# User:WillNess

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

− | + | ||

− | I like ''[http://ideone.com/qpnqe this]'': | + | |

<haskell> | <haskell> | ||

− | + | -- 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 | ||

</haskell> | </haskell> | ||

− | <code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]]. | + | <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: | ||

+ | |||

+ | ::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}</math> | ||

+ | using standard definition | ||

+ | ::::<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>   :) :) . | ||

+ | |||

+ | Trial division sieve is: | ||

+ | |||

+ | ::::<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> | ||

+ | |||

+ | 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. |

## 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.