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**Additional info for C#: Your Visual Blueprint for Building .Net Applications (With CD-ROM)**

**Example text**

THE PRIME FACTORIZATION ALGORITHM 19 2. (Bool,String) 3. [(Bool,String)] 4. ([Bool],String) 5. Bool -> Bool Test your answers by means of the Hugs command :t. 19 Use the Hugs command :t to find the types of the following predefined functions: 1. head 2. last 3. init 4. fst 5. (++) 6. flip 7. flip (++) Next, supply these functions with arguments of the expected types, and try to guess what these functions do. 7 The Prime Factorization Algorithm Let n be an arbitrary natural number > 1. A prime factorization of n is a list of prime numbers p1 , .

The function ldf used in the definition √ of ld looks for a prime divisor of n by checking k|n for all k with √ 2 k n. In fact, it is enough to check p|n for the primes p with 2 p n. Here are functions ldp and ldpf that perform this more efficient check: ldp :: Integer -> Integer ldp n = ldpf primes1 n ldpf :: [Integer] -> Integer ldpf (p:ps) n | rem n p == 0 | p^2 > n | otherwise -> Integer = p = n = ldpf ps n ldp makes a call to primes1, the list of prime numbers. This is a first illustration of a ‘lazy list’.

19 Use the Hugs command :t to find the types of the following predefined functions: 1. head 2. last 3. init 4. fst 5. (++) 6. flip 7. flip (++) Next, supply these functions with arguments of the expected types, and try to guess what these functions do. 7 The Prime Factorization Algorithm Let n be an arbitrary natural number > 1. A prime factorization of n is a list of prime numbers p1 , . . , pj with the property that p1 · · · · · pj = n. We will show that a prime factorization of every natural number n > 1 exists by producing one by means of the following method of splitting off prime factors: WHILE n = 1 DO BEGIN p := LD(n); n := n END p Here := denotes assignment or the act of giving a variable a new value.