----------------------------------------------------------------
perfect.lhs -- a halting problem in haskell
----------------------------------------------------------------
Perfect number equals the sum of its proper divisors.
Numbers 6, 28, 496, 8128, 33550336... are perfect, but it is
not known if there are any odd perfect numbers.
----------------------------------------------------------------

So does oddPerfect halt or not halt?

----------------------------------------------------------------
Find the smallest odd perfect number
----------------------------------------------------------------

> oddPerfect :: Integer
> oddPerfect = findIt 1
>  where 
>   findIt x 
>    | perfect x = x
>    | otherwise = findIt (x+2)

----------------------------------------------------------------
Check for a perfect number
----------------------------------------------------------------

> perfect :: Integer -> Bool
> perfect x
>  | x < 2 = False
>  | otherwise = sum (properDivisors 1 x) == x
>  where
>   properDivisors :: Integer -> Integer -> [Integer]
>   properDivisors n x
>    | n > x `div` 2 = []
>    | x `mod` n == 0 = n:(properDivisors (n+1) x)
>    | otherwise = properDivisors (n+1) x

-- end of file perfect.lhs