God may not play dice with the universe, but something strange is going on with the prime numbers.

Transcription

1 Primes: Definitions God may not play dice with the universe, but something strange is going on with the prime numbers. P. Erdös (attributed by Carl Pomerance) Def: A prime integer is a number whose only factorizations are into itself and 1. Def: A composite is a number n which has divisors other than ±n and ±1. Robert Campbell (UMBC) 1. Number Theory February 16, / 15

2 More Definitions Def: A unit is a number n which which divides 1. Def: n and m are associates if n = um, where u is a unit. Integers: Z Primes are 2, 3, 5,... (and -2, -3, -5,...) Units are ±1 The numbers 2 and -2 are associates, as are 3 and -3, etc Gaussian Integers: Z[ 1] Units are ±1 and ± 1 The numbers 2, 2, 2i, 2i are associates Primes are 3, 7, (1 ± i), (1 ± 2i), (2 ± i),... (and their associates) Robert Campbell (UMBC) 1. Number Theory February 16, / 15

3 How Many Primes? Thm: There are an infinite number of primes. [Euclid, IX.20] proof: Assume not (proof by contradiction) Thus the set of positive primes is finite: {p i i = 1,..., N} Add one to the product of primes: P = ( i N p i) + 1 This number is strictly greater than any of the primes None of the primes divides it (p j prime = p j i N p i, so if p j ( i N p i) + 1, then p j 1) As no prime divides P it must itself be prime... contradicting our construction of P Thus there cannot be a finite number of primes. Robert Campbell (UMBC) 1. Number Theory February 16, / 15

4 Finding Primes How do you find primes? Strategy: Sieve for them. Given a large block of candidate primes Efficiently exclude the composites Strategy: Test for them. Given a likely candidate integer n, run a test which: Shows that it is prime (primality test) Shows that it is composite (compositeness aka pseudoprimality test) Robert Campbell (UMBC) 1. Number Theory February 16, / 15

5 Sieve of Eratosthenes Eratosthenes of Cyrene ( BC) Algorithm: Sieve of Eratosthenes 1 Write down the numbers from 2 to N 2 Start at 2 3 Until you reach N 4 Repeat 1 Let p be the first number not crossed out: p is prime Cross out all multiples of p Robert Campbell (UMBC) 1. Number Theory February 16, / 15

6 Primes and Divisibility Euclid s Lemma: If p is prime and p divides ab, then either p divides a or p divides b [Euclid VII.30] proof: Assume p does not divide a and show that p must divide b. If p does not divide a then gcd(p, a) = 1. So x, y such that ax + py = 1 (Bézout s Identity) Thus b = axb + pyb But p ab, so p axb, and obviously p pyb. Thus p (axb + pyb) = p b. Robert Campbell (UMBC) 1. Number Theory February 16, / 15

7 Prime Factorization Thm: Any integer n > 1 is either prime or factors into a product of primes proof: (Inductive Proof) True for all 1 < k 2, i.e. 2, as it is prime. Make the inductive assumption that it is true for all k less than some bound n and prove that it is true for n. Case n prime: The conclusion is trivially true Case n not prime: So m > 1 so that m n Let n = ml, so l < n and m < n So both m and l are either primes or products of primes. (Ind Hyp) Let m = q j and l = p i Thus n = ml = ( q j )( p i ), a product of primes. Robert Campbell (UMBC) 1. Number Theory February 16, / 15

8 Unique Factorization proof: Only need to prove the uniqueness of factorization. 2 factors uniquely as 2 = 2 1 Assume uniqueness of factorization of integers < n. If n is prime we are done, so assume that n is composite. Suppose n has two factorizations: n = p i = q j Need to prove that {p i } and {q j } are equal up to reordering. As p 1 n we have p 1 q j. k such that p 1 q k (and hence p 1 = q k ) (Euclid s Lemma) So n/p 1 = n/q k. But n/p 1 < n so it has a unique factorization. (inductive hypothesis) Thus n/p 1 = n/q k has the unique factorization i 1 p i Thus {p i : i 1} and {q j : j k} are equal up to reordering. Thus {p i } = {p 1 } {p i : i 1} and {q j } = {q k } {q j : j k} are equal Robert Campbell (UMBC) 1. Number Theory February 16, / 15 Thm: (Fundamental Theorem of Arithmetic) Any natural number n > 1 factors into a product of primes which is unique up to reordering.

9 Distribution of Primes I Thm: (Prime Number Theorem) If the number of primes less than x is denoted π(x), then asymptotically π(x) x log(x) x (log(x) B) Conjecture: π(x) (Legendre, 1796) Conjecture: π(x) li(x) := x 0 (Gauss, 1800?) Proved by Hadamard and de la Vallée-Poussin (1896) dt log(t) Robert Campbell (UMBC) 1. Number Theory February 16, / 15

10 Distribution of Primes II Lemma: If x is a product of primes of the form p i = 4k i + 1, then x has the form x = 4k + 1 proof: Let a and b (not necessarily prime) have the form 4k + 1 So a = 4k a + 1 and b = 4k b + 1 ab = (4k a + 1)(4k b + 1) = 16k a k b + 4k a + 4k b + 1 = 4(4k a k b + k a + k b ) + 1 So ab has the form ab = 4k + 1 where k = 4k a k b + k a + k b Thm: There are an infinite number of primes of the form 4k + 3 proof: Assume that the set of primes of the form 4k + 3 is finite - {p 1, p 2,..., p k } Let m := 4p 1 p 2...p k 1, which has form 4q + 3 As m is odd, every prime p dividing m is odd, so p = 4k + 1 or p = 4k + 3 Not every divisor of m has form p = 4k + 1, as then m would have form m = 4n + 1 So at least one divisor of m has the form p = 4k + 3, so p = p i for some i So p (4p 1...p k m) = 1, a contradiction So there are an infinite number of primes of form p = 4k + 3 Robert Campbell (UMBC) 1. Number Theory February 16, / 15

11 Distribution of Primes III Thm: (Dirichlet s Thm) If gcd(a, b) = 1 then there are an infinite number of primes of the form p = ax + b Robert Campbell (UMBC) 1. Number Theory February 16, / 15

12 Mersenne Primes Def: A Mersenne number is an integer of the form M n = 2 n 1. If M n is prime it is called a Mersenne prime. M 2 = = 3, M 3 = = 7, M 4 = = 15 = (3)(5), M 5 = = 31, M 6 = = 63 = (3 2 )(7), M 7 = = 127 n composite = M n composite. Lucas-Lehmer test for primality Current record prime: 2 232,582,657 1 (Sept 4, 2006) Conjectures & Open Questions: Are there an infinite number of Mersenne primes? Are there any odd perfect numbers? If p is prime is M p square free? Applications: Even Perfect Numbers Finite Fields Robert Campbell (UMBC) 1. Number Theory February 16, / 15

13 Fermat Primes Def: A Fermat number is an integer of the form F n = 2 2n + 1. If F n is prime it is called a Fermat prime. F 1 = = = 5, F 2 = = = 17, F 3 = = = 257, F 4 = = = 65537, F 5 = = = = (641)( ) Conjectures & Open Questions: Are all F n composite for n > 4? Are there an infinite number of Fermat primes? Applications: Compass & Straightedge Constructions Robert Campbell (UMBC) 1. Number Theory February 16, / 15

14 Primes in the Gaussian Integers Robert Campbell (UMBC) 1. Number Theory February 16, / 15

15 Primes in Z[ 5] Robert Campbell (UMBC) 1. Number Theory February 16, / 15