Elementary Number Theory WISB321 = F.Beukers 2012 Department of Mathematics UU ELEMENTARY NUMBER THEORY Frits Beukers Fall semester 2013 Contents 1 Integers and the Euclidean algorithm 4 1.1 Integers . 4 1.2 Greatest common divisors . 7 1.3 Euclidean algorithm for Z ...................... 8 1.4 Fundamental theorem of arithmetic . 10 1.5 Exercises . 12 2 Arithmetic functions 15 2.1 Definitions, examples . 15 2.2 Convolution, M¨obiusinversion . 18 2.3 Exercises . 20 3 Residue classes 21 3.1 Basic properties . 21 3.2 Chinese remainder theorem . 22 3.3 Invertible residue classes . 24 3.4 Periodic decimal expansions . 27 3.5 Exercises . 29 4 Primality and factorisation 33 4.1 Prime tests and compositeness tests . 33 4.2 A polynomial time primality test . 38 4.3 Factorisation methods . 39 4.4 The quadratic sieve . 41 4.5 Cryptosystems, zero-knowledge proofs . 44 5 Quadratic reciprocity 47 5.1 The Legendre symbol . 47 5.2 Quadratic reciprocity . 48 5.3 A group theoretic proof . 51 5.4 Applications . 52 5.5 Jacobi symbols, computing square roots . 55 5.6 Class numbers . 59 1 2 CONTENTS 5.7 Exercises . 60 6 Dirichlet characters and Gauss sums 62 6.1 Characters . 62 6.2 Gauss sums, Jacobi sums . 65 6.3 Applications . 67 6.4 Exercises . 71 7 Sums of squares, Waring's problem 72 7.1 Sums of two squares . 72 7.2 Sums of more than two squares . 74 7.3 The 15-theorem . 77 7.4 Waring's problem . 78 7.5 Exercises . 81 8 Continued fractions 82 8.1 Introduction . 82 8.2 Continued fractions for quadratic irrationals . 85 8.3 Pell's equation . 88 8.4 Archimedes's Cattle Problem . 90 8.5 Cornacchia's algorithm . 91 8.6 Exercises . 93 9 Diophantine equations 94 9.1 General remarks . 94 9.2 Pythagorean triplets . 94 9.3 Fermat's equation . 96 9.4 Mordell's equation . 98 9.5 The `abc'-conjecture . 100 9.6 The equation xp + yq = zr . 102 9.7 Mordell's conjecture . 105 9.8 Exercises . 105 10 Prime numbers 107 10.1 Introductory remarks . 107 10.2 Elementary methods . 111 10.3 Exercises . 114 11 Irrationality and transcendence 117 11.1 Irrationality . 117 11.2 Transcendence . 120 11.3 Irrationality of ζ(3) . 122 11.4 Exercises . 124 F.Beukers, Elementary Number Theory CONTENTS 3 12 Solutions to selected problems 125 13 Appendix: Elementary algebra 143 13.1 Finite abelian groups . 143 13.2 Euclidean domains . 146 13.3 Gaussian integers . 147 13.4 Quaternion integers . 147 13.5 Polynomials . 150 F.Beukers, Elementary Number Theory Chapter 1 Integers and the Euclidean algorithm 1.1 Integers Roughly speaking, number theory is the mathematics of the integers. In any systematic treatment of the integers we would have to start with the so-called Peano-axioms for the natural numbers, define addition, multiplication and order- ing on them and then deduce their elementary properties such as the commuta- tive, associatative and distributive properties. However, because most students are very familiar with the usual rules of manipulation of integers, we prefer to shortcut this axiomatic approach. Instead we simply formulate the basic rules which form the basis of our course. After all, we like to get as quickly to the parts which make number theory such a beautiful branch of mathematics. We start with the natural numbers N : 1; 2; 3; 4; 5;::: On N we have an addition (+) and multiplication (× or ·) law and a well-ordering (>; <; ≥; ≤). By a well-ordering we mean that 1. For any distinct a; b 2 N we have either a > b or a < b. 2. From a < b and b < c follows a < c 3. There is a smallest element, namely 1. So a ≥ 1 for all a 2 N. We shall assume that we are all familiar with the usual rules of addition and multiplication. 1. For all a; b 2 N: a + b = b + a and ab = ba (commutativity of addition and multiplication). 4 1.1. INTEGERS 5 2. For all a; b; c 2 N:(a + b) + c = a + (b + c) and (ab)c = a(bc) (associativity of addition and multiplication) 3. For all a; b; c 2 N: a(b + c) = ab + ac (distributive law). 4. For all a 2 N: 1 · a = a. 5. For all a 2 N: a + 1 > a. 6. For all a; b; c 2 N: b > c ) a + b > a + c and b ≥ c ) ab ≥ ac. 7. For all a; b 2 N: a > b ) there exists c 2 N such that a = b + c. We shall also use the following fact. Theorem 1.1.1 Every non-empty subset of N has a smallest element. Then there is the principle of induction. Theorem 1.1.2 Let S ⊂ N and suppose that 1. 