{-# LANGUAGE ScopedTypeVariables #-}
import Euler

-- Pentagonal numbers are generated by the formula, P[n]=n(3n−1)/2.
--
-- Find the pair of pentagonal numbers, P[j] and P[k], for which their sum and
-- difference are pentagonal and D = |P[k] − P[j]| is minimised; what is the value
-- of D?

pentagonals = map (\n -> n * (3*n - 1) `div` 2) [1..]
isPentagonal x = let n = (sqrt (24 * fromIntegral x + 1) + 1) / 6 in n == fromIntegral (floor n) && n > 0

main = print $ minimum $ do
  -- Assumption: The solution will not require considering pentagonals over 10^7.
  b <- takeWhile (< 10000000) pentagonals
  a <- takeWhile (< b) pentagonals
  guard $ isPentagonal (a + b)
  guard $ isPentagonal (b - a)
  return $ b - a