1The authors do indeed realize that every person on their list of acknowledgments is deceased. But that mortality status did not prevent any of these 35 people from giving thoughtful comments and spiritual inspiration to co-authors aBa and Post in the writing of this book! Section CONTENTS Page 7 About the authors: Co-author aBa received his PhD from the University of Iowa and then completed a 3-year postdoctoral position at Bowdoin College in Maine. He is now an associate professor at the University of Wisconsin-Eau Claire (UWEC). His mathematical interests include combinatorics as related to areas such as representation theory and algebraic geometry/topology. His research interests with UWEC undergraduates have included lattice point visibility, complex reflection groups, and number theory in the Eisenstein integer ring. Co-author Post received her Bachelors degree in Mathematics at UWEC with a minor in Computer Science. She is currently exploring the computer science industry as an associate developer. Her research interests include combinatorics as related to permutation statistics in the hyperoctahedral group and topology as related to knot theory focusing on virtual spatial trivalent graphs. Post and aBa (in Summer 2018 at Math In The Woods) Section 0 PRELIMINARIES Page 9 0 Preliminaries 0.1 Math Symbols Used In This Book Below is a list of symbols with their associated meanings that we will come across in this course. Symbol Meaning N set of natural numbers (we exclude 0) Z set of integers Q set of rational numbers R set of real numbers C set of complex numbers [ union \ intersection ` disjoint union n! n factorial n k binomial coefficient th Tn n triangular number a j b a divides b a - b a does not divide b gcd(a; b) greatest common divisor of a and b lcm(a; b) least common multiple of a and b a ≡ b (mod n) a is congruent to b modulo n th Mn n Mersenne number th Fn n Fermat number τ(n) number of positive divisors of n σ(n) sum of positive divisors of n φ(n) Euler's phi function !(n) omega function Ω(n) Omega function P djn sum over divisors d of n µ(n) M¨obiusmu function L (Zn; ) set of integers modulo n (U(n); ) group of units in Zn a b Legendre symbol (pronounced \a on b") Section 0 PRELIMINARIES Page 10 We also have some common abbreviations used in proofs for the most part as follows. Abbreviations Meaning 8 for all 9 there exists 2 is an element of BWOC by way of contradiction WLOG without loss of generality TFAE the following are equivalent s.t. such that : or equivalently j such that (used in set-theoretic nota- tion) =) implies ELFS exercise left for students WWTS we want to show Q.E.D. quod erat demonstrandum [end of proof] WARNING!!!!: Caution Alert! Often, it is very inappropriate to use abbreviations or math symbols to replace English words. For example, here are some atrocities in math writing: In the monomial x2, the exponent of x = 2. There exists an x 2 R s.t. x2 = x. If 9 two numbers (WLOG x and y) that are even 2 Z, then x + y being odd =) a contradiction. That last example is enough to make unborn babies cry in the womb. A variety of ways ELFS (Exercise Left For Students) may appear: [You Do!] { generally we will all do this together. [You Verify] or equivalently [Confirm This!], usually for minor calculations. [Let's Discuss!] { meaning of this is obvious. Section 0 PRELIMINARIES Page 11 Throughout the text the following two characters will arise at times to make certain observations or ask poignant questions. We call them the expressionless people. Lastly, a very important symbol we will use in proofs is what we call the WWTS bubble. When doing a proof, it forces you to place your thoughts of what you want show at the beginning of the proof, but the BUBBLE around it reminds you that this is not true yet. Often students make the mistake of using what they want to show in their proof and taking for granted its truth. The bubble may help. For example, if we were tasked with proving the following theorem: Theorem 0.1. For every real number x, there exists a real number y such that x + y = 0.
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