# Difference between revisions of "User:WillNess"

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− | I'm interested in Haskell. |
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<haskell> |
<haskell> |
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g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) [] |
g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) [] |
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[[x*x, x*x+2*x..] | x <- xs]) |
[[x*x, x*x+2*x..] | x <- xs]) |
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− | | k < c = k : gaps (k+2) s -- == minus [k,k+2..] (c:t), k<=c, |
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fix g = xs where xs = g xs -- global defn to avoid space leak |
fix g = xs where xs = g xs -- global defn to avoid space leak |
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+ | | True = gaps (k+2) t |
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</haskell> |
</haskell> |
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<code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. <code>union</code> and more at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]]. |
<code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. <code>union</code> and more at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]]. |
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− | The |
+ | The constructive definition of primes is the Sieve of Eratosthenes: |

::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}</math> |
::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}</math> |
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− | where |
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+ | using standard definition |
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::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math>   . . . or,  <math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math>   :) :) . |
::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math>   . . . or,  <math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math>   :) :) . |
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− | Trial division sieve: |
+ | Trial division sieve is: |

::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}</math> |
::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}</math> |
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− | If you're put off by self-referentiality, just replace <math>\mathbb{S}</math> or <math>\mathbb{T}</math> on the right-hand side of equations with <math>\mathbb{N}_{2}</math>. |
+ | If you're put off by self-referentiality, just replace <math>\mathbb{S}</math> or <math>\mathbb{T}</math> on the right-hand side of equations with <math>\mathbb{N}_{2}</math>, but even ancient Greeks knew better. |

## Revision as of 16:54, 19 November 2011

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:

using standard definition

- . . . or, :) :) .

Trial division sieve is:

If you're put off by self-referentiality, just replace or on the right-hand side of equations with , but even ancient Greeks knew better.