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:

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

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.