arithmoi 0.9.0.0 → 0.10.0.0

raw patch · 74 files changed

+1343/−1405 lines, 74 filesdep +constraintsdep ~semiringsPVP ok

version bump matches the API change (PVP)

Dependencies added: constraints

Dependency ranges changed: semirings

API changes (from Hackage documentation)

Files

• Changes +59/−0
• Math/NumberTheory/ArithmeticFunctions/Inverse.hs +13/−9
• Math/NumberTheory/ArithmeticFunctions/Moebius.hs +2/−3
• Math/NumberTheory/ArithmeticFunctions/NFreedom.hs +7/−6
• Math/NumberTheory/ArithmeticFunctions/SieveBlock.hs +109/−38
• Math/NumberTheory/ArithmeticFunctions/SieveBlock/Unboxed.hs +0/−131
• Math/NumberTheory/Curves/Montgomery.hs +4/−2
• Math/NumberTheory/Euclidean.hs +41/−142
• Math/NumberTheory/Euclidean/Coprimes.hs +42/−27
• Math/NumberTheory/Moduli/Chinese.hs +4/−4
• Math/NumberTheory/Moduli/Class.hs +9/−1
• Math/NumberTheory/Moduli/DiscreteLogarithm.hs +29/−24
• Math/NumberTheory/Moduli/Equations.hs +13/−11
• Math/NumberTheory/Moduli/PrimitiveRoot.hs +26/−120
• Math/NumberTheory/Moduli/Singleton.hs +314/−0
• Math/NumberTheory/Moduli/Sqrt.hs +23/−29
• Math/NumberTheory/Moduli/SqrtOld.hs +0/−232
• Math/NumberTheory/MoebiusInversion.hs +52/−32
• Math/NumberTheory/MoebiusInversion/Int.hs +3/−1
• Math/NumberTheory/Powers/General.hs +9/−9
• Math/NumberTheory/Prefactored.hs +9/−6
• Math/NumberTheory/Primes.hs +9/−4
• Math/NumberTheory/Primes/Counting/Impl.hs +3/−2
• Math/NumberTheory/Primes/Factorisation.hs +2/−1
• Math/NumberTheory/Primes/Factorisation/Certified.hs +11/−13
• Math/NumberTheory/Primes/Factorisation/Montgomery.hs +24/−21
• Math/NumberTheory/Primes/Factorisation/TrialDivision.hs +1/−1
• Math/NumberTheory/Primes/Sieve.hs +3/−1
• Math/NumberTheory/Primes/Small.hs +20/−0
• Math/NumberTheory/Primes/Testing/Certificates.hs +2/−1
• Math/NumberTheory/Primes/Testing/Certificates/Internal.hs +5/−5
• Math/NumberTheory/Quadratic/EisensteinIntegers.hs +28/−18
• Math/NumberTheory/Quadratic/GaussianIntegers.hs +27/−18
• Math/NumberTheory/Recurrencies.hs +0/−17
• Math/NumberTheory/Recurrencies/Bilinear.hs +0/−38
• Math/NumberTheory/Recurrencies/Linear.hs +0/−14
• Math/NumberTheory/SmoothNumbers.hs +42/−33
• Math/NumberTheory/UniqueFactorisation.hs +0/−13
• Math/NumberTheory/Utils.hs +8/−6
• Math/NumberTheory/Utils/DirichletSeries.hs +3/−2
• Math/NumberTheory/Zeta.hs +10/−7
• Math/NumberTheory/Zeta/Dirichlet.hs +2/−5
• Math/NumberTheory/Zeta/Hurwitz.hs +5/−5
• Math/NumberTheory/Zeta/Riemann.hs +2/−5
• arithmoi.cabal +14/−16
• benchmark/Math/NumberTheory/DiscreteLogarithmBench.hs +14/−10
• benchmark/Math/NumberTheory/InverseBench.hs +3/−1
• benchmark/Math/NumberTheory/PrimesBench.hs +1/−1
• benchmark/Math/NumberTheory/PrimitiveRootsBench.hs +16/−5
• benchmark/Math/NumberTheory/SequenceBench.hs +3/−2
• benchmark/Math/NumberTheory/SmoothNumbersBench.hs +6/−7
• test-suite/Math/NumberTheory/ArithmeticFunctions/InverseTests.hs +3/−2
• test-suite/Math/NumberTheory/ArithmeticFunctions/SieveBlockTests.hs +0/−20
• test-suite/Math/NumberTheory/ArithmeticFunctionsTests.hs +8/−8
• test-suite/Math/NumberTheory/EisensteinIntegersTests.hs +15/−33
• test-suite/Math/NumberTheory/EuclideanTests.hs +76/−24
• test-suite/Math/NumberTheory/GaussianIntegersTests.hs +17/−5
• test-suite/Math/NumberTheory/Moduli/ChineseTests.hs +8/−5
• test-suite/Math/NumberTheory/Moduli/ClassTests.hs +1/−1
• test-suite/Math/NumberTheory/Moduli/DiscreteLogarithmTests.hs +12/−8
• test-suite/Math/NumberTheory/Moduli/EquationsTests.hs +2/−1
• test-suite/Math/NumberTheory/Moduli/PrimitiveRootTests.hs +44/−46
• test-suite/Math/NumberTheory/Moduli/SingletonTests.hs +46/−0
• test-suite/Math/NumberTheory/Moduli/SqrtTests.hs +7/−6
• test-suite/Math/NumberTheory/MoebiusInversion/IntTests.hs +0/−52
• test-suite/Math/NumberTheory/MoebiusInversionTests.hs +12/−10
• test-suite/Math/NumberTheory/PrefactoredTests.hs +2/−2
• test-suite/Math/NumberTheory/Primes/CountingTests.hs +1/−1
• test-suite/Math/NumberTheory/Primes/FactorisationTests.hs +9/−8
• test-suite/Math/NumberTheory/Recurrences/BilinearTests.hs +1/−1
• test-suite/Math/NumberTheory/SmoothNumbersTests.hs +30/−27
• test-suite/Math/NumberTheory/TestUtils.hs +6/−38
• test-suite/Math/NumberTheory/TestUtils/Wrappers.hs +8/−3
• test-suite/Test.hs +3/−5

@@ -1,3 +1,62 @@+0.10.0.0+    This release supports GHC 8.0, 8.2, 8.4, 8.6 and 8.8.++    Breaking changes:++        Move 'Euclidean' type class to 'semirings' package (#168).+        Embrace the new 'Semiring' -> 'GcdDomain' -> 'Euclidean' hierarchy+        of classes, refining 'Num' and 'Integral' constraints.++        Deprecate 'Math.NumberTheory.Primes.Factorisation', use+        'Math.NumberTheory.Primes.factorise' instead. Deprecate+        'Math.NumberTheory.Primes.Sieve', use 'Enum' instance instead.+        Deprecate 'Math.NumberTheory.Primes.Factorisation.Certified' and+        'Math.NumberTheory.Primes.Testing.Certificates'.++        Remove deprecated earlier 'Math.NumberTheory.Recurrencies.*'+        and 'Math.NumberTheory.UniqueFactorisation' modules.+        Use 'Math.NumberTheory.Recurrences.*' and 'Math.NumberTheory.Primes'+        instead.++        Remove deprecated earlier an old interface of 'Math.NumberTheory.Moduli.Sqrt'.++        Reshuffle exports from 'Math.NumberTheory.Zeta', do not advertise+        its submodules as available to import.++        Add a proxy argument storing vector's flavor to+        'Math.NumberTheory.MoebiusInversion.{generalInversion,totientSum}'.+        Deprecate 'Math.NumberTheory.MoebiusInversion.Int'.++        Deprecate 'Math.NumberTheory.SmoothNumbers.{fromSet,fromSmoothUpperBound}'.+        Use 'Math.NumberTheory.SmoothNumbers.fromList' instead.+        Deprecate 'Math.NumberTheory.SmoothNumbers.smoothOverInRange' in favor+        of 'smoothOver' and 'Math.NumberTheory.SmoothNumbers.smoothOverInRange'+        in favor of 'isSmooth'.++        'solveQuadratic' and 'sqrtsMod' require an additional argument: a singleton+        linking a type-level modulo with a term-level factorisation (#169).++    New features:++        The machinery of cyclic groups, primitive roots and discrete logarithms+        has been completely overhauled and rewritten using singleton types (#169).++        There is also a new singleton type, linking a type-level modulo with+        a term-level factorisation. It allows both to have a nicely-typed API+        of `Mod m` and avoid repeating factorisations (#169).++        Refer to a brand new module 'Math.NumberTheory.Moduli.Singleton' for details.++        Add a new function 'factorBack'.++    Improvements:++        Add 'Ord SomeMod' instance (#165).++        Generalize 'sieveBlock' to handle any flavor of 'Vector' (#164).++        Add Semiring and Ring instances for Eisenstein and Gaussian integers.+ 0.9.0.0     This release supports GHC 8.0, 8.2, 8.4 and 8.6. 

