University Taylor Computer Science and Engineering Bill Toll Spring 2009
COS 382 Language Structures Haskell The Sieve of Eratosthenes ASSIGNED: May 4, 2009 DUE: May 13, 2009 SUBMISSION: Submit your Haskell program as a .hs file through moodle. NOTE: This assignment is to be done individually.
Overview
The purpose of this assignment is to write Haskell functions to generate prime numbers. We will take advantage of the lazy evaluation feature of Haskell to generate inifinite lists used in a sieve approach. This assignment is based on an assignment used by Dr. Neil Leslie in the Programming Languages course at Victoria University of Wellington in New Zealand. The Sieve of Eratosthenes is an ancient technique to generate prime numbers dating to the 3rd century BC. The algorithm uses no division but rather generates composite numbers and defines primes as non-composites. A prime number is a number with no divisors other than 1 and itself. A composite number is one with divisors other than 1 and itself.
Technique
We will use 3 lists. The first is a list of all possible primes. The second is a list of composite numbers discovered at a given time. The third is a list of primes discovered at a given time. Our approach will be to look for primes in the sequence {2, 3, 4, 5, . . . } We begin with an empty list of composites and an empty list of primes. Look at the first item in the sequence (2). Since it is not in our list of composites, it is prime. Therefore, add 2 to the the list of primes and add all multiples 2 of (beginning at 4) to our list of composites. Next, look at the first number in the sequence {3, 4, 5, . . . } Since 3 is not in the list of composites, it is prime and is added to the list of primes, All multiples of 3 (beginning with 9) are added to the list of composites. Next, check 4. It is in the list of composites and is not added to the list or primes, nor are any numbers added to the list of composites.
Requirements
Write the following Haskell functions. Prods:: Int [Int] prods n generates the infinite list of products of n beginning with n^2 prods 3 [9, 12, 15, 18, 21, . . . ] mix:: [Int] [Int] [Int] mix takes 2 infinite lists (each assumed to be in order) and merges them into a single infinite list in order with no duplicates mix (prods 5) (prods 7) [25, 30, 35, 40, 45, 49, 50, 55, 56, 60, 63, . . . ] sieve:: [Int] [Int] [Int] sieve implements the Sieve of Eratosthenes. The first ...