User:WillNess

I like this one-liner:

--   infinite folding due to Richard Bird
--   double staged primes production due to Melissa O'Neill
--   tree folding idea Heinrich Apfelmus / Dave Bayer 
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 via Y combinator

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:

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

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.