@@ -9,6 +9,8 @@ -- <https://www.emis.de/journals/JIS/VOL19/Alekseyev/alek5.pdf Computing the Inverses, their Power Sums, and Extrema for Euler’s Totient and Other Multiplicative Functions> -- by M. A. Alekseyev. +{-# LANGUAGE CPP                 #-}+{-# LANGUAGE FlexibleContexts    #-} {-# LANGUAGE RankNTypes          #-} {-# LANGUAGE ScopedTypeVariables #-} @@ -25,13 +27,16 @@   ) where  import Prelude hiding (rem, quot)+import Data.Bits (Bits) import Data.List import Data.Map (Map) import qualified Data.Map as M import Data.Maybe import Data.Ord (Down(..))+#if __GLASGOW_HASKELL__ < 803 import Data.Semigroup-import Data.Semiring (Semiring(..))+#endif+import Data.Semiring (Semiring(..), Mul(..)) import Data.Set (Set) import qualified Data.Set as S import Numeric.Natural@@ -41,7 +46,6 @@ import Math.NumberTheory.Logarithms import Math.NumberTheory.Powers import Math.NumberTheory.Primes-import Math.NumberTheory.Primes.Sieve (primes) import Math.NumberTheory.Utils.DirichletSeries (DirichletSeries) import qualified Math.NumberTheory.Utils.DirichletSeries as DS import Math.NumberTheory.Utils.FromIntegral@@ -87,7 +91,7 @@  -- | See section 5.2 of the paper. invSigma-  :: forall a. (Euclidean a, Integral a, UniqueFactorisation a)+  :: forall a. (Euclidean a, Integral a, UniqueFactorisation a, Enum (Prime a), Bits a)   => [(Prime a, Word)]   -- ^ Factorisation of a value of the sum-of-divisors function   -> [PrimePowers a]@@ -129,7 +133,7 @@     pksSmall :: Map (Prime a) (Set Word)     pksSmall = M.fromDistinctAscList       [ (p, pows)-      | p <- takeWhile ((< lim) . unPrime) primes+      | p <- [nextPrime 2 .. precPrime lim]       , let pows = doPrime p       , not (null pows)       ]@@ -156,7 +160,7 @@ -- This allows us to crop resulting Dirichlet series (see 'filter' calls -- in 'invertFunction' below) at the end of each batch, saving time and memory. strategy-  :: forall a c. (Euclidean c, Ord c)+  :: forall a c. (GcdDomain c, Ord c)   => ArithmeticFunction a c   -- ^ Arithmetic function, which we aim to inverse   -> [(Prime c, Word)]@@ -177,7 +181,7 @@       -> ([PrimePowers a], (Maybe (Prime c, Word), [PrimePowers a]))     go ts (p, k) = (rs, (Just (p, k), qs))       where-        predicate (PrimePowers q ls) = any (\l -> g (f q l) `rem` unPrime p == 0) ls+        predicate (PrimePowers q ls) = any (\l -> isJust $ g (f q l) `divide` unPrime p) ls         (qs, rs) = partition predicate ts  -- | Main workhorse.@@ -298,7 +302,7 @@ -- >>> unMaxWord (inverseSigma MaxWord 120) -- 95 inverseSigma-  :: (Semiring b, Euclidean a, UniqueFactorisation a, Integral a)+  :: (Semiring b, Euclidean a, UniqueFactorisation a, Integral a, Enum (Prime a), Bits a)   => (a -> b)   -> a   -> b@@ -363,8 +367,8 @@  -- | Helper to extract a set of preimages for 'inverseTotient' or 'inverseSigma'. asSetOfPreimages-  :: (Euclidean a, Integral a)+  :: (Ord a, Semiring a)   => (forall b. Semiring b => (a -> b) -> a -> b)   -> a   -> S.Set a-asSetOfPreimages f = S.mapMonotonic getProduct . f (S.singleton . Product)+asSetOfPreimages f = S.mapMonotonic getMul . f (S.singleton . Mul)

@@ -37,8 +37,7 @@ import Unsafe.Coerce  import Math.NumberTheory.Powers.Squares (integerSquareRoot)-import Math.NumberTheory.Primes (unPrime)-import Math.NumberTheory.Primes.Sieve (primes)+import Math.NumberTheory.Primes import Math.NumberTheory.Utils.FromIntegral (wordToInt)  import Math.NumberTheory.Logarithms@@ -161,7 +160,7 @@     -- Bit fiddling in 'mapper' is correct only     -- if all sufficiently small (<= 191) primes has been sieved out.     ps :: [Int]-    ps = takeWhile (<= (191 `max` integerSquareRoot highIndex)) $ map unPrime primes+    ps = map unPrime [nextPrime 2 .. precPrime (191 `max` integerSquareRoot highIndex)]      mapper :: Int -> Word8 -> Word8     mapper ix val

@@ -7,6 +7,7 @@ -- N-free number generation. -- +{-# LANGUAGE FlexibleContexts    #-} {-# LANGUAGE ScopedTypeVariables #-}  module Math.NumberTheory.ArithmeticFunctions.NFreedom@@ -17,13 +18,13 @@  import Control.Monad                         (forM_) import Control.Monad.ST                      (runST)+import Data.Bits (Bits) import Data.List                             (scanl') import qualified Data.Vector.Unboxed         as U import qualified Data.Vector.Unboxed.Mutable as MU  import Math.NumberTheory.Powers.Squares      (integerSquareRoot)-import Math.NumberTheory.Primes              (unPrime)-import Math.NumberTheory.Primes.Sieve        (primes)+import Math.NumberTheory.Primes import Math.NumberTheory.Utils.FromIntegral  (wordToInt)  -- | Evaluate the `Math.NumberTheory.ArithmeticFunctions.isNFree` function over a block.@@ -42,7 +43,7 @@ -- >>> sieveBlockNFree 2 1 10 -- [True,True,True,False,True,True,True,False,False,True] sieveBlockNFree-  :: forall a . Integral a+  :: forall a. (Integral a, Enum (Prime a), Bits a, UniqueFactorisation a)   => Word   -- ^ Power whose @n@-freedom will be checked.   -> a@@ -82,14 +83,14 @@     highIndex = lowIndex + len - 1      ps :: [a]-    ps = takeWhile (<= integerSquareRoot highIndex) $ map unPrime primes+    ps = if highIndex < 4 then [] else map unPrime [nextPrime 2 .. precPrime (integerSquareRoot highIndex)]  -- | For a given nonnegative integer power @n@, generate all @n@-free -- numbers in ascending order, starting at @1@. -- -- When @n@ is @0@ or @1@, the resulting list is @[1]@. nFrees-    :: forall a. Integral a+    :: forall a. (Integral a, Bits a, UniqueFactorisation a, Enum (Prime a))     => Word     -- ^ Power @n@ to be used to generate @n@-free numbers.     -> [a]@@ -125,7 +126,7 @@ -- -- As with @nFrees@, passing @n = 0, 1@ results in an empty list. nFreesBlock-    :: forall a . Integral a+    :: forall a . (Integral a, Bits a, UniqueFactorisation a, Enum (Prime a))     => Word     -- ^ Power @n@ to be used to generate @n@-free numbers.     -> a

