===== Least divisor function =====
==== Function ====

| @#FFBB00: definiendum | @#FFBB00: $ \mathrm{ld}:\mathbb N^+\to\{1\}\cup\mathrm{Prime\ number}  $ |
| @#FFBB00: definiendum | @#FFBB00: $ \mathrm{ld}(n):=\mathrm{min}\left(\mathrm{divisors}(n)\right)  $ |

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=== Code ===
== Haskell ==
<code haskell>
divides :: Integral a => a -> a -> Bool
divides d n = rem n d == 0
</code>

<code haskell>
ld :: Integral a => a -> a
ld n = ldf 2 n

ldf :: Integral a => a -> a -> a
ldf k n | divides k n = k 
      	| k^2 > n     = n
	| otherwise   = ldf (k+1) n
</code>

using [[Set of divisors function]]:
<code haskell>
divides :: Integral a => a -> a -> Bool
divides d n = rem n d == 0
</code>
=== Theorems ===
If $n$ isn't a prime, then $n$ divided by the //least// divisor is some number bigger than $\mathrm{ld}(n)$ and hence
^ $\mathrm{ld}(n)^2\le n$ ^

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=== Subset of ===
[[Surjective function]]
=== Requirements ===
[[Prime number]], [[Set of divisors function]]
=== Code reference ===
[[Set of divisors function]]