# User:WillNess

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

− | + | ||

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

<haskell> | <haskell> | ||

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primes = 2 : g (fix g) | primes = 2 : g (fix g) | ||

where | where | ||

− | g | + | g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) [] |

− | [[ | + | [[x*x, x*x+2*x..] | x <- xs]) |

− | + | ||

− | + | ||

− | + | ||

− | fix g = xs where xs = g xs | + | fix g = xs where xs = g xs -- global defn to avoid space leak |

+ | |||

+ | gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c, | ||

+ | | k < c = k : gaps (k+2) s -- fused to avoid a space leak | ||

+ | | True = gaps (k+2) t | ||

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

<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]]. | <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 16:54, 19 November 2011

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

-- inifinte folding idea due to Richard Bird -- double staged production idea due to Melissa O'Neill -- tree folding idea Dave Bayer / simplified formulation Will Ness primes = 2 : g (fix g) where g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) [] [[x*x, x*x+2*x..] | x <- xs]) fix g = xs where xs = g xs -- global defn to avoid space leak gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c, | k < c = k : gaps (k+2) s -- fused to avoid a space leak | True = gaps (k+2) t

`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.