@@ -9,7 +9,6 @@ --  {-# LANGUAGE BangPatterns        #-}-{-# LANGUAGE CPP                 #-} {-# LANGUAGE MagicHash           #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE UnboxedTuples       #-}@@ -24,23 +23,60 @@   , sieveBlockMoebius   ) where -import Control.Monad (forM_)+import Control.Monad (forM_, when) import Control.Monad.ST (runST)+import Data.Bits import Data.Coerce+import qualified Data.Vector.Generic as G+import qualified Data.Vector.Generic.Mutable as MG import qualified Data.Vector as V-import qualified Data.Vector.Mutable as MV+import qualified Data.Vector.Unboxed as U+import qualified Data.Vector.Unboxed.Mutable as MU import GHC.Exts  import Math.NumberTheory.ArithmeticFunctions.Class-import Math.NumberTheory.ArithmeticFunctions.Moebius (sieveBlockMoebius)-import Math.NumberTheory.ArithmeticFunctions.SieveBlock.Unboxed-import Math.NumberTheory.Logarithms (integerLogBase')-import Math.NumberTheory.Primes.Sieve (primes)+import Math.NumberTheory.ArithmeticFunctions.Moebius (Moebius, sieveBlockMoebius)+import Math.NumberTheory.Logarithms (wordLog2, integerLogBase')+import Math.NumberTheory.Primes import Math.NumberTheory.Primes.Types import Math.NumberTheory.Powers.Squares (integerSquareRoot)-import Math.NumberTheory.Utils (splitOff#)+import Math.NumberTheory.Utils (splitOff) import Math.NumberTheory.Utils.FromIntegral (wordToInt, intToWord) +-- | A record, which specifies a function to evaluate over a block.+--+-- For example, here is a configuration for the totient function:+--+-- > SieveBlockConfig+-- >   { sbcEmpty                = 1+-- >   , sbcFunctionOnPrimePower = \p a -> (unPrime p - 1) * unPrime p ^ (a - 1)+-- >   , sbcAppend               = (*)+-- >   }+data SieveBlockConfig a = SieveBlockConfig+  { sbcEmpty                :: a+    -- ^ value of a function on 1+  , sbcFunctionOnPrimePower :: Prime Word -> Word -> a+    -- ^ how to evaluate a function on prime powers+  , sbcAppend               :: a -> a -> a+    -- ^ how to combine values of a function on coprime arguments+  }++-- | Create a config for a multiplicative function from its definition on prime powers.+multiplicativeSieveBlockConfig :: Num a => (Prime Word -> Word -> a) -> SieveBlockConfig a+multiplicativeSieveBlockConfig f = SieveBlockConfig+  { sbcEmpty                = 1+  , sbcFunctionOnPrimePower = f+  , sbcAppend               = (*)+  }++-- | Create a config for an additive function from its definition on prime powers.+additiveSieveBlockConfig :: Num a => (Prime Word -> Word -> a) -> SieveBlockConfig a+additiveSieveBlockConfig f = SieveBlockConfig+  { sbcEmpty                = 0+  , sbcFunctionOnPrimePower = f+  , sbcAppend               = (+)+  }+ -- | 'runFunctionOverBlock' @f@ @x@ @l@ evaluates an arithmetic function -- for integers between @x@ and @x+l-1@ and returns a vector of length @l@. -- It completely avoids factorisation, so it is asymptotically faster than@@ -61,7 +97,7 @@   -> Word   -> Word   -> V.Vector a-runFunctionOverBlock (ArithmeticFunction f g) = (V.map g .) . sieveBlock SieveBlockConfig+runFunctionOverBlock (ArithmeticFunction f g) = (G.map g .) . sieveBlock SieveBlockConfig   { sbcEmpty                = mempty   , sbcAppend               = mappend   , sbcFunctionOnPrimePower = coerce f@@ -72,10 +108,6 @@ -- -- Based on Algorithm M of <https://arxiv.org/pdf/1305.1639.pdf Parity of the number of primes in a given interval and algorithms of the sublinear summation> by A. V. Lelechenko. See Lemma 2 on p. 5 on its algorithmic complexity. For the majority of use-cases its time complexity is O(x^(1+ε)). ----- 'sieveBlock' is similar to 'sieveBlockUnboxed' up to flavour of 'Data.Vector',--- but is typically 7x-10x slower and consumes 3x memory.--- Use unboxed version whenever possible.--- -- For example, following code lists smallest prime factors: -- -- >>> sieveBlock (SieveBlockConfig maxBound (\p _ -> unPrime p) min) 2 10@@ -86,12 +118,14 @@ -- >>> sieveBlock (SieveBlockConfig [] (\p k -> [(unPrime p, k)]) (++)) 2 10 -- [[(2,1)],[(3,1)],[(2,2)],[(5,1)],[(2,1),(3,1)],[(7,1)],[(2,3)],[(3,2)],[(2,1),(5,1)],[(11,1)]] sieveBlock-  :: SieveBlockConfig a+  :: forall v a.+     G.Vector v a+  => SieveBlockConfig a   -> Word   -> Word-  -> V.Vector a-sieveBlock _ _ 0 = V.empty-sieveBlock (SieveBlockConfig empty f append) lowIndex' len' = runST $ do+  -> v a+sieveBlock _ _ 0 = G.empty+sieveBlock (SieveBlockConfig empty f append) !lowIndex' len' = runST $ do      let lowIndex :: Int         lowIndex = wordToInt lowIndex'@@ -99,35 +133,72 @@         len :: Int         len = wordToInt len' -    as <- V.unsafeThaw $ V.enumFromN lowIndex' len-    bs <- MV.replicate len empty--    let highIndex :: Int+        highIndex :: Int         highIndex = lowIndex + len - 1 +        highIndex' :: Word+        highIndex' = intToWord highIndex+         ps :: [Int]-        ps = takeWhile (<= integerSquareRoot highIndex) $ map unPrime primes+        ps = if highIndex < 4 then [] else map unPrime [nextPrime 2 .. precPrime (integerSquareRoot highIndex)] -    forM_ ps $ \p -> do+    as <- MU.replicate len 1+    bs <- MG.replicate len empty -      let p# :: Word#-          !p'@(W# p#) = intToWord p+    let doPrime 2 = do+          let fs = V.generate (wordLog2 highIndex')+                (\k -> f (Prime 2) (intToWord k + 1))+              npLow  = (lowIndex' + 1) `shiftR` 1+              npHigh = highIndex'      `shiftR` 1+          forM_ [npLow .. npHigh] $ \np@(W# np#) -> do+            let ix = wordToInt (np `shiftL` 1) - lowIndex :: Int+                tz = I# (word2Int# (ctz# np#))+            MU.unsafeModify as (\x -> x `shiftL` (tz + 1)) ix+            MG.unsafeModify bs (\y -> y `append` V.unsafeIndex fs tz) ix -          fs = V.generate-            (integerLogBase' (toInteger p) (toInteger highIndex))-            (\k -> f (Prime p') (intToWord k + 1))+        doPrime p = do+          let p' = intToWord p+              f0 = f (Prime p') 1+              logp = integerLogBase' (toInteger p) (toInteger highIndex) - 1+              fs = V.generate logp (\k -> f (Prime p') (intToWord k + 2))+              npLow  = (lowIndex + p - 1) `quot` p+              npHigh = highIndex          `quot` p -          offset :: Int-          offset = negate lowIndex `mod` p+          forM_ [npLow .. npHigh] $ \np -> do+            let !(I# ix#) = np * p - lowIndex+                (q, r) = np `quotRem` p+            if r /= 0+            then do+              MU.unsafeModify as (\x -> x * p')        (I# ix#)+              MG.unsafeModify bs (\y -> y `append` f0) (I# ix#)+            else do+              let (pow, _) = splitOff p q+              MU.unsafeModify as (\x -> x * p' ^ (pow + 2))                          (I# ix#)+              MG.unsafeModify bs (\y -> y `append` V.unsafeIndex fs (wordToInt pow)) (I# ix#) -      forM_ [offset, offset + p .. len - 1] $ \ix -> do-        W# a# <- MV.unsafeRead as ix-        let !(# pow#, a'# #) = splitOff# p# (a# `quotWord#` p#)-        MV.unsafeWrite as ix (W# a'#)-        MV.unsafeModify bs (\y -> y `append` V.unsafeIndex fs (I# (word2Int# pow#))) ix+    forM_ ps doPrime      forM_ [0 .. len - 1] $ \k -> do-      a <- MV.unsafeRead as k-      MV.unsafeModify bs (\b -> if a /= 1 then b `append` f (Prime a) 1 else b) k+      a <- MU.unsafeRead as k+      let a' = intToWord (k + lowIndex)+      when (a /= a') $+        MG.unsafeModify bs (\b -> b `append` f (Prime $ a' `quot` a) 1) k -    V.unsafeFreeze bs+    G.unsafeFreeze bs++-- | This is 'sieveBlock' specialized to unboxed vectors.+--+-- >>> sieveBlockUnboxed (SieveBlockConfig 1 (\_ a -> a + 1) (*)) 1 10+-- [1,2,2,3,2,4,2,4,3,4]+sieveBlockUnboxed+  :: U.Unbox a+  => SieveBlockConfig a+  -> Word+  -> Word+  -> U.Vector a+sieveBlockUnboxed = sieveBlock++{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Int  -> Word -> Word -> U.Vector Int  #-}+{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Word -> Word -> Word -> U.Vector Word #-}+{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Bool -> Word -> Word -> U.Vector Bool #-}+{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Moebius -> Word -> Word -> U.Vector Moebius #-}

