diff --git a/Math/NumberTheory/Moduli/SquareRoots.hs b/Math/NumberTheory/Moduli/SquareRoots.hs --- a/Math/NumberTheory/Moduli/SquareRoots.hs +++ b/Math/NumberTheory/Moduli/SquareRoots.hs @@ -1,18 +1,19 @@ -- | -- Module: Math.NumberTheory.Moduli.SquareRoots --- Copyright: (c) 2015 by Dr. Lars Brünjes +-- Copyright: (c) 2016 by Dr. Lars Brünjes -- Licence: MIT --- Maintainer: Dr. Lars Brünjes +-- Maintainer: Dr. Lars Brünjes -- Stability: Provisional -- Portability: portable -- -- This module provides a function to find all square roots of a number modulo another number. -module Math.NumberTheory.Moduli.SquareRoots ( - sqrts ) where +module Math.NumberTheory.Moduli.SquareRoots + ( sqrts + ) where -import Data.List (sort) +import Data.List (sort) import Math.NumberTheory.Primes.Factorisation (factorise) -import Math.NumberTheory.Moduli (chineseRemainder, sqrtModPPList) +import Math.NumberTheory.Moduli (chineseRemainder, sqrtModPPList) chineseRemainders :: [([Integer], Integer)] -> [Integer] chineseRemainders = fst . go where diff --git a/Math/NumberTheory/Moduli/SquareRoots/Test.hs b/Math/NumberTheory/Moduli/SquareRoots/Test.hs --- a/Math/NumberTheory/Moduli/SquareRoots/Test.hs +++ b/Math/NumberTheory/Moduli/SquareRoots/Test.hs @@ -1,8 +1,9 @@ -module Math.NumberTheory.Moduli.SquareRoots.Test ( - prop_sqrtsPP, - prop_sqrts) where +module Math.NumberTheory.Moduli.SquareRoots.Test + ( prop_sqrtsPP + , prop_sqrts + ) where -import Data.Numbers.Primes (primes) +import Data.Numbers.Primes (primes) import Math.NumberTheory.Moduli.SquareRoots (sqrts) import Test.QuickCheck @@ -19,7 +20,7 @@ e <- choose (1, 20) return $ PrimePower (primes !! indexP, e) - shrink (PrimePower (p, e)) = + shrink (PrimePower (p, e)) = map PrimePower $ [(p', e) | p' <- takeWhile (< p) primes] ++ [(p, e') | e' <- [1, e - 1]] newtype ProblemPP = ProblemPP (Integer, PrimePower) deriving (Show, Eq) @@ -30,14 +31,14 @@ pp <- arbitrary a <- choose (0, evalPP pp - 1) return $ ProblemPP (a, pp) - + shrink (ProblemPP (a, pp)) = map ProblemPP $ [(a', pp) | a' <- [0, a - 1]] ++ [(a, pp') | pp' <- shrink pp, evalPP pp' > a] newtype Problem = Problem (Integer, Integer) deriving (Show, Eq) instance Arbitrary Problem where - + arbitrary = do m <- suchThat arbitrary (> 0) a <- choose (0, m - 1) diff --git a/Math/NumberTheory/Pell.hs b/Math/NumberTheory/Pell.hs --- a/Math/NumberTheory/Pell.hs +++ b/Math/NumberTheory/Pell.hs @@ -1,31 +1,32 @@ -- | -- Module: Math.NumberTheory.Pell --- Copyright: (c) 2015 by Dr. Lars Brünjes +-- Copyright: (c) 2016 by Dr. Lars Brünjes -- Licence: MIT --- Maintainer: Dr. Lars Brünjes +-- Maintainer: Dr. Lars Brünjes -- Stability: Provisional -- Portability: portable -- --- This module provides a function to solve generalized Pell Equations, +-- This module provides a function to solve generalized Pell Equations, -- using the "LMM Algorithm" described by John P. Robertson in --- . +-- . -- A /generalized Pell Equation/ is a diophantine equation of the form -- @x^2 - dy^2 = n@, where @d@ is a positive integer which is not a square -- and where @n@ is a non-zero integer. -- We are looking for solutions @(x,y)@, where @x@ and @y@ are non-negative integers. -module Math.NumberTheory.Pell ( - Solution, - solve ) where +module Math.NumberTheory.Pell + ( Solution + , solve + ) where -import Control.Arrow ((***)) -import Data.List (sort, nub) -import Data.Ratio ((%)) -import Data.Set (toList) -import Math.NumberTheory.Moduli.SquareRoots (sqrts) -import Math.NumberTheory.Pell.PQa (PQa(..), period) -import Math.NumberTheory.Powers.Squares (isSquare, integerSquareRoot) -import Math.NumberTheory.Primes.Factorisation (divisors) - +import Control.Arrow ((***)) +import Data.List (sort, nub) +import Data.Ratio ((%)) +import Data.Set (toList) +import Math.NumberTheory.Moduli.SquareRoots (sqrts) +import Math.NumberTheory.Pell.PQa (PQa(..), period) +import Math.NumberTheory.Powers.Squares (isSquare, integerSquareRoot) +import Math.NumberTheory.ArithmeticFunctions (divisors) + fmzs :: Integer -> Integer -> [(Integer, Integer, [Integer])] fmzs d n = map (\f -> let m = n `div` (f * f) in (f, m, zs m)) $ filter (\f -> (n `mod` (f * f)) == 0) $ toList $ divisors n where @@ -36,12 +37,12 @@ am = abs m am2 = floor (am % 2) norm x = if x <= am2 then x else x - am - + getRS :: Integer -> Integer -> Integer -> Maybe (Integer, Integer) -getRS d m z = +getRS d m z = let (_, pqas) = period z (abs m) d - qrs = zipWith (\x y -> (q x, g y, b y)) (tail pqas) pqas + qrs = zipWith (\x y -> (q x, g y, b y)) (tail pqas) pqas rss = map (\(_, r, s) -> (r, s)) $ filter (\(q', _, _) -> abs q' == 1) qrs in case rss of @@ -99,15 +100,15 @@ toMinimal (x, y) = minimum $ map (abs *** abs) $ filter (\(x', y') -> x' * y' >= 0) [(x, y), mul d (r, s) (x, y), mul d (r, -s) (x, y)] -- |@solve d n@ calculates all non-negative integer solutions of the generalized Pell Equation --- x^2 - @d@y^2 = @n@, +-- x^2 - @d@y^2 = @n@, -- where @d@ must be a positive integer which is not a square, -- and @n@ must be a non-zero integer. solve :: Integer -> Integer -> [Solution] -solve d n +solve d n | d <= 0 = error $ "D must be positive, but D == " ++ show d ++ "." | isSquare d = error $ "D must not be a square, but D == " ++ show (integerSquareRoot d) ++ "^2." | n == 0 = error "N must not be zero." - | otherwise = case getMinimalReps d n of + | otherwise = case getMinimalReps d n of (_, []) -> [] ((r, s), xys) -> go xys where go xys' = normalize xys' ++ go (step xys') diff --git a/Math/NumberTheory/Pell/PQa.hs b/Math/NumberTheory/Pell/PQa.hs --- a/Math/NumberTheory/Pell/PQa.hs +++ b/Math/NumberTheory/Pell/PQa.hs @@ -1,27 +1,29 @@ -module Math.NumberTheory.Pell.PQa ( - PQa(..), - pqa, - reduced, - period) where +module Math.NumberTheory.Pell.PQa + ( PQa(..) + , pqa + , reduced + , period + ) where -import Data.Ratio ((%)) +import Data.Ratio ((%)) import Math.NumberTheory.Powers.Squares (isSquare, integerSquareRoot) -data PQa = PQa { - a :: Integer, - b :: Integer, - g :: Integer, - a' :: Integer, - p :: Integer, - q :: Integer } deriving Show - +data PQa = PQa + { a :: Integer + , b :: Integer + , g :: Integer + , a' :: Integer + , p :: Integer + , q :: Integer + } deriving Show + pqa :: Integer -> Integer -> Integer -> [PQa] pqa p0 q0 d | q0 == 0 = error "Q0 must not be zero." | d <= 0 = error "D must be positive." | isSquare d = error $ "D must not be a square, but D == " ++ show dd ++ "^2." - | (p0 * p0 - d) `mod` q0 /= 0 = error $ "P0^2 must be equivalent to D modulo Q0, but " - ++ show p0 ++ "^2 == " ++ show (p0 `mod` q0) ++ " /= " ++ show (d `mod` q0) + | (p0 * p0 - d) `mod` q0 /= 0 = error $ "P0^2 must be equivalent to D modulo Q0, but " + ++ show p0 ++ "^2 == " ++ show (p0 `mod` q0) ++ " /= " ++ show (d `mod` q0) ++ " == " ++ show d ++ " (mod " ++ show q0 ++ ")" | otherwise = go p0 q0 (PQa 0 1 (-p0) undefined undefined undefined) (PQa 1 0 q0 undefined undefined undefined) where @@ -43,7 +45,7 @@ reduced x y dd | y > 0 = (dd >= y - x) && (dd < x + y) && (dd >= x) | otherwise = (dd < y - x) && (dd >= x + y) && (dd < x) - + period :: Integer -> Integer -> Integer -> (Int, [PQa]) period p0 q0 d = u [] 0 $ pqa p0 q0 d where dd = integerSquareRoot d diff --git a/Math/NumberTheory/Pell/Test.hs b/Math/NumberTheory/Pell/Test.hs --- a/Math/NumberTheory/Pell/Test.hs +++ b/Math/NumberTheory/Pell/Test.hs @@ -1,35 +1,21 @@ module Math.NumberTheory.Pell.Test where -import Distribution.TestSuite.QuickCheck (Test, testProperty, testGroup) +import Distribution.TestSuite.QuickCheck (Test, testProperty, testGroup) import Math.NumberTheory.Moduli.SquareRoots.Test (prop_sqrtsPP, prop_sqrts) -import Math.NumberTheory.Pell.Test.Reduced (prop_reduced) -import Math.NumberTheory.Pell.Test.Solve (Problem (..), prop_solves) +import Math.NumberTheory.Pell.Test.Reduced (prop_reduced) +import Math.NumberTheory.Pell.Test.Solve (Problem (..), prop_solves) tests :: IO [Test] -tests = return - [ - testGroup "SquareRoots" - [ - testProperty "sqrtsPP" prop_sqrtsPP, - testProperty "sqrts" prop_sqrts - ], - testGroup "Pell" - [ - testProperty "reduced" prop_reduced, - testProperty "solves 7 9" (prop_solves 100 $ Problem 7 9), - testProperty "solves 5 (-4)" (prop_solves 100 $ Problem 5 (-4)), - testProperty "solves 2 (-7)" (prop_solves 100 $ Problem 2 (-7)), - testProperty "solves" (prop_solves 100000) - ] - ] - --- main :: IO () --- main = do --- test prop_sqrtsPP --- test prop_sqrts --- test prop_reduced --- test (prop_solves 100 $ Problem 7 9) --- test (prop_solves 100 $ Problem 5 (-4)) --- test (prop_solves 100 $ Problem 2 (-7)) --- test (prop_solves 100000) --- +tests = return + [ testGroup "SquareRoots" + [ testProperty "sqrtsPP" prop_sqrtsPP + , testProperty "sqrts" prop_sqrts + ] + , testGroup "Pell" + [ testProperty "reduced" prop_reduced + , testProperty "solves 7 9" (prop_solves 100 $ Problem 7 9) + , testProperty "solves 5 (-4)" (prop_solves 100 $ Problem 5 (-4)) + , testProperty "solves 2 (-7)" (prop_solves 100 $ Problem 2 (-7)) + , testProperty "solves" (prop_solves 100000) + ] + ] diff --git a/Math/NumberTheory/Pell/Test/Reduced.hs b/Math/NumberTheory/Pell/Test/Reduced.hs --- a/Math/NumberTheory/Pell/Test/Reduced.hs +++ b/Math/NumberTheory/Pell/Test/Reduced.hs @@ -1,9 +1,11 @@ -module Math.NumberTheory.Pell.Test.Reduced ( prop_reduced ) where +module Math.NumberTheory.Pell.Test.Reduced + ( prop_reduced + ) where -import Math.NumberTheory.Pell.PQa (reduced) +import Math.NumberTheory.Pell.PQa (reduced) import Math.NumberTheory.Powers.Squares (isSquare, integerSquareRoot) import Test.QuickCheck - + data Triple = Triple Integer Integer Integer deriving (Show, Eq) isProperTriple :: Triple -> Bool @@ -19,27 +21,27 @@ y = (pp - dd) / qq in (x, y) - + isReduced :: Triple -> Bool isReduced t = let (x, y) = toDouble t in (x > 1) && (-1 < y) && (y < 0) - + genTriple :: Gen Triple genTriple = flip suchThat isProperTriple $ do p <- scale (* 2) arbitrary q <- scale (* 2) arbitrary d <- scale (* 3) arbitrary return $ Triple p q d - + instance Arbitrary Triple where arbitrary = oneof $ map (suchThat genTriple) [isReduced, not . isReduced] shrink (Triple p q d) = filter isProperTriple $ [Triple p' q d | p' <- shrink p] ++ [Triple p q' d | q' <- shrink q] ++ [Triple p q d' | d' <- shrink d] - + reduced' :: Triple -> Bool reduced' (Triple p q d) = reduced p q (integerSquareRoot d) - + prop_reduced :: Triple -> Property prop_reduced t@(Triple _ _ d) = counterexample (show $ toDouble t) $ @@ -48,4 +50,4 @@ classify (d <= 100) "d <= 100" $ classify (d > 100) "d > 100" $ reduced' t === isReduced t - + diff --git a/Math/NumberTheory/Pell/Test/Solve.hs b/Math/NumberTheory/Pell/Test/Solve.hs --- a/Math/NumberTheory/Pell/Test/Solve.hs +++ b/Math/NumberTheory/Pell/Test/Solve.hs @@ -1,10 +1,11 @@ -module Math.NumberTheory.Pell.Test.Solve ( - Problem (..), - prop_solves, - naive) where +module Math.NumberTheory.Pell.Test.Solve + ( Problem (..) + , prop_solves + , naive + ) where -import Control.Monad (liftM2) -import Math.NumberTheory.Pell (solve) +import Control.Monad (liftM2) +import Math.NumberTheory.Pell (solve) import Math.NumberTheory.Powers.Squares (isSquare, integerSquareRoot) import Test.QuickCheck @@ -27,21 +28,21 @@ x <- suchThat arbitrary (> 0) let y = integerSquareRoot x elements [-y, y] - + shrinkN :: Integer -> [Integer] shrinkN n = filter (/= 0) $ shrink n - + instance Arbitrary Problem where arbitrary = liftM2 Problem genD genN shrink (Problem d n) = [Problem d' n | d' <- shrinkD d] ++ [Problem d n' | n' <- shrinkN n] - + naive :: Integer -> Integer -> Integer -> [(Integer, Integer)] -naive maxY d n = [(integerSquareRoot $ n + d * y * y, y) | y <- [0..maxY], isSquare $ n + d * y * y] +naive maxY d n = [(integerSquareRoot $ n + d * y * y, y) | y <- [0..maxY], isSquare $ n + d * y * y] prop_solves :: Integer -> Problem -> Property prop_solves limit (Problem d n) = classify (n == 1) "n == 1" $ - classify (n == (-1)) "n == -1" $ + classify (n == (-1)) "n == -1" $ classify (n == 4) "n == 4" $ classify (n == (-4)) "n == -4" $ classify (abs n `notElem` [1, 4]) "|n| /= 1, 4" $ diff --git a/Math/NumberTheory/Pell/Test/Utils.hs b/Math/NumberTheory/Pell/Test/Utils.hs --- a/Math/NumberTheory/Pell/Test/Utils.hs +++ b/Math/NumberTheory/Pell/Test/Utils.hs @@ -1,5 +1,6 @@ -module Math.NumberTheory.Pell.Test.Utils ( - (~~) ) where +module Math.NumberTheory.Pell.Test.Utils + ( (~~) + ) where import Test.QuickCheck diff --git a/dist/build/test-pellStub/test-pellStub-tmp/test-pellStub.hs b/dist/build/test-pellStub/test-pellStub-tmp/test-pellStub.hs deleted file mode 100644 --- a/dist/build/test-pellStub/test-pellStub-tmp/test-pellStub.hs +++ /dev/null @@ -1,5 +0,0 @@ -module Main ( main ) where -import Distribution.Simple.Test.LibV09 ( stubMain ) -import Math.NumberTheory.Pell.Test ( tests ) -main :: IO () -main = stubMain tests diff --git a/pell.cabal b/pell.cabal --- a/pell.cabal +++ b/pell.cabal @@ -2,47 +2,46 @@ -- see http://haskell.org/cabal/users-guide/ name: pell -version: 0.1.0.0 +version: 0.1.1.0 synopsis: Package to solve the Generalized Pell Equation. description: Finds all solutions of the generalized Pell Equation. homepage: https://github.com/brunjlar/pell license: MIT license-file: LICENSE author: Lars Bruenjes -maintainer: lbrunjes@gmx.de -copyright: (c) 2015 by Dr. Lars Brünjes +maintainer: brunjlar@gmail.com +copyright: (c) 2016 by Dr. Lars Brünjes category: Math, Algorithms, Number Theory build-type: Simple extra-source-files: README.md cabal-version: >=1.20.0 library - exposed-modules: Math.NumberTheory.Pell, Math.NumberTheory.Moduli.SquareRoots + exposed-modules: Math.NumberTheory.Pell + , Math.NumberTheory.Moduli.SquareRoots other-modules: Math.NumberTheory.Pell.PQa - -- other-extensions: - build-depends: base >=4.7 && <4.8, - arithmoi, - containers - -- hs-source-dirs: + build-depends: base >=4.7 && <5 + , arithmoi + , containers default-language: Haskell2010 Test-Suite test-pell type: detailed-0.9 test-module: Math.NumberTheory.Pell.Test - other-modules: Math.NumberTheory.Moduli.SquareRoots, - Math.NumberTheory.Moduli.SquareRoots.Test, - Math.NumberTheory.Pell, - Math.NumberTheory.Pell.PQa, - Math.NumberTheory.Pell.Test.Reduced, - Math.NumberTheory.Pell.Test.Solve, - Math.NumberTheory.Pell.Test.Utils - build-depends: base >= 4.7 && <4.8, - arithmoi, - containers, - QuickCheck >= 2.8, - primes, - Cabal >= 1.20.0, - cabal-test-quickcheck + other-modules: Math.NumberTheory.Moduli.SquareRoots + , Math.NumberTheory.Moduli.SquareRoots.Test + , Math.NumberTheory.Pell + , Math.NumberTheory.Pell.PQa + , Math.NumberTheory.Pell.Test.Reduced + , Math.NumberTheory.Pell.Test.Solve + , Math.NumberTheory.Pell.Test.Utils + build-depends: base >= 4.7 && <5 + , arithmoi + , containers + , QuickCheck >= 2.8 + , primes + , Cabal >= 1.20.0 + , cabal-test-quickcheck default-language: Haskell2010 source-repository head @@ -52,4 +51,4 @@ source-repository this type: git location: https://github.com/brunjlar/pell - tag: 0.1.0.0 + tag: 0.1.1.0