User:WillNess

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.