@@ -1,131 +0,0 @@--- |--- Module:      Math.NumberTheory.ArithmeticFunctions.SieveBlock.Unboxed--- Copyright:   (c) 2017 Andrew Lelechenko--- Licence:     MIT--- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>------ Bulk evaluation of arithmetic functions without factorisation--- of arguments.-----{-# LANGUAGE BangPatterns        #-}-{-# LANGUAGE MagicHash           #-}-{-# LANGUAGE ScopedTypeVariables #-}-{-# LANGUAGE UnboxedTuples       #-}--module Math.NumberTheory.ArithmeticFunctions.SieveBlock.Unboxed-  ( SieveBlockConfig(..)-  , multiplicativeSieveBlockConfig-  , additiveSieveBlockConfig-  , sieveBlockUnboxed-  ) where--import Control.Monad (forM_)-import Control.Monad.ST (runST)-import qualified Data.Vector.Unboxed as V-import qualified Data.Vector.Unboxed.Mutable as MV-import GHC.Exts--import Math.NumberTheory.ArithmeticFunctions.Moebius (Moebius)-import Math.NumberTheory.Logarithms (integerLogBase')-import Math.NumberTheory.Primes.Sieve (primes)-import Math.NumberTheory.Primes.Types (Prime(..))-import Math.NumberTheory.Powers.Squares (integerSquareRoot)-import Math.NumberTheory.Utils (splitOff#)-import Math.NumberTheory.Utils.FromIntegral (wordToInt, intToWord)---- | A record, which specifies a function to evaluate over a block.------ For example, here is a configuration for the totient function:------ > SieveBlockConfig--- >   { sbcEmpty                = 1--- >   , sbcFunctionOnPrimePower = \p a -> (unPrime p - 1) * unPrime p ^ (a - 1)--- >   , sbcAppend               = (*)--- >   }-data SieveBlockConfig a = SieveBlockConfig-  { sbcEmpty                :: a-    -- ^ value of a function on 1-  , sbcFunctionOnPrimePower :: Prime Word -> Word -> a-    -- ^ how to evaluate a function on prime powers-  , sbcAppend               :: a -> a -> a-    -- ^ how to combine values of a function on coprime arguments-  }---- | Create a config for a multiplicative function from its definition on prime powers.-multiplicativeSieveBlockConfig :: Num a => (Prime Word -> Word -> a) -> SieveBlockConfig a-multiplicativeSieveBlockConfig f = SieveBlockConfig-  { sbcEmpty                = 1-  , sbcFunctionOnPrimePower = f-  , sbcAppend               = (*)-  }---- | Create a config for an additive function from its definition on prime powers.-additiveSieveBlockConfig :: Num a => (Prime Word -> Word -> a) -> SieveBlockConfig a-additiveSieveBlockConfig f = SieveBlockConfig-  { sbcEmpty                = 0-  , sbcFunctionOnPrimePower = f-  , sbcAppend               = (+)-  }---- | Evaluate a function over a block in accordance to provided configuration.--- Value of @f@ at 0, if zero falls into block, is undefined.------ Based on Algorithm M of <https://arxiv.org/pdf/1305.1639.pdf Parity of the number of primes in a given interval and algorithms of the sublinear summation> by A. V. Lelechenko. See Lemma 2 on p. 5 on its algorithmic complexity. For the majority of use-cases its time complexity is O(x^(1+ε)).------ For example, here is an analogue of divisor function 'Math.NumberTheory.ArithmeticFunctions.tau':------ >>> sieveBlockUnboxed (SieveBlockConfig 1 (\_ a -> a + 1) (*)) 1 10--- [1,2,2,3,2,4,2,4,3,4]-sieveBlockUnboxed-  :: V.Unbox a-  => SieveBlockConfig a-  -> Word-  -> Word-  -> V.Vector a-sieveBlockUnboxed _ _ 0 = V.empty-sieveBlockUnboxed (SieveBlockConfig empty f append) lowIndex' len' = runST $ do--    let lowIndex :: Int-        lowIndex = wordToInt lowIndex'--        len :: Int-        len = wordToInt len'--    as <- V.unsafeThaw $ V.enumFromN lowIndex' len-    bs <- MV.replicate len empty--    let highIndex :: Int-        highIndex = lowIndex + len - 1--        ps :: [Int]-        ps = takeWhile (<= integerSquareRoot highIndex) $ map unPrime primes--    forM_ ps $ \p -> do--      let p# :: Word#-          !p'@(W# p#) = intToWord p--          fs = V.generate-            (integerLogBase' (toInteger p) (toInteger highIndex))-            (\k -> f (Prime p') (intToWord k + 1))--          offset :: Int-          offset = negate lowIndex `mod` p--      forM_ [offset, offset + p .. len - 1] $ \ix -> do-        W# a# <- MV.unsafeRead as ix-        let !(# pow#, a'# #) = splitOff# p# (a# `quotWord#` p#)-        MV.unsafeWrite as ix (W# a'#)-        MV.unsafeModify bs (\y -> y `append` V.unsafeIndex fs (I# (word2Int# pow#))) ix--    forM_ [0 .. len - 1] $ \k -> do-      a <- MV.unsafeRead as k-      MV.unsafeModify bs (\b -> if a /= 1 then b `append` f (Prime a) 1 else b) k--    V.unsafeFreeze bs--{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Int  -> Word -> Word -> V.Vector Int  #-}-{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Word -> Word -> Word -> V.Vector Word #-}-{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Bool -> Word -> Word -> V.Vector Bool #-}-{-# SPECIALIZE sieveBlockUnboxed :: SieveBlockConfig Moebius -> Word -> Word -> V.Vector Moebius #-}

@@ -4,7 +4,8 @@ -- Licence:     MIT -- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com> ----- Arithmetic on Montgomery elliptic curve.+-- Arithmetic on Montgomery elliptic curves.+-- This is an internal module, exposed only for purposes of testing. --  {-# LANGUAGE BangPatterns        #-}@@ -15,6 +16,7 @@ {-# LANGUAGE ScopedTypeVariables #-}  {-# OPTIONS_GHC -fno-warn-type-defaults #-}+{-# OPTIONS_HADDOCK hide #-}  module Math.NumberTheory.Curves.Montgomery   ( Point@@ -68,7 +70,7 @@   Point _ 0 == Point _ 0 = True   Point _ 0 == _         = False   _         == Point _ 0 = False-  p@(Point x1 z1) == Point x2 z2 = let n = pointN p in x1 * z2 `mod` n == x2 * z1 `mod` n+  p@(Point x1 z1) == Point x2 z2 = let n = pointN p in (x1 * z2 - x2 * z1) `rem` n == 0  -- | For debugging. instance (KnownNat a24, KnownNat n) => Show (Point a24 n) where

