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:

$\mathbb{S}={\mathbb{N}}_2\setminus {\bigcup }_{p\in \mathbb{S}}\{pq:q\in {\mathbb{N}}_p\}$

using standard definition

${\mathbb{N}}_k=\{n\in \mathbb{N}:n\ge k\}$   . . . or,  ${\mathbb{N}}_k=\{k\}\bigcup {\mathbb{N}}_{k+1}$   :) :) .

Trial division sieve is:

$\mathbb{T}=\{n\in {\mathbb{N}}_2:(\mathrm{\forall }p\in \mathbb{T})(2\le p\le \sqrt{n}\Rightarrow \mathrm{\neg }(p\mid n))\}$

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.