Primes and Factorization

Transcription

1 Primes and Factorization 1

2 A prime number is an integer greater than 1 with no proper divisors. The list begins 2, 3, 5, 7, 11, 13, 19,... See for a wealth of information about primes. 2

3 Primes can be found via the Sieve of Eratosthenes. In the following array, striking out all multiples of 2, 3, 5, 7, 11, 13, and 17 leaves all primes from 2 up to

4 Reason: At each step, the next number not struck out has no prime divisor smaller than it is; it must therefore be prime. There is no need to look at prime factors beyond 17 because bracket = 189 and 19 2 = 361

5 The list goes on and on, though there are arbitrarily large gaps between primes E.g., there are no primes in the range 100! + 2, 100! + 3, 100! + 4,...,100! because these numbers are, respectively, divisible by 2, 3,...,99, 100 4

6 Gaps between successive primes gap from prime n to prime n n 5

7 Density of Primes With π(n) = number of primes less than or equal to n the following graph shows π(n)/n y = π(n) 0.1 n for 2 n n 6

8 Prime density compared with 1 ln n y y = π(n)/n y = 1/ln(n) n Prime number theorem π(n) n 1 ln n for large n 7

9 Legendre (1798) and Gauss (1793) suggested the result. 8

10 Hadamard and de la Vallée Poussin proved the results. Landau (1903) simplified the proof, and Erdös and Selberg (1949) gave a purely number theoretic proof. 9

11 Example Approximately how many primes are there in the range from 1 to a google? 10

12 Solution 1google = By the Prime Number Theorem, π( ) ln( ) = ln 10 = =

13 Prime Determination Example Is prime? 12

14 Solution If is composite, then = k m, then k and m cannot both be greater than = Consequently, trial division by 2, 3, 5, 7, 11,... up to this number will either result in a zero remainder, in which case the number is composite, or, result in non-zero remainders for all these divisions, in which case has no proper factors and is therefore prime. 13

15 12997 = = = = Thus with a relatively small number of divisions, we know that is composite. 14

16 General trial division procedure: to test n for primeness, find n and divide by primes in the range 2 to n until either a zero remainder occurs or until n has been divided by all of these primes without a zero remainder. 15

17 Probabilistic Primality Testing Large prime numbers, such as n = are raw material for some public-key cryptographic methods. 16

18 Given a number such as N = how do we test for primeness? 17

19 N so trial division by all the integers from 2 to would take roughly this many arithmetic steps. If a computer can perform 1 division in one billionth of a second (10 9 sec), then these computations would take = sec = yr, which is many orders of magnitude larger than the expected life of the universe. 18

20 A probabilistic prime test algorithm, such as the Miller-Rabin Test is used to show that the probability N is prime is very close to 1. 19

21 There is an algorithm for determining whether a positive integer with x binary digits is prime whose time comlexity is O(x 7.5 ). This was shown by Agrawal, Kayal, and Saxena in Thus prime testing is a polynomial time problem. 20

22 Largest Known Primes See is the largest known prime as of It was discovered by Michael Cameron, George Woltman, Scott Kurowski, et al. in 2001 as part of GIMPS. It is a Mersenne prime, one of the form 2 p 1, where p is prime. It has digits in its binary representation, and ln(2)/ ln(10) + 1 = digits in its decimal representation. 21

23 Infinitude of Primes In c. 500 B.C., Euclid proved that there are infinitely many primes. Euclid s proof: Suppose, to the contrary, that there are finitely many primes, 2, 3, 5, 7,...,P, where P is the last prime. Then are all composite. P +1,P +2,P +3,... But consider the number N =( P )+1 22

24 This number is not divisible by 2, 3, 5,..., P (remainder is 1 in each case), so by the Fundamental Theorem of Arithmetic, N must itself be prime. However, N is bigger than P, which means N must be composite. This contradiction shows that the assumption of a last prime P was wrong: there is no last prime; there are infinitely many primes. 23

25 Other interesting number theory research In August 2002, Agrawal, Kayal, and Saxena discovered a polynomial time algorithm for determining whether a given integer is prime. See 24

26 Factoring FTA: Every positive integer is factorable into a product of powers of primes. Apart from the order of the factors, the factorization is unique. E.g., = as you can check by multiplying the numbers of the right-hand side. The prime checking procedure can be extended to a sure-fire if inefficient method of factoring an integer n. 25

27 Trial division factorization of an integer n For each prime p in the range 2 to n, check if n MOD p is 0. If YES, then n = p m; test m for divisibility by p in the same way until a non-zero remainder occurs. The number of times a zero remainder occurs is the power of p contained in n. if NO, then go on to the next prime. 26

28 Faster Factorization Algorithms Pollard p 1 method Quadratic Sieve Other fast algorithms exist, but none is known that in general factors integers in polynomial time of the size of the integer. That is, factoring in general is believed to be inherently a time-consuming process. Consequence: If two distinct large (say 100 digits in the decimal representation) primes p and q are multiplied together to form n, then an individual presented with the product n will have a low probability of factoring the number to recover p and q in a reasonable amount of time. 27