@@ -14,151 +14,50 @@ {-# LANGUAGE ScopedTypeVariables        #-}  module Math.NumberTheory.Euclidean-  ( Euclidean(..)+  ( GcdDomain(..)+  , Euclidean(..)   , WrappedIntegral(..)+  , extendedGCD+  , isUnit   ) where  import Prelude hiding (divMod, div, gcd, lcm, mod, quotRem, quot, rem)-import qualified Prelude as P--import GHC.Exts-import GHC.Integer.GMP.Internals-import Numeric.Natural---- | A class to represent a Euclidean domain,--- which is basically an 'Integral' without 'toInteger'.-class (Eq a, Num a) => Euclidean a where-  -- | When restriced to a subring of the Euclidean domain @a@ isomorphic to-  -- @Integer@, this function should match @quotRem@ for Integers.-  quotRem :: a -> a -> (a, a)-  -- | When restriced to a subring of the Euclidean domain @a@ isomorphic to-  -- @Integer@, this function should match @divMod@ for Integers.-  divMod  :: a -> a -> (a, a)--  quot :: a -> a -> a-  quot x y = fst (quotRem x y)--  rem :: a -> a -> a-  rem x y = snd (quotRem x y)--  div :: a -> a -> a-  div x y = fst (divMod x y)--  mod :: a -> a -> a-  mod x y = snd (divMod x y)--  -- | @'gcd' x y@ is the greatest number that divides both @x@ and @y@.-  gcd :: a -> a -> a-  gcd x y =  gcd' (abs x) (abs y)-    where-      gcd' :: a -> a -> a-      gcd' a 0  =  a-      gcd' a b  =  gcd' b (abs (a `mod` b))--  -- | @'lcm' x y@ is the smallest number that both @x@ and @y@ divide.-  lcm :: a -> a -> a-  lcm _ 0 =  0-  lcm 0 _ =  0-  lcm x y =  abs ((x `quot` (gcd x y)) * y)--  -- | Test whether two numbers are coprime.-  coprime :: a -> a -> Bool-  coprime x y = gcd x y == 1--  -- | Calculate the greatest common divisor of two numbers and coefficients-  --   for the linear combination.-  ---  --   For signed types satisfies:-  ---  -- > case extendedGCD a b of-  -- >   (d, u, v) -> u*a + v*b == d-  -- >                && d == gcd a b-  ---  --   For unsigned and bounded types the property above holds, but since @u@ and @v@ must also be unsigned,-  --   the result may look weird. E. g., on 64-bit architecture-  ---  -- > extendedGCD (2 :: Word) (3 :: Word) == (1, 2^64-1, 1)-  ---  --   For unsigned and unbounded types (like 'Numeric.Natural.Natural') the result is undefined.-  ---  --   For signed types we also have-  ---  -- > abs u < abs b || abs b <= 1-  -- >-  -- > abs v < abs a || abs a <= 1-  ---  --   (except if one of @a@ and @b@ is 'minBound' of a signed type).-  extendedGCD :: a -> a -> (a, a, a)-  extendedGCD a b = (d, x * signum a, y * signum b)-    where-      (d, x, y) = eGCD 0 1 1 0 (abs a) (abs b)-      eGCD !n1 o1 !n2 o2 r s-        | s == 0    = (r, o1, o2)-        | otherwise = case r `quotRem` s of-                        (q, t) -> eGCD (o1 - q*n1) n1 (o2 - q*n2) n2 s t--coprimeIntegral :: Integral a => a -> a -> Bool-coprimeIntegral x y = (odd x || odd y) && P.gcd x y == 1---- | Wrapper around 'Integral', which has an 'Euclidean' instance.-newtype WrappedIntegral a = WrappedIntegral { unWrappedIntegral :: a }-  deriving (Eq, Ord, Show, Num, Integral, Real, Enum)--instance Integral a => Euclidean (WrappedIntegral a) where-  quotRem = P.quotRem-  divMod  = P.divMod-  quot    = P.quot-  rem     = P.rem-  div     = P.div-  mod     = P.mod-  gcd     = P.gcd-  lcm     = P.lcm-  coprime = coprimeIntegral--instance Euclidean Int where-  quotRem = P.quotRem-  divMod  = P.divMod-  quot    = P.quot-  rem     = P.rem-  div     = P.div-  mod     = P.mod-  gcd (I# x) (I# y) = I# (gcdInt x y)-  lcm     = P.lcm-  coprime = coprimeIntegral--instance Euclidean Word where-  quotRem = P.quotRem-  divMod  = P.divMod-  quot    = P.quot-  rem     = P.rem-  div     = P.div-  mod     = P.mod-  gcd (W# x) (W# y) = W# (gcdWord x y)-  lcm     = P.lcm-  coprime = coprimeIntegral+import Data.Euclidean+import Data.Maybe+import Data.Semiring (Semiring(..), isZero) -instance Euclidean Integer where-  quotRem = P.quotRem-  divMod  = P.divMod-  quot    = P.quot-  rem     = P.rem-  div     = P.div-  mod     = P.mod-  gcd     = gcdInteger-  lcm     = lcmInteger-  coprime = coprimeIntegral-  -- Blocked by GHC bug-  -- https://ghc.haskell.org/trac/ghc/ticket/15350-  -- extendedGCD = gcdExtInteger+-- | Check whether an element is a unit of the ring.+isUnit :: (Eq a, GcdDomain a) => a -> Bool+isUnit x = not (isZero x) && isJust (one `divide` x) --- | Beware that 'extendedGCD' does not make any sense for 'Natural'.-instance Euclidean Natural where-  quotRem = P.quotRem-  divMod  = P.divMod-  quot    = P.quot-  rem     = P.rem-  div     = P.div-  mod     = P.mod-  gcd     = P.gcd-  lcm     = P.lcm-  coprime = coprimeIntegral+-- | Calculate the greatest common divisor of two numbers and coefficients+--   for the linear combination.+--+--   For signed types satisfies:+--+-- > case extendedGCD a b of+-- >   (d, u, v) -> u*a + v*b == d+-- >                && d == gcd a b+--+--   For unsigned and bounded types the property above holds, but since @u@ and @v@ must also be unsigned,+--   the result may look weird. E. g., on 64-bit architecture+--+-- > extendedGCD (2 :: Word) (3 :: Word) == (1, 2^64-1, 1)+--+--   For unsigned and unbounded types (like 'Numeric.Natural.Natural') the result is undefined.+--+--   For signed types we also have+--+-- > abs u < abs b || abs b <= 1+-- >+-- > abs v < abs a || abs a <= 1+--+--   (except if one of @a@ and @b@ is 'minBound' of a signed type).+extendedGCD :: (Eq a, Num a, Euclidean a) => a -> a -> (a, a, a)+extendedGCD a b = (d, x * signum a, y * signum b)+  where+    (d, x, y) = eGCD 0 1 1 0 (abs a) (abs b)+    eGCD !n1 o1 !n2 o2 r s+      | s == 0    = (r, o1, o2)+      | otherwise = case r `quotRem` s of+                      (q, t) -> eGCD (o1 - q*n1) n1 (o2 - q*n2) n2 s t