1 2 S 2. For all a 2 N: a 2 S ) a + 1 2 S Then S = N. Theorem 1.1.2 follows from Theorem 1.1.1 in the following way. Let S be as in Theorem 1.1.2 and consider the complement Sc. This set is either empty, in which case Theorem 1.1.2 is proven, or Sc is non-empty. Let us assume the latter. Theorem 1.1.1 states that Sc has a smallest element, which we denote by a. If a = 1, then a 62 S, violating the first condition of Theorem 1.1.2. If a > 1 then a − 1 62 Sc. Hence a − 1 2 S and a 62 S, violating the second condition. We conclude that Sc is empty, hence S = N. We call a subset S ⊂ N finite if there exists m 2 N such that s < m for all s 2 S. There are two concepts which partially invert addition and multiplication. 1. Subtraction Let a; b 2 N and a > b. Then there exists a unique c 2 N such that a = b + c. We call c the difference between a and b. Notation; a − b. 2. Divisibility We say that the natural number b divides a if there exists c 2 N such that a = bc. Notation: bja, and b is called a divisor of a. There are many well-known, almost obvious, properties which are not mentioned in the above rules, but which nevertheless follow in a more or less straightforward way. As an exercise you might try to prove the following properties. F.Beukers, Elementary Number Theory 6 CHAPTER 1. INTEGERS AND THE EUCLIDEAN ALGORITHM 1. For all a; b; c 2 N: a + b = a + c ) b = c 2. For all a; b; c 2 N: ab = ac ) b = c. 3. For all a; b; d 2 N: dja; djb ) dj(a + b) 4. Any a 2 N has finitely many divisors. 5. Any finite set of natural numbers has a biggest element. Although divison of one number by another usually fails we do have the concept of division with remainder. Theorem 1.1.3 (Euclid) Let a; b 2 N with a > b. Then either bja or there exist q; r 2 N such that a = bq + r; r < b: Moreover, q; r are uniquely determined by these (in)equalities. Proof. Suppose b does not divide a. Consider all multiples of b which are less than a. This is a non-empty set, since b < a. Choose the largest multiple and call it bq. Then clearly a − bq < b. Conversely, if we have a multiple qb such that a − bq < b then qb is the largest b-multiple < a. Our theorem follows by taking r = a − bq. 2 Another important concept in the natural numbers are prime numbers. These are natural numbers p > 1 that have only the trivial divisors 1; p. Here are the first few: 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31;::: Most of us have heard about them at a very early age. We also learnt that there are infinitely many of them and that every integer can be written in a unique way as a product of primes. These are properties that are not mentioned in our rules. So one has to prove them, which turns out to be not entirely trivial. This is the beginning of number theory and we will take these proofs up in this chapter. In the history of arithmetic the number 0 was introduced after the natural numbers as the symbol with properties 0 · a = 0 for all a and a + 0 = a for all a. Then came the negative numbers -1,-2,-3,::: with the property that −1 + 1 = 0; −2 + 2 = 0;:::. Their rules of addition and multiplication are uniquely determined if we insist that these rules obey the commutative, associate and distributive laws of addition and multiplication. Including the infamous "minus times minus is plus" which causes so many high school children great headaches. Also in the history of mathematics we see that negative numbers and their arithmetic were only generally accepted at a surprisingly late age, the beginning of the 19th century. F.Beukers, Elementary Number Theory 1.2. GREATEST COMMON DIVISORS 7 From now on we will assume that we have gone through all these formal intro- ductions and we are ready to work with the set of integers Z, which consists of the natural numbers, their opposites and the number 0. The main role of Z is to have extended N to a system in which the operation of subtraction is well-defined for any two elements. One may proceed further by extending Z to a system in which also element (=6 0) divides any other.