Daniel A. Klain Essentials of Number Theory Essentials of Number Theory by Daniel A. Klain Daniel A. Klain Essentials of Number Theory Preliminary Edition last updated March 22, 2020 Theory Essentials of Number by Daniel A. Klain Copyright c 2018 by Daniel A. Klain Permission is granted to copy or distribute this work for non-commercial purposes, without adaptation or modification, as long as credit is given to the original author. The original book title, author credit, and this copyright notice must be retained in all copies. Contents Preface 6 1 What is number theory? 7 2 Integers and arithmetic 9 3 Some useful algebraic identities 15 4 Divisibility 19 5 Representations of numbers 23 6 Common divisors 27 7 Relatively prime integers 32 8 Solving linear Diophantine equations 36 9 Prime numbers and unique factorization 44 10 Modular arithmetic 53 11 The Chinese remainder theorem Theory 62 12 Divisibility tests 67 13 Checksums 70 14 Pollard’sEssentials Rho of Number 73 15 Zp and Fermat’s theorem 78 16 Units in Zn and Euler’s function 82 17 Elementary cryptography 90 18 Lagrange’s root theorem by Daniel A. Klain101 19 Polynomial equations and Hensel’s lemma 104 20 Primitive roots 109 21 The existence of primitive roots 115 22 Quadratic residues 120 23 The law of quadratic reciprocity 126 24 Proof of quadratic reciprocity 129 25 Quadratic residues over composite moduli 133 3 26 Jacobi symbols 140 27 Computing square roots mod p 145 28 Sums of squares 152 29 Pseudorandom numbers 160 30 Elementary primality testing 167 31 Advanced primality testing 170 32 Continued fractions 176 33 Infinite continued fractions 186 34 Recommended reading 192 35 Answers, hints, and solutions to selected exercises 194 References 201 Index 202 Theory Essentials of Number by Daniel A. Klain 4 Kein’ Musik ist ja nicht auf Erden, Die uns’rer verglichen kann werden. Essentials of Number Theory by Daniel A. Klain Preface This is an undergraduate level introduction to classical number theory, covering traditional topics (from discoveries of the ancient Greeks, to the work of Fermat, Euler, and Gauss), along with a few sections that outline newer applications of number theory made possible by 20th century computer science. While familiarity with calculus and linear algebra may be helpful for reasons of mathematical maturity, most of the material in this book is accessible to read- ers having a solid background in high school level mathematics. Sections are kept sufficiently short and focused for single session of reading. The first 26 sections (omitting sections 13, 14, and 17 with no loss of continuity) form a core introduction to the subject, providing the basic tools needed for further study. Selections can be made from among the remaining sections (13, 14, 17, 27–33) for applications and further topics. The book is also designed with extracurricular readers in mind: a student using this textbook for a reading course can read every section in the order provided. As with all mathematical studies, the exercises are of paramount importance. Section 35 contains hints, simple answers, and, in some cases, full solutions to selected exercises. Theory The first edition of any textbook is likely to want correction and refinement. I hope to find and correct errors and to fill some omissions for a more polished future edition. Comments are most welcome. I am gratefulEssentials to my students at UMass of Lowell Number for their patience with earlier drafts of this book, as well as for their comments and corrections. I am also very grateful to Tanya Khovanova for carefully proofreading several sections and for making many detailed and helpful suggestions. Finally, I am grateful to Michael Artin, for introducing me to this subject, and to Glenn Stevens, for showing me that classical number theory is a deep, exciting, and accessible part of mathematics that every studentby of the sciences Daniel can enjoy. A. Klain Dan Klain June 2017 6 1 What is number theory? At first glance the term “number theory” seems mysteriously broad. Isn’t all of mathematics about numbers? Is this just another name for mathematics in general? A more cautious reader might note that geometry and logic (for ex- ample) aren’t really about numbers, even if numbers are sometimes used. But even leaving out these topics, the “study of numbers” still sounds overly broad. Indeed, the term number theory is traditional, and refers exclusively to the study of whole numbers; that is, the numbers we count with: 1, 2, 3, 4, . along with the daring addition of 0, and, when convenient, the negative integers. Excluded from consideration are fractions, real numbers, and complex