@@ -8,6 +8,7 @@  {-# LANGUAGE CPP                 #-} {-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE TupleSections       #-}  module Math.NumberTheory.Euclidean.Coprimes   ( splitIntoCoprimes@@ -20,9 +21,11 @@ import Prelude hiding (gcd, quot, rem) import Data.Coerce import Data.List (tails, mapAccumL)+import Data.Maybe #if __GLASGOW_HASKELL__ < 803 import Data.Semigroup #endif+import Data.Semiring (Semiring(..), isZero)  import Math.NumberTheory.Euclidean @@ -33,48 +36,59 @@   }   deriving (Eq, Show) -doPair :: (Euclidean a, Eq b, Num b) => a -> b -> a -> b -> (a, a, [(a, b)])-doPair x xm y ym = case gcd x y of-  1 -> (x, y, [])-  g -> (x', y', concat rests)+unsafeDivide :: GcdDomain a => a -> a -> a+unsafeDivide x y = case x `divide` y of+  Nothing -> error "violated prerequisite of unsafeDivide"+  Just z  -> z++doPair :: (Eq a, GcdDomain a, Eq b, Num b) => a -> b -> a -> b -> (a, a, [(a, b)])+doPair x xm y ym+  | isUnit g  = (x, y, [])+  | otherwise = (x', y', concat rests)     where-      (x', g', xgs) = doPair (x `quot` g) xm g (xm + ym)-      xgs' = if g' == 1 then xgs else ((g', xm + ym) : xgs)+      g = gcd x y -      (y', rests) = mapAccumL go (y `quot` g) xgs'-      go w (t, tm) = (w', if t' == 1 then acc else (t', tm) : acc)+      (x', g', xgs) = doPair (x `unsafeDivide` g) xm g (xm + ym)+      xgs' = if isUnit g' then xgs else ((g', xm + ym) : xgs)++      (y', rests) = mapAccumL go (y `unsafeDivide` g) xgs'+      go w (t, tm) = (w', if isUnit t' then acc else (t', tm) : acc)         where           (w', t', acc) = doPair w ym t tm -_propDoPair :: (Euclidean a, Integral b) => a -> b -> a -> b -> Bool+_propDoPair :: (Eq a, Num a, GcdDomain a, Integral b) => a -> b -> a -> b -> Bool _propDoPair x xm y ym-  =  x `rem` x' == 0-  && y `rem` y' == 0+  =  isJust (x `divide` x')+  && isJust (y `divide` y')   && coprime x' y'   && all (coprime x') (map fst rest)   && all (coprime y') (map fst rest)-  && all (/= 1) (map fst rest)+  && all (not . isUnit) (map fst rest)   && and [ coprime s t | (s, _) : ts <- tails rest, (t, _) <- ts ]-  && (x ^ xm) * (y ^ ym) == (x' ^ xm) * (y' ^ ym) * product (map (\(r, k) -> r ^ k) rest)+  && abs ((x ^ xm) * (y ^ ym)) == abs ((x' ^ xm) * (y' ^ ym) * product (map (\(r, k) -> r ^ k) rest))   where     (x', y', rest) = doPair x xm y ym  insertInternal   :: forall a b.-     (Euclidean a, Eq b, Num b)+     (Eq a, GcdDomain a, Eq b, Num b)   => a   -> b   -> Coprimes a b   -> (Coprimes a b, Coprimes a b)-insertInternal 0   _ = const (Coprimes [(0, 1)], Coprimes [])-insertInternal xx xm = coerce (go ([], []) xx)+insertInternal xx xm+  | isZero xx && xm == 0 = (, Coprimes [])+  | isZero xx            = const (Coprimes [(zero, 1)], Coprimes [])+  | otherwise            = coerce (go ([], []) xx)   where     go :: ([(a, b)], [(a, b)]) -> a -> [(a, b)] -> ([(a, b)], [(a, b)])-    go (old, new) 1 rest = (rest ++ old, new)+    go (old, new) x rest+      | isUnit x = (rest ++ old, new)     go (old, new) x [] = (old, (x, xm) : new)-    go _ _ ((0, _) : _) = ([(0, 1)], [])+    go _ _ ((x, _) : _)+      | isZero x = ([(zero, 1)], [])     go (old, new) x ((y, ym) : rest)-      | y' == 1   = go (old, xys ++ new) x' rest+      | isUnit y' = go (old, xys ++ new) x' rest       | otherwise = go ((y', ym) : old, xys ++ new) x' rest       where         (x', y', xys) = doPair x xm y ym@@ -83,10 +97,11 @@ -- -- >>> singleton 210 1 -- Coprimes {unCoprimes = [(210,1)]}-singleton :: (Eq a, Num a, Eq b, Num b) => a -> b -> Coprimes a b-singleton 0 0 = Coprimes []-singleton 1 _ = Coprimes []-singleton a b = Coprimes [(a, b)]+singleton :: (Eq a, GcdDomain a, Eq b, Num b) => a -> b -> Coprimes a b+singleton a b+  | isZero a && b == 0 = Coprimes []+  | isUnit a           = Coprimes []+  | otherwise          = Coprimes [(a, b)]  -- | Add a non-zero number with its multiplicity to 'Coprimes'. --@@ -94,17 +109,17 @@ -- Coprimes {unCoprimes = [(7,1),(5,2),(3,3),(2,4)]} -- >>> insert 2 4 (insert 7 1 (insert 5 2 (singleton 4 3))) -- Coprimes {unCoprimes = [(7,1),(5,2),(2,10)]}-insert :: (Euclidean a, Eq b, Num b) => a -> b -> Coprimes a b -> Coprimes a b+insert :: (Eq a, GcdDomain a, Eq b, Num b) => a -> b -> Coprimes a b -> Coprimes a b insert x xm ys = Coprimes $ unCoprimes zs <> unCoprimes ws   where     (zs, ws) = insertInternal x xm ys -instance (Euclidean a, Eq b, Num b) => Semigroup (Coprimes a b) where+instance (Eq a, GcdDomain a, Eq b, Num b) => Semigroup (Coprimes a b) where   (Coprimes xs) <> ys = Coprimes $ unCoprimes zs <> foldMap unCoprimes wss     where       (zs, wss) = mapAccumL (\vs (x, xm) -> insertInternal x xm vs) ys xs -instance (Euclidean a, Eq b, Num b) => Monoid (Coprimes a b) where+instance (Eq a, GcdDomain a, Eq b, Num b) => Monoid (Coprimes a b) where   mempty  = Coprimes []   mappend = (<>) @@ -120,5 +135,5 @@ -- Coprimes {unCoprimes = [(28,1),(33,1),(5,2)]} -- >>> splitIntoCoprimes [(360, 1), (210, 1)] -- Coprimes {unCoprimes = [(7,1),(5,2),(3,3),(2,4)]}-splitIntoCoprimes :: (Euclidean a, Eq b, Num b) => [(a, b)] -> Coprimes a b+splitIntoCoprimes :: (Eq a, GcdDomain a, Eq b, Num b) => [(a, b)] -> Coprimes a b splitIntoCoprimes = foldl (\acc (x, xm) -> insert x xm acc) mempty

@@ -31,7 +31,7 @@   , chineseRemainder2   ) where -import Prelude hiding (mod, quot, gcd, lcm)+import Prelude hiding (rem, quot, gcd, lcm)  import Control.Monad (foldM) import Data.Foldable@@ -54,7 +54,7 @@ -- Just 5 -- >>> chineseCoprime (3, 4) (5, 6) -- Nothing -- moduli must be coprime-chineseCoprime :: Euclidean a => (a, a) -> (a, a) -> Maybe a+chineseCoprime :: (Integral a, Euclidean a) => (a, a) -> (a, a) -> Maybe a chineseCoprime (n1, m1) (n2, m2) = case d of   1 -> Just $ ((1 - u * m1) * n1 + (1 - v * m2) * n2) `mod` (m1 * m2)   _ -> Nothing@@ -76,9 +76,9 @@ -- Just 11 -- >>> chinese (3, 4) (2, 6) -- Nothing-chinese :: forall a. Euclidean a => (a, a) -> (a, a) -> Maybe a+chinese :: forall a. (Integral a, GcdDomain a, Euclidean a) => (a, a) -> (a, a) -> Maybe a chinese (n1, m1) (n2, m2)-  | n1 `mod` g == n2 `mod` g+  | (n1 - n2) `rem` g == 0   = chineseCoprime (n1 `mod` m1', m1') (n2 `mod` m2', m2')   | otherwise   = Nothing

@@ -264,9 +264,17 @@   InfMod  :: Rational -> SomeMod  instance Eq SomeMod where-  SomeMod mx == SomeMod my = getMod mx == getMod my && getVal mx == getVal my+  SomeMod mx == SomeMod my =+    getMod mx == getMod my && getVal mx == getVal my   InfMod rx  == InfMod ry  = rx == ry   _          == _          = False++instance Ord SomeMod where+  SomeMod mx `compare` SomeMod my =+    getMod mx `compare` getMod my <> getVal mx `compare` getVal my+  SomeMod{} `compare` InfMod{} = LT+  InfMod{} `compare` SomeMod{} = GT+  InfMod rx `compare` InfMod ry = rx `compare` ry  instance Show SomeMod where   show = \case

