COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

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1 COMPSCI 230 Discrete Math January 24, 2017 COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

2 Outline 1 Prime Numbers The Sieve of Eratosthenes Python Implementations GCD and Co-Primes COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

3 Primes Natural numbers greater than 1 with no proper divisors 1, 2, 3, 6 6 but 1, 7 7 Numbers in red are proper divisors Fundamental theorem of arithmetic: 1 < n N n = p i i where the p i s are prime Any natural number greater than 1 can be written as a product of primes Example: = Fundamental in number theory Applied in cryptography, electrical communications, computer chips,... COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

4 Problems About Primes Generate all primes up to n (sieve) Test whether n is prime Is prime? Factor n into primes = You did factor, which subsumes test, so you know both There are much more efficient methods [Composite = not prime] COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

5 The Sieve of Eratosthenes The Sieve of Eratosthenes To generate all primes up to n: List all integers from 2 to n Let p = 2 (smallest prime); set it aside Cross p, 2p, 3p, from the list First remaining number must be prime Why? Primes before it are potential factors, but we removed all their multiples Set p to that, and repeat until no more numbers are left COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

6 The Sieve of Eratosthenes Example: n = COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

7 The Sieve of Eratosthenes The Sieve of Eratosthenes Speedups exist Takes large amounts of storage for nontrivial n Each number is touched at least once Takes at least n steps COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

8 Python Implementations Primes in Python: Caveat There is a pyprimes module in Python that implements several algorithms to generate primes, test for primality, and factor into primes Please do explore pyprimes if interested However, you are required to study how to accomplish the simpler versions of these tasks on your own, without pyprimes COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

9 Python Implementations The Sieve in Python def sieve(n): primes = [] integers = list(range(2, n+1)) while integers: p = integers[0] primes += [p] integers = [k for k in integers if k % p] return primes >>> sieve(40) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37] COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

10 Python Implementations Alternative Implementation Current: Generate all integers up to n For each new prime, delete its multiples Alternative: for k in range(2, n+1): Keep k if it is not a multiple of one of the primes found so far Does not require to make all integers first range(2, n+1) is a generator-like class, not a list Think of a generator as a function that returns the next item if and when you ask for it Main advantage: No long list of integers to store COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

11 Python Implementations Python Generators We would like to say def primesuntiltired(): print('hit return for the next prime,', 'anything else and return to stop') for p in allprimes(): if input() == '': print(p, end = '') else: break allprimes is a generator Can also say prime = allprimes() next(prime) gives 2 next(prime) gives 3... COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

12 Python Implementations Generators A generator is a reentrant function that returns a value every time it is suspended Execution can be suspended and a value returned. If the function is called again, the function is resumed where it had been suspended If there is no more code to execute in the generator, a StopIteration exception is raised [For some generators, this may never happen] Generators enable lazy evaluation COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

13 Python Implementations A Python Prime Generator def allprimes(): primes = [] k = 2 while True: isprime = True for p in primes: if k % p == 0: isprime = False break if isprime: primes += [k] yield k k += 1 Designed as an (infinite) loop At yield: Suspend allprimes Remember state Return argument of yield If allprimes is called again, resume execution right after yield, with the old state No StopIteration is ever raised in this case COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

14 GCD and Co-Primes Greatest Common Divisor Let m, n be positive integers (m, n > 0) GCD(m, n) is the largest positive integer g such that g m and g n GCD(120, 700) = 20 GCD(24, 72) = 24 GCD(a, b) = GCD(b, a) GCD(1, n) = 1 GCD(1, 1) = 1 COMPSCI 230 Discrete Math Prime Numbers January 24, / 15

15 GCD and Co-Primes Euclid s Algorithm Computes GCD(x, y) for x, y > 0 Based on the following recursion (FDM pages 18-19, without proof for now): GCD(x, y) = { y GCD(y, x MOD y) if y x otherwise Examples: GCD(15, 4) = GCD(4, 3) = GCD(3, 1) = 1 GCD(4, 15) = GCD(15, 4) = GCD(4, 3) = = 1 COMPSCI 230 Discrete Math Prime Numbers January 24, / 15