numbers. Those abstractions, while called numbers in ordinary language, are traditionally studied in courses on Analysis. While the whole numbers have their origins in counting, number theory is not about counting either. The study of advanced techniques in counting1 is a field of mathematics all its own, called CombinatoricsTheory. Number theory then is the pure study of whole numbers and their relations to one another, especially with regards to addition and multiplication, both of which will always transform whole numbers into whole numbers. For a sense of what this means,Essentials consider the following questions of about Number whole numbers: • Is the sum of two odd numbers even or odd? What about the product? • If we divide n by 3, we have 2 left over. If we divide the same number n by 17, we have 9 leftover. What are the possible values for n? • Can a power of two ever end in the digits “...324”? • When can a positive integer n be written as a sum of two integer squares? • + 1 + 1 + ··· +by1 Daniel> A. Klain Is the number 1 2 3 n ever a whole number if n 1? • Does the equation 12x − 57y = 39 have integer solutions x and y? What if 39 is replaced with 38? Number theory, as described so far, may seem a rather abstract topic to spend months (years?) studying. Indeed, because of its ostensible purity and great dis- tance from industrial or scientific applications, number theory was once known as the “Queen of Mathematics.” 1For example, if four married couples are seated at a round table, how many ways can they be arranged, alternating by gender, so that no one sits next to his or her spouse? 7 8 1 What is number theory? This is no longer the case. While still considered an exemplar of abstract math- ematical elegance, number theory now provides concrete applications to infor- mation theory and computer science, including cryptography, data compression, error-correcting codes, and pseudorandom number generation, to name just a few examples. All the same, the most compelling reason to study number theory is for its unique combination of simplicity and mystifying complexity, which provides a setting for mathematical beauty, surprise, and the sudden clarity that can be so thrilling to those who enjoy mathematics. Theory Essentials of Number by Daniel A. Klain 2 Integers and arithmetic Our primary setting is the set Z of integers; that is, whole numbers, positive and negative: Z = f..., −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, . .g. It is also convenient to denote by N the set of natural numbers; that is, the positive integers: N = f1, 2, 3, 4, 5, 6, 7, 8, . .g. We will assume familiarity with properties of the integers that the reader should recall from grade school. In particular, the integers have an ordering, and they are closed under addition, subtraction, and multiplication. Moreover, we will also assume that N is closed under addition and multiplication. The following identities summarize the algebraic properties of the integers. Theorem 2.1 (Basic properties of integer arithmetic). Let a, b, c 2 Z. • a + b = b + a Theory(addition is commutative) • a + (b + c) = (a + b) + c (addition is associative) • a + 0 = a (zero is the additive identity) • a + (−a) = 0 (every integer has an additive inverse) • abEssentials= ba of Number(multiplication is commutative) • a(bc) = (ab)c (multiplication is associative) • a · 1 = a (1 is the multiplicative identity) • a(b + c) = ab + ac (distributive law) • If ab = 0 then either a = 0 or bby= 0 (or both).Daniel(integral A. domain) Klain The last property in the list above implies the following Cancellation Law. Proposition 2.2. Let a, b, c 2 Z, and suppose that a 6= 0. If ab = ac then b = c. Proof. Since ab = ac, we have ab − ac = 0, so that a(b − c) = 0. Since a 6= 0, the last property in Theorem 2.1 implies that b − c = 0, so that b = c. q Notice what we did not say in the proof above. We did not talk about “dividing both sides by a”. Instead, the proof used properties of addition, subtraction, 9 10 2 Integers and arithmetic and multiplication, without any direct reference to division. The reason for this circumlocution will become evident as the theory unfolds. § For a, b 2 Z we write a < b if a is less than b. Denote a ≤ b if either a is less than b or a = b. The following identities summarize some of the order properties of the integers. Theorem 2.3 (Order properties of Z). Let a, b, c 2 Z. • a ≤ a (reflexive property) • If a ≤ b and b ≤ a then a = b (antisymmetric property) • If a ≤ b and b ≤ c then a ≤ c (transitive property) • If a, b 2 Z then a ≤ b or b ≤ a (total ordering) • If a ≥ b then a + c ≥ b + c. • If a ≥ b and c ≥ 0 then ac ≥ bc.