@@ -6,9 +6,14 @@ --  {-# LANGUAGE BangPatterns        #-}+{-# LANGUAGE CPP                 #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE ViewPatterns        #-} +#if __GLASGOW_HASKELL__ < 801+{-# OPTIONS_GHC -fno-warn-incomplete-patterns #-}+#endif+ module Math.NumberTheory.Moduli.DiscreteLogarithm   ( discreteLogarithm   ) where@@ -21,36 +26,36 @@ import GHC.TypeNats.Compat  import Math.NumberTheory.Moduli.Chinese       (chineseRemainder2)-import Math.NumberTheory.Moduli.Class         (KnownNat, MultMod(..), Mod, getVal)+import Math.NumberTheory.Moduli.Class         (MultMod(..), Mod, getVal) import Math.NumberTheory.Moduli.Equations     (solveLinear)-import Math.NumberTheory.Moduli.PrimitiveRoot (PrimitiveRoot(..), CyclicGroup(..))+import Math.NumberTheory.Moduli.PrimitiveRoot (PrimitiveRoot(..))+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.Powers.Squares       (integerSquareRoot) import Math.NumberTheory.Primes  (unPrime)  -- | Computes the discrete logarithm. Currently uses a combination of the baby-step -- giant-step method and Pollard's rho algorithm, with Bach reduction.-discreteLogarithm :: KnownNat m => PrimitiveRoot m -> MultMod m -> Natural-discreteLogarithm a b = discreteLogarithm' (getGroup a) (multElement $ unPrimitiveRoot a) (multElement b)--discreteLogarithm'-  :: KnownNat m-  => CyclicGroup Natural  -- ^ group structure (must be the multiplicative group mod m)-  -> Mod m                -- ^ a-  -> Mod m                -- ^ b-  -> Natural              -- ^ result-discreteLogarithm' cg a b =-  case cg of-    CG2-      -> 0-      -- the only valid input was a=1, b=1-    CG4-      -> if b == 1 then 0 else 1-      -- the only possible input here is a=3 with b = 1 or 3-    CGOddPrimePower       (toInteger . unPrime -> p) k-      -> discreteLogarithmPP p k (getVal a) (getVal b)-    CGDoubleOddPrimePower (toInteger . unPrime -> p) k-      -> discreteLogarithmPP p k (getVal a `rem` p^k) (getVal b `rem` p^k)-      -- we have the isomorphism t -> t `rem` p^k from (Z/2p^kZ)* -> (Z/p^kZ)*+--+-- >>> :set -XDataKinds+-- >>> import Data.Maybe+-- >>> let cg = fromJust cyclicGroup :: CyclicGroup Integer 13+-- >>> let rt = fromJust (isPrimitiveRoot cg 2)+-- >>> let x  = fromJust (isMultElement 11)+-- >>> discreteLogarithm cg rt x+-- 7+discreteLogarithm :: CyclicGroup Integer m -> PrimitiveRoot m -> MultMod m -> Natural+discreteLogarithm cg (multElement . unPrimitiveRoot -> a) (multElement -> b) = case cg of+  CG2+    -> 0+    -- the only valid input was a=1, b=1+  CG4+    -> if getVal b == 1 then 0 else 1+    -- the only possible input here is a=3 with b = 1 or 3+  CGOddPrimePower (unPrime -> p) k+    -> discreteLogarithmPP p k (getVal a) (getVal b)+  CGDoubleOddPrimePower (unPrime -> p) k+    -> discreteLogarithmPP p k (getVal a `rem` p^k) (getVal b `rem` p^k)+    -- we have the isomorphism t -> t `rem` p^k from (Z/2p^kZ)* -> (Z/p^kZ)*  -- Implementation of Bach reduction (https://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/CSD-84-186.pdf) {-# INLINE discreteLogarithmPP #-}

@@ -15,10 +15,12 @@   , solveQuadratic   ) where +import Data.Constraint import GHC.Integer.GMP.Internals  import Math.NumberTheory.Moduli.Chinese import Math.NumberTheory.Moduli.Class+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.Moduli.Sqrt import Math.NumberTheory.Primes import Math.NumberTheory.Utils (recipMod)@@ -56,21 +58,21 @@ -- | Find all solutions of ax² + bx + c ≡ 0 (mod m). -- -- >>> :set -XDataKinds--- >>> solveQuadratic (1 :: Mod 32) 0 (-17) -- solving x² - 17 ≡ 0 (mod 32)+-- >>> solveQuadratic sfactors (1 :: Mod 32) 0 (-17) -- solving x² - 17 ≡ 0 (mod 32) -- [(9 `modulo` 32),(25 `modulo` 32),(7 `modulo` 32),(23 `modulo` 32)] solveQuadratic-  :: KnownNat m-  => Mod m   -- ^ a+  :: SFactors Integer m+  -> Mod m   -- ^ a   -> Mod m   -- ^ b   -> Mod m   -- ^ c   -> [Mod m] -- ^ list of x-solveQuadratic a b c-  = map fromInteger-  $ fst-  $ combine-  $ map (\(p, n) -> (solveQuadraticPrimePower a' b' c' p n, unPrime p ^ n))-  $ factorise-  $ getMod a+solveQuadratic sm a b c = case proofFromSFactors sm of+  Sub Dict ->+    map fromInteger+    $ fst+    $ combine+    $ map (\(p, n) -> (solveQuadraticPrimePower a' b' c' p n, unPrime p ^ n))+    $ unSFactors sm   where     a' = getVal a     b' = getVal b@@ -123,7 +125,7 @@     (_, False)   -> [1]     _            -> [] solveQuadraticPrime a b c p-  | a `mod` p' == 0+  | a `rem` p' == 0   = solveLinear' p' b c   | otherwise   = map (\n -> n * recipModInteger (2 * a) p' `mod` p')

