{- Copyright (C) 2011 Dr. Alistair Ward This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see <http://www.gnu.org/licenses/>. -} {- | [@AUTHOR@] Dr. Alistair Ward [@DESCRIPTION@] Exports functions related to /perfect powers/. -} module Factory.Math.PerfectPower( -- * Functions maybeSquareNumber, -- ** Predicates isPerfectPower -- isPerfectPowerInt ) where import qualified Data.IntSet import qualified Data.Set import qualified Factory.Math.Power as Math.Power {- | * Returns @(Just . sqrt)@ if the specified integer is a /square number/ (AKA /perfect square/). * <https://en.wikipedia.org/wiki/Square_number>. * <http://mathworld.wolfram.com/SquareNumber.html>. * @(Math.Power.square . sqrt)@ is expensive, so the modulus of the operand is tested first, in an attempt to prove it isn't a /perfect square/. The set of tests, and the valid moduli within each test, are ordered to maximize the rate of failure-detection. -} maybeSquareNumber :: Integral i => i -> Maybe i maybeSquareNumber i -- | i < 0 = Nothing -- This function is performance-sensitive, but this test is neither strictly nor frequently required. | all (\(modulus, valid) -> rem i modulus `elem` valid) [ -- -- Distribution of moduli amongst perfect squares Cumulative failure-detection. (16, [0,1,4,9]), -- All moduli are equally likely. 75% (9, [0,1,4,7]), -- Zero occurs 33%, the others only 22%. 88% (17, [1,2,4,8,9,13,15,16,0]), -- Zero only occurs 5.8%, the others 11.8%. 94% -- These additional tests, aren't always cost-effective. (13, [1,3,4,9,10,12,0]), -- Zero only occurs 7.7%, the others 15.4%. 97% (7, [1,2,4,0]), -- Zero only occurs 14.3%, the others 28.6%. 98% (5, [1,4,0]) -- Zero only occurs 20%, the others 40%. 99% -- ] && fromIntegral iSqrt == sqrt' = Just iSqrt -- CAVEAT: erroneously True for 187598574531033120 (187598574531033121 is square). ] && Math.Power.square iSqrt == i = Just iSqrt | otherwise = Nothing where sqrt' :: Double sqrt' = sqrt $ fromIntegral i iSqrt = round sqrt' {- | * An integer @(> 1)@ which can be expressed as an integral power @(> 1)@ of a smaller /natural/ number. * CAVEAT: /zero/ and /one/ are normally excluded from this set. * <https://en.wikipedia.org/wiki/Perfect_power>. * <http://mathworld.wolfram.com/PerfectPower.html>. * A generalisation of the concept of /perfect squares/, in which only the exponent '2' is significant. -} isPerfectPower :: Integral i => i -> Bool isPerfectPower i | i < Math.Power.square 2 = False | otherwise = i `Data.Set.member` foldr ( \n set -> if n `Data.Set.member` set then set -- else Data.Set.union set . Data.Set.fromDistinctAscList . takeWhile (<= i) . iterate (* n) $ Math.Power.square n else foldr Data.Set.insert set . takeWhile (<= i) . iterate (* n) $ Math.Power.square n -- Faster. ) Data.Set.empty [2 .. round $ sqrt (fromIntegral i :: Double)] {-# NOINLINE isPerfectPower #-} {-# RULES "isPerfectPower/Int" isPerfectPower = isPerfectPowerInt #-} -- | A specialisation of 'isPerfectPower'. isPerfectPowerInt :: Int -> Bool isPerfectPowerInt i | i < Math.Power.square 2 = False | otherwise = i `Data.IntSet.member` foldr ( \n set -> if n `Data.IntSet.member` set then set else foldr Data.IntSet.insert set . takeWhile (<= i) . iterate (* n) $ Math.Power.square n ) Data.IntSet.empty [2 .. round $ sqrt (fromIntegral i :: Double)]