Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads . 33 6.3 Diffie-Hellman key exchange . 34 6.4 RSA . 36 7 Quadratic Residues 37 7.1 Which numbers are squares? . 37 7.2 Euler's criterion . 38 7.3 Exercises due March 16 . 40 8 Quadratic Reciprocity 40 8.1 The Legendre symbol . 40 8.2 Some reciprocity laws . 41 8.3 The main quadratic reciprocity law . 42 8.4 The Jacobi symbol . 44 8.5 Exercises due March 23 . 45 9 The Gaussian integers 46 9.1 Motivation and definitions . 46 9.2 The division algorithm and the gcd . 48 9.3 Unique factorization in Z[i]................... 49 9.4 The factorization of rational primes in Z[i]........... 49 9.5 Exercises due March 30 . 50 2 10 Unique factorization and its applications 51 10.1 Pythagorean triples, revisited . 51 10.2 A cubic Diophantinep equation . 51 10.3 The system Z[ −2] ....................... 52 10.4 Examples of the failure of unique factorization . 53 10.5 The Eisenstein integers . 54 10.6 Exercises due April 13 . 56 11 Some analytic number theory 57 P 11.1 p 1=p diverges . 58 11.2 Classes of primes, and their infinitude . 60 P 11.3 p≡±1 (mod 4) 1=p diverges . 61 11.4 Exercises due April 20 . 63 12 Continued fractions and Pell's equation 64 12.1 A closer look at the Euclidean algorithm . 64 12.2 Continued fractions in the large . 67 12.3 Real quadratic irrationalsp and their continued fractions . 68 12.4 Pell's equation and Z[ d].................... 70 12.5 The fundamental unit . .p . 71 12.6 The question of unique factorization for Z[ d]........ 73 12.7 Exercises due April 27 . 74 13 Lagrange's four square theorem 74 13.1 Hamiltonian quaternions . 75 13.2 The Lipschitz quaternions . 77 13.3 The Hurwitz quaternions . 78 13.4 Hurwitz primes . 80 13.5 The end of the proof . 81 1 Basic properties of the integers 1.1 Definitions: Z and Q Number theory is the study of the integers: :::; −3; −2; −1; 0; 1; 2; 3;::: We use the symbol Z to stand for the set of integers. (Z stands for German Zahl, meaning number.) Now might be a good time to review some set- theoretic notations: 3 2 pZ is a true statement, meaning that 3 is a member of the integers, whereas 7 62 Z. 3 We observe that integers can be added, subtracted, and multiplied to produce other integers, but the same cannot be said for division. When we divide integers we create rational numbers, such as 3=7 and −2=3. We write the set of rational numbers as Q, for quotient. The failure of integers to divide each other evenly is so important that we have special notation to express it: for integers a and b, we write ajb to mean that b=a is an integer. In other words, ajb means that there exists c 2 Z such that b = ac. In this case we say that a is a divisor of b, and that b is a multiple of a. Example 1.1.1. The divisors of 12 are 1,2,3,4,6,12 and their negatives. A divisor of a positive integer n is proper if it's positive and not equal to n itself. Thus the proper divisors of 12 are just 1,2,3,4,6. Example 1.1.2. 1 is a divisor of every integer, as is −1. Also, every integer divides 0, since 0 = 0 · a for every a. However, the only multiple of 0 is 0 itself. Proposition 1.1.3. Suppose that a; b; c 2 Z. If ajb and bjc, then ajc. Proof. There exists integers m; n such that b = am and c = bn. Then c = amn, so ajc. The above proposition says that the relation ajb is transitive. Proposition 1.1.4. Suppose a; b; d; x; y 2 Z. If dja and djb, then djax + by. We remark that ax + by is called a linear combination of a and b. Proof. Write a = dm and b = dn, then ax+by = d(mx+ny), so djax+by. A positive integer is prime if it has no proper divisors other than 1. By convention, 1 is not counted as prime. Theorem 1.1.5 (Euclid). There are infinitely many primes. Proof. If there we finitely many primes, then we could list all of them as 1 p1; : : : ; pn. The number N = p1 ··· pn + 1 is divisible by some prime , which must be one of our enumerated primes, say pi. Then pijN but also pijp1 ··· pn. Thus pij(N − p1 ··· pn) = 1, which is absurd. 1Strictly speaking, we don't know this fact yet, but for now we'll take it for granted. 4 Therefore we are guaranteed to never run out of primes. As of January 2018 the largest known prime is 277;232;917 − 1: This is a Mersenne prime, meaning a prime which is one less than a power of two. It is not known if there are infinitely many Mersenne primes. 1.2 The well-ordering principle How do we know that every integer n > 1 is divisible by a prime? An argument might go this way: if n isn't itself prime, then it has a proper divisor n1 > 1. If n1 isn't prime, then it has a proper divisor n2 > 1, and so on. The result is that we get a strictly decreasing sequence of positive integers n > n1 > n2 > : : : , which cannot go on indefinitely. This fact, obvious that it may be, is quite important. We give it a name: The well- ordering principle. Axiom 1.2.1 (The well-ordering principle). 2 A strictly decreasing sequence of positive integers cannot go on indefinitely. Rather than attempt to prove this statement, we take it as an axiom of the system of integers. 1.3 The division algorithm We noted before that the integers are not closed under division. But there is a familiar operation among integers: you can divide one by another to obtain a quotient and a remainder. For instance, when 39 is divided by 5, the quotient is 7 and the remainder is 4. We can check this by verifying that 39 = 5 · 7 + 4. When this is done, the remainder must be less than the number you divided by. It would be incorrect to say that 5 goes into 39 with a quotient of 6 and a remainder of 9, even though 39 = 5 · 6 + 9 is also true. Theorem 1.3.1 (The division algorithm). Let a; b 2 Z, with b > 0. There exists a unique pair of integers q; r 2 Z such that a = bq + r and that 0 ≤ r < b. Of course, if the remainder r is 0, then a = bq and therefore bja. 2There is another formulation: every nonempty subset of the positive integers has a least element. The two formulations are equivalent. 5 Proof. We'll assume that a is positive, the other cases are similar. Consider the sequence a, a − b, a − 2b, a − 3b; : : : . By the well-ordering principle, these cannot all be nonnegative integers. So there is a least one which is nonnegative, call it r = a − bq. If r > b, then a − b(q + 1) = r − b > 0, which contradicts our assumption that r was the least element of our sequence. Therefore r ≤ b. That handles the existence part of the theorem. For uniqueness: if there were another pair q0; r0 such that a = bq + r = bq0 + r0, then r − r0 = b(q0 − q) would be a multiple of b, but since 0 ≤ r; r0 < b, this can only happen if r = r0, which implies q = q0 as well. This proof gives a hint to the \algorithm" part of the division algorithm: to divide 5 into 39, keep subtracting 5 from 39 to get 34, 29, 24, 19, 14, 9, 4, at which point we cannot subtract anymore and 4 is the remainder.