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I'm interested in Haskell.
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A perpetual Haskell newbie. I like ''[http://ideone.com/qpnqe this semi-one-liner]'':
 
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I like ''[http://ideone.com/qpnqe this]'':
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<haskell>
 
<haskell>
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primes = 2 : g (fix g)  
 
primes = 2 : g (fix g)  
 
   where                 
 
   where                 
     g ps = 3 : gaps 5 (foldi (\(q:qs) -> (q:) . union qs)  
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     g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) []
                             [[p*p, p*p+2*p..] | p <- ps])
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                             [[x*x, x*x+2*x..] | x <- xs])
    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
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fix g = xs where xs = g xs       -- global defn to avoid space leak
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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 to avoid a space leak
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  | 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]].
  
Also, the math formula for Sieve of Eratosthenes,
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The constructive definition of primes is the Sieve of Eratosthenes:
  
::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{n p:n \in \mathbb{N}_{p}\}</math>  
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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> &emsp; . . . or, &ensp;<math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> &emsp; :)&emsp;:) .
  
where
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Trial division sieve is:
 
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::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math> &emsp; . . . or, &ensp;<math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> &emsp; :)&emsp;:) .
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Trial division sieve:
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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>
  
::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\not\exists p \in \mathbb{T}) (p\leq \sqrt{n} \and 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.

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:

\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}

using standard definition

\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}   . . . or,  \textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}   :) :) .

Trial division sieve is:

\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}

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