# User:WillNess

A perpetual Haskell newbie. I like this one-liner:

--   infinite folding idea due to Richard Bird
--   double staged production idea due to Melissa O'Neill
--   tree folding idea Dave Bayer / improved tree structure
--     Heinrich Apfelmus / simplified formulation Will Ness
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

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.