# User:WillNess

### From HaskellWiki

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− | I am a newbie, interested in Haskell. |
+ | A perpetual Haskell newbie. I like ''[http://ideone.com/qpnqe this one-liner]'': |

− | |||

− | I like ''[http://ideone.com/qpnqe this]'': |
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<haskell> |
<haskell> |
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− | -- inifinte folding idea due to Richard Bird |
+ | -- infinite folding idea due to Richard Bird |

-- double staged production idea due to Melissa O'Neill |
-- double staged production idea due to Melissa O'Neill |
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− | -- tree folding idea Dave Bayer / simplified formulation Will Ness |
+ | -- tree folding idea Dave Bayer / improved tree structure |

− | primes = 2 : g (fix g) |
+ | -- Heinrich Apfelmus / simplified formulation Will Ness |

− | where |
+ | primes = 2 : _Y ((3:) . gaps 5 |

− | g ps = 3 : gaps 5 (foldi (\(q:qs) -> (q:) . union qs) |
+ | . foldi (\(x:xs) -> (x:) . union xs) [] |

− | [[p*p, p*p+2*p..] | p <- ps]) |
+ | . map (\p-> [p*p, p*p+2*p..])) |

− | gaps k s@(c:t) |
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− | | k < c = k : gaps (k+2) s -- | k<=c = minus [k,k+2..] s |
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− | | True = gaps (k+2) t -- fused to avoid a space leak |
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− | fix g = xs where xs = g xs -- global defn to avoid space leak |
+ | _Y g = g (_Y g) -- multistage production |

+ | |||

+ | gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c, |
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+ | | k < c = k : gaps (k+2) s -- fused for better performance |
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+ | | otherwise = gaps (k+2) t -- k==c |
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</haskell> |
</haskell> |
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− | <code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. More at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]]. |
+ | <code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. <code>union</code> and more at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]]. |

+ | |||

+ | The constructive definition of primes is the Sieve of Eratosthenes: |
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+ | |||

+ | ::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}</math> |
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+ | using standard definition |
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+ | ::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math>   . . . or,  <math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math>   :) :) . |
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+ | |||

+ | Trial division sieve is: |
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+ | |||

+ | ::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}</math> |
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+ | |||

+ | If you're put off by self-referentiality, just replace <math>\mathbb{S}</math> or <math>\mathbb{T}</math> on the right-hand side of equations with <math>\mathbb{N}_{2}</math>, but even ancient Greeks knew better. |

## Latest revision as of 09:30, 6 August 2013

A perpetual Haskell newbie. I like *this one-liner*:

-- infinite folding idea due to Richard Bird -- double staged production idea due to Melissa O'Neill -- tree folding idea Dave Bayer / improved tree structure -- Heinrich Apfelmus / simplified formulation Will Ness primes = 2 : _Y ((3:) . gaps 5 . foldi (\(x:xs) -> (x:) . union xs) [] . map (\p-> [p*p, p*p+2*p..])) _Y g = g (_Y g) -- multistage production gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c, | k < c = k : gaps (k+2) s -- fused for better performance | otherwise = gaps (k+2) t -- k==c

`foldi`

is on Tree-like folds page. `union`

and more at Prime numbers.

The constructive definition of primes is the Sieve of Eratosthenes:

using standard definition

- . . . or, :) :) .

Trial division sieve is:

If you're put off by self-referentiality, just replace or on the right-hand side of equations with , but even ancient Greeks knew better.