# Difference between revisions of "99 questions/31 to 41"

(P39 solved.) |
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== Problem 39 == |
== Problem 39 == |
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− | <Problem description> |
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+ | A list of prime numbers. |
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⚫ | |||

+ | Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range. |
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− | Example: |
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− | <example in lisp> |
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⚫ | |||

Example in Haskell: |
Example in Haskell: |
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− | <example in Haskell> |
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+ | P29> primesR 10 20 |
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+ | [11,13,17,19] |
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</pre> |
</pre> |
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− | Solution: |
+ | Solution 1: |

<haskell> |
<haskell> |
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⚫ | |||

+ | primesR :: Integral a => a -> a -> [a] |
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+ | primesR a b = filter isPrime [a..b] |
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⚫ | |||

+ | |||

+ | If we are challenged to give all primes in the range between a and b we simply take all number from a up to b and filter the primes out. |
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+ | |||

+ | Solution 2: |
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+ | <haskell> |
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+ | primes :: Integral a => [a] |
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+ | primes = let sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ] in sieve [2..] |
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+ | |||

+ | primesR :: Integral a => a -> a -> [a] |
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+ | primesR a b = takeWhile (<= b) $ dropWhile (< a) primes |
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</haskell> |
</haskell> |
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− | <description of implementation> |
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+ | Another way to compute the claimed list is done by using the ''Sieve of Eratosthenes''. The <hask>primes</hask> function generates a list of all (!) prime numbers using this algorithm and <hask>primesR</hask> filter the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nice the ''Sieve of Eratosthenes'' can be implemented in Haskell :)] |
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== Problem 40 == |
== Problem 40 == |

## Revision as of 19:17, 12 December 2006

These are Haskell translations of Ninety Nine Lisp Problems.

If you want to work on one of these, put your name in the block so we know someone's working on it. Then, change n in your block to the appropriate problem number, and fill in the <Problem description>,<example in lisp>,<example in Haskell>,<solution in haskell> and <description of implementation> fields.

## Problem 31

Determine whether a given integer number is prime.

Example: * (is-prime 7) T Example in Haskell: P31> isPrime 7 True

Solution:

```
isPrime :: Integral a => a -> Bool
isPrime p = all (\n -> p `mod` n /= 0 ) $ takeWhile (\n -> n*n <= x) [2..]
```

Well, a natural number p is a prime number iff no natural number n with n >= 2 and n^2 <= p is a divisor of p. That's exactly what is implemented: we take the list of all integral numbers starting with 2 as long as their square is at most p and check that for all these n there is a remainder concerning the division of p by n.

## Problem 32

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>

## Problem 33

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>

## Problem 34

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>

## Problem 35

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>

## Problem 36

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>

## Problem 37

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>

## Problem 38

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>

## Problem 39

A list of prime numbers.

Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range.

Example in Haskell: P29> primesR 10 20 [11,13,17,19]

Solution 1:

```
primesR :: Integral a => a -> a -> [a]
primesR a b = filter isPrime [a..b]
```

If we are challenged to give all primes in the range between a and b we simply take all number from a up to b and filter the primes out.

Solution 2:

```
primes :: Integral a => [a]
primes = let sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ] in sieve [2..]
primesR :: Integral a => a -> a -> [a]
primesR a b = takeWhile (<= b) $ dropWhile (< a) primes
```

Another way to compute the claimed list is done by using the *Sieve of Eratosthenes*. The `primes`

function generates a list of all (!) prime numbers using this algorithm and `primesR`

filter the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nice the *Sieve of Eratosthenes* can be implemented in Haskell :)]

## Problem 40

<Problem description>

Example: <example in lisp> Example in Haskell: <example in Haskell>

Solution:

```
<solution in haskell>
```

<description of implementation>