@@ -14,130 +14,41 @@ {-# LANGUAGE StandaloneDeriving   #-} {-# LANGUAGE TupleSections        #-} {-# LANGUAGE UndecidableInstances #-}+{-# LANGUAGE ViewPatterns         #-} +#if __GLASGOW_HASKELL__ < 801+{-# OPTIONS_GHC -fno-warn-incomplete-patterns #-}+#endif+ module Math.NumberTheory.Moduli.PrimitiveRoot-  ( -- * Cyclic groups-    CyclicGroup(..)-  , cyclicGroupFromModulo-  , cyclicGroupToModulo-  , groupSize-    -- * Primitive roots-  , PrimitiveRoot+  ( -- * Primitive roots+    PrimitiveRoot   , unPrimitiveRoot-  , getGroup   , isPrimitiveRoot-  , isPrimitiveRoot'   ) where -#if __GLASGOW_HASKELL__ < 803-import Data.Semigroup-#endif- import Math.NumberTheory.ArithmeticFunctions (totient)-import qualified Math.NumberTheory.Euclidean as E-import Math.NumberTheory.Euclidean.Coprimes as Coprimes (singleton)-import Math.NumberTheory.Moduli.Class (getNatMod, getNatVal, KnownNat, Mod, MultMod, isMultElement)-import Math.NumberTheory.Powers.General (highestPower)+import Math.NumberTheory.Moduli.Class hiding (powMod)+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.Powers.Modular-import Math.NumberTheory.Prefactored import Math.NumberTheory.Primes -import Control.DeepSeq import Control.Monad (guard)-import GHC.Generics-import Numeric.Natural---- | A multiplicative group of residues is called cyclic,--- if there is a primitive root @g@,--- whose powers generates all elements.--- Any cyclic multiplicative group of residues--- falls into one of the following cases.-data CyclicGroup a-  = CG2 -- ^ Residues modulo 2.-  | CG4 -- ^ Residues modulo 4.-  | CGOddPrimePower       (Prime a) Word-  -- ^ Residues modulo @p@^@k@ for __odd__ prime @p@.-  | CGDoubleOddPrimePower (Prime a) Word-  -- ^ Residues modulo 2@p@^@k@ for __odd__ prime @p@.-  deriving (Eq, Show, Generic)--instance NFData a => NFData (CyclicGroup a)---- | Check whether a multiplicative group of residues,--- characterized by its modulo, is cyclic and, if yes, return its form.------ >>> cyclicGroupFromModulo 4--- Just CG4--- >>> cyclicGroupFromModulo (2 * 13 ^ 3)--- Just (CGDoubleOddPrimePower (Prime 13) 3)--- >>> cyclicGroupFromModulo (4 * 13)--- Nothing-cyclicGroupFromModulo-  :: (Ord a, Integral a, UniqueFactorisation a)-  => a-  -> Maybe (CyclicGroup a)-cyclicGroupFromModulo = \case-  2 -> Just CG2-  4 -> Just CG4-  n-    | n <= 1    -> Nothing-    | odd n     -> uncurry CGOddPrimePower       <$> isPrimePower n-    | odd halfN -> uncurry CGDoubleOddPrimePower <$> isPrimePower halfN-    | otherwise -> Nothing-    where-      halfN = n `quot` 2--isPrimePower-  :: (Integral a, UniqueFactorisation a)-  => a-  -> Maybe (Prime a, Word)-isPrimePower n = (, k) <$> isPrime m-  where-    (m, k) = highestPower n---- | Extract modulo and its factorisation from--- a cyclic multiplicative group of residues.------ >>> cyclicGroupToModulo CG4--- Prefactored {prefValue = 4, prefFactors = Coprimes {unCoprimes = [(2,2)]}}------ >>> import Data.Maybe--- >>> cyclicGroupToModulo (CGDoubleOddPrimePower (fromJust (isPrime 13)) 3)--- Prefactored {prefValue = 4394, prefFactors = Coprimes {unCoprimes = [(13,3),(2,1)]}}-cyclicGroupToModulo-  :: E.Euclidean a-  => CyclicGroup a-  -> Prefactored a-cyclicGroupToModulo = fromFactors . \case-  CG2                       -> Coprimes.singleton 2 1-  CG4                       -> Coprimes.singleton 2 2-  CGOddPrimePower p k       -> Coprimes.singleton (unPrime p) k-  CGDoubleOddPrimePower p k -> Coprimes.singleton 2 1-                            <> Coprimes.singleton (unPrime p) k+import Data.Constraint  -- | 'PrimitiveRoot' m is a type which is only inhabited -- by <https://en.wikipedia.org/wiki/Primitive_root_modulo_n primitive roots> of m.-data PrimitiveRoot m = PrimitiveRoot+newtype PrimitiveRoot m = PrimitiveRoot   { unPrimitiveRoot :: MultMod m -- ^ Extract primitive root value.-  , getGroup        :: CyclicGroup Natural -- ^ Get cyclic group structure.   }   deriving (Eq, Show) --- | 'isPrimitiveRoot'' @cg@ @a@ checks whether @a@ is--- a <https://en.wikipedia.org/wiki/Primitive_root_modulo_n primitive root>--- of a given cyclic multiplicative group of residues @cg@.------ >>> let Just cg = cyclicGroupFromModulo 13--- >>> isPrimitiveRoot' cg 1--- False--- >>> isPrimitiveRoot' cg 2--- True+-- https://en.wikipedia.org/wiki/Primitive_root_modulo_n#Finding_primitive_roots isPrimitiveRoot'   :: (Integral a, UniqueFactorisation a)-  => CyclicGroup a+  => CyclicGroup a m   -> a   -> Bool--- https://en.wikipedia.org/wiki/Primitive_root_modulo_n#Finding_primitive_roots isPrimitiveRoot' cg r =   case cg of     CG2                       -> r == 1@@ -157,23 +68,18 @@ -- a <https://en.wikipedia.org/wiki/Primitive_root_modulo_n primitive root>. -- -- >>> :set -XDataKinds--- >>> isPrimitiveRoot (1 :: Mod 13)+-- >>> import Data.Maybe+-- >>> isPrimitiveRoot (fromJust cyclicGroup) (1 :: Mod 13) -- Nothing--- >>> isPrimitiveRoot (2 :: Mod 13)--- Just (PrimitiveRoot {unPrimitiveRoot = MultMod {multElement = (2 `modulo` 13)}, getGroup = CGOddPrimePower (Prime 13) 1})------ This function is a convenient wrapper around 'isPrimitiveRoot''. The latter--- provides better control and performance, if you need them.+-- >>> isPrimitiveRoot (fromJust cyclicGroup) (2 :: Mod 13)+-- Just (PrimitiveRoot {unPrimitiveRoot = MultMod {multElement = (2 `modulo` 13)}}) isPrimitiveRoot-  :: KnownNat n-  => Mod n-  -> Maybe (PrimitiveRoot n)-isPrimitiveRoot r = do-  r' <- isMultElement r-  cg <- cyclicGroupFromModulo (getNatMod r)-  guard $ isPrimitiveRoot' cg (getNatVal r)-  return $ PrimitiveRoot r' cg---- | Calculate the size of a given cyclic group.-groupSize :: (E.Euclidean a, UniqueFactorisation a) => CyclicGroup a -> Prefactored a-groupSize = totient . cyclicGroupToModulo+  :: (Integral a, UniqueFactorisation a)+  => CyclicGroup a m+  -> Mod m+  -> Maybe (PrimitiveRoot m)+isPrimitiveRoot cg r = case proofFromCyclicGroup cg of+  Sub Dict -> do+    r' <- isMultElement r+    guard $ isPrimitiveRoot' cg (fromIntegral (getNatVal r))+    return $ PrimitiveRoot r'

@@ -0,0 +1,314 @@+-- |+-- Module:      Math.NumberTheory.Moduli.Singleton+-- Copyright:   (c) 2019 Andrew Lelechenko+-- Licence:     MIT+-- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>+--+-- Singleton data types.+--++{-# LANGUAGE CPP                 #-}+{-# LANGUAGE DataKinds           #-}+{-# LANGUAGE DeriveGeneric       #-}+{-# LANGUAGE FlexibleInstances   #-}+{-# LANGUAGE GADTs               #-}+{-# LANGUAGE KindSignatures      #-}+{-# LANGUAGE LambdaCase          #-}+{-# LANGUAGE PatternSynonyms     #-}+{-# LANGUAGE PolyKinds           #-}+{-# LANGUAGE RankNTypes          #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE StandaloneDeriving  #-}+{-# LANGUAGE TupleSections       #-}+{-# LANGUAGE TypeOperators       #-}+{-# LANGUAGE ViewPatterns        #-}++module Math.NumberTheory.Moduli.Singleton+  ( -- * SFactors singleton+    SFactors+  , sfactors+  , someSFactors+  , unSFactors+  , proofFromSFactors+    -- * CyclicGroup singleton+  , CyclicGroup+  , cyclicGroup+  , cyclicGroupFromFactors+  , cyclicGroupFromModulo+  , proofFromCyclicGroup+  , pattern CG2+  , pattern CG4+  , pattern CGOddPrimePower+  , pattern CGDoubleOddPrimePower+    -- * SFactors \<=\> CyclicGroup+  , cyclicGroupToSFactors+  , sfactorsToCyclicGroup+    -- * Some wrapper+  , Some(..)+  ) where++import Control.DeepSeq+import Data.Constraint+import Data.List+import qualified Data.Map as M+import Data.Proxy+#if __GLASGOW_HASKELL__ < 803+import Data.Semigroup+#endif+import GHC.Generics+import GHC.TypeNats.Compat+import Numeric.Natural+import Unsafe.Coerce++import Math.NumberTheory.Powers+import Math.NumberTheory.Primes+import Math.NumberTheory.Primes.Types++-- | Wrapper to hide an unknown type-level natural.+data Some (a :: Nat -> *) where+  Some :: a m -> Some a++-- | From "Data.Constraint.Nat".+newtype Magic n = Magic (KnownNat n => Dict (KnownNat n))++-- | This singleton data type establishes a correspondence+-- between a modulo @m@ on type level+-- and its factorisation on term level.+newtype SFactors a (m :: Nat) = SFactors+  { unSFactors :: [(Prime a, Word)]+  -- ^ Factors of @m@.+  } deriving (Show, Generic)++instance Eq (SFactors a m) where+  _ == _ = True++instance Ord (SFactors a m) where+  _ `compare` _ = EQ++instance NFData a => NFData (SFactors a m)++instance Ord a => Eq (Some (SFactors a)) where+  Some (SFactors xs) == Some (SFactors ys) =+    xs == ys++instance Ord a => Ord (Some (SFactors a)) where+  Some (SFactors xs) `compare` Some (SFactors ys) =+    xs `compare` ys++instance Show a => Show (Some (SFactors a)) where+  showsPrec p (Some x) = showsPrec p x++instance NFData a => NFData (Some (SFactors a)) where+  rnf (Some x) = rnf x++-- | Create a singleton from a type-level positive modulo @m@,+-- passed in a constraint.+--+-- >>> :set -XDataKinds+-- >>> