Cut the search space for problem 23 in half.

Problem23.hs

@@ -1,4 +1,22 @@
-import Data.List
+-- A perfect number is a number for which the sum of its proper divisors is
+-- exactly equal to the number. For example, the sum of the proper divisors of
+-- 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect
+-- number.
+--
+-- A number n is called deficient if the sum of its proper divisors is less
+-- than n and it is called abundant if this sum exceeds n.
+--
+-- As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest
+-- number that can be written as the sum of two abundant numbers is 24. By
+-- mathematical analysis, it can be shown that all integers greater than 28123
+-- can be written as the sum of two abundant numbers. However, this upper limit
+-- cannot be reduced any further by analysis even though it is known that the
+-- greatest number that cannot be expressed as the sum of two abundant numbers
+-- is less than this limit.
+--
+-- Find the sum of all the positive integers which cannot be written as the sum
+-- of two abundant numbers.
+
 import Data.Maybe
 import Euler
 
@@ -9,6 +27,6 @@ abundant :: [Int]
 abundant = filter isAbundant [1..]
 
 abundantPair :: Int -> Maybe (Int, Int)
-abundantPair n = listToMaybe [ (p,q) | p <- takeWhile (<n) abundant, let q = n - p, isAbundant q ]
+abundantPair n = listToMaybe [ (p,q) | p <- takeWhile (<= (n `div` 2)) abundant, let q = n - p, isAbundant q ]
 
 main = print $ sum $ filter (isNothing . abundantPair) [1..28123]