AN INTRODUCTION TO NUMBER THEORY J. J. P. Veerman Portland State University An Introduction to Number Theory (Veerman) This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. 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This text was compiled on 09/22/2021 TABLE OF CONTENTS These notes are intended for a graduate course in Number Theory. No prior familiarity with number theory is assumed. Chapters 1-6 represent approximately 1 trimester of the course. Eventually we intend to publish a full year (3 trimesters) course on number theory. 1: A QUICK TOUR OF NUMBER THEORY 1.1: DIVISORS AND CONGRUENCES 1.2: RATIONAL AND IRRATIONAL NUMBERS 1.3: ALGEBRAIC AND TRANSCENDENTAL NUMBERS 1.4: COUNTABLE AND UNCOUNTABLE SETS 1.5: EXERCISES 2: THE FUNDAMENTAL THEOREM OF ARITHMETIC 2.1: BÉZOUT'S LEMMA 2.2: COROLLARIES OF BE ́ZOUT’S LEMMA 2.3: THE FUNDAMENTAL THEOREM OF ARITHMETIC 2.4: COROLLARIES OF THE FUNDAMENTAL THEOREM OF ARITHMETIC 2.5: THE RIEMANN HYPOTHESIS 2.6: EXERCISES 3: LINEAR DIOPHANTINE EQUATIONS 3.1: THE EUCLIDEAN ALGORITHM 3.2: A PARTICULAR SOLUTION OF AX+BY = C 3.3: SOLUTION OF THE HOMOGENEOUS EQUATION AX+BY = 0 3.4: THE GENERAL SOLUTION OF AX+BY = C 3.5: RECURSIVE SOLUTION OF X AND Y IN THE DIOPHANTINE EQUATION 3.6 EXERCISES 4: NUMBER THEORETIC FUNCTIONS 4.1: MULTIPLICATIVE FUNCTIONS 4.2: ADDITIVE FUNCTIONS 4.3: MOBIUS INVERSION 4.4: EULER’S PHI OR TOTIENT FUNCTION 4.5: DIRICHLET AND LAMBERT SERIES 4.6: EXERCISE 5: MODULAR ARITHMETIC AND PRIMES 5.1: MODULAR ARITHMETIC 5.2: EULER'S METHOD 5.3: FERMAT’S LITTLE THEOREM AND PRIMALITY TESTING 5.4: FERMET AND MERSENNE PRIMES 5.5: DIVISION IN Zb 5.6: EXERCISES 6: CONTINUED FRACTIONS 6.1: THE GAUSS MAP 6.2: NEW PAGE 6.3: NEW PAGE 6.4: NEW PAGE 6.5: NEW PAGE 6.6: NEW PAGE BACK MATTER INDEX 1 9/22/2021 GLOSSARY 2 9/22/2021 CHAPTER OVERVIEW 1: A QUICK TOUR OF NUMBER THEORY 1.1: DIVISORS AND CONGRUENCES 1.2: RATIONAL AND IRRATIONAL NUMBERS 1.3: ALGEBRAIC AND TRANSCENDENTAL NUMBERS 1.4: COUNTABLE AND UNCOUNTABLE SETS 1.5: EXERCISES 1 9/22/2021 1.1: Divisors and Congruences Definition 1.1 Given two numbers a and b. A multiple b of a is a number that satisfies b = ac. A divisor a of b is an integer that satisfies ac = b where c is an integer. We write a|b. This reads as a divides b or a is a divisor of b. Definition 1.2 Let a and b non-zero. The greatest common divisor of two integers a and b is the maximum of the numbers that are divisors of both a and b. It is denoted by gcd(a, b). The least common multiple of a and b is the least of the positive numbers that are multiples of both a and b. It is denoted by lcm(a, b). Note that for any a and b in Z, gcd(a, b) ≥ 1, as 1 is a divisor of every integer. Similarly lcm(a, b) ≤ |ab|. Definition 1.3 A number a > 1 is prime in N if its only divisors in N are a and 1 (the so-called trivial divisors). A number a > 1 is composite if it has more than 2 divisors. (The number 1 is neither.) −− Figure 1. Eratosthenes’ sieve up to n = 30. All multiples of a less than √31 are cancelled. The remainder are the primes less than n = 31. An equivalent definition of prime is a natural number with precisely two (distinct) divisors. Eratosthenes’ sieve is a simple and ancient method to generate a list of primes for all numbers less than, say, 225. First, list all integers from 2 to 225. Start by circling the number 2 and crossing out all its remaining multiples: 4, 6, 8, etcetera. At each step, circle the smallest −u−n−marked number and cross out all its remaining multiples in the list. It turns out that we need to sieve out only multiples of √225 = 15 and less (see exercise 2.4). This method is illustrated if Figure 1. When done, the primes are those numbers that are circled or unmarked in the list. Definition 1.4 Let a and b in Z. Then a and b are relatively prime if gcd(a, b) = 1. Definition 1.5 Let a and b in Z and m ∈ N. Then a is congruent to b modulo m if a +my = b for some y ∈ Z or m|(b −a). We write J. J. P. Veerman 1.1.1 9/8/2021 https://math.libretexts.org/@go/page/60294 a =m b or a = b mod m or a ∈ b +mZ Definition 1.6 The residue of a modulo m is the (unique) integer r in {0, ⋯ , m −1} such that a =m r. It is denoted by Resm(a). These notions are cornerstones of much of number theory as we will see. But they are also very common in all kinds of applications. For in- stance, our expressions for the time on the clock are nothing but counting modulo 12 or 24. To figure out how many hours elapse between 4pm and 3am next morning is a simple exercise in working with modular arithmetic, that is: computations involving congruences. J. J. P. Veerman 1.1.2 9/8/2021 https://math.libretexts.org/@go/page/60294 1.2: Rational and Irrational Numbers We start with a few results we need in the remainder of this subsection. Theorem 1.7: Well-ordering Principle Any non-empty set S in N ∪ {0} has a smallest element. Proof Suppose this is false. Pick s1 ∈ S . Then there is another natural number s2 in S such that s2 ≤ s1 −1 . After a finite number of steps, we pass zero, implying that S has elements less than 0 in it. This is a contradiction. Note that any non-empty set S of integers with a lower bound can be transformed by addition of a integer b ∈ N0 into a non- empty S +b in N0 . Then S +b has a lower bound, and therefore so does S. Furthermore, a non-empty set S of integers with a upper bound can also be transformed into a non-empty −S +b in N0. Here, −S stands for the collection of elements of S multiplied by −1. Thus we have the following corollary of the well-ordering principle. Corollary 1.8 Let be a non-empty set S in Z with a lower (upper) bound. Then S has a smallest (largest) element. Definition 1.9 An element x ∈ R is called rational if it satisfies qx −p = 0 where p and q ≠ 0 are integers. Otherwise it is called an irrational number. The set of rational numbers is denoted by Q. p The usual way of expressing this, is that a rational number can be written as q . The advantage of expressing a rational number as the solution of a degree 1 polynomial, however, is that it naturally leads to Definition 1.12. Theorem 1.10 Any interval in R contains an element of Q. We say that Q is dense in R. Proof I = (a, b) b > a R n > 1 Let with any interval in . From Corollary 1.8 we see that there is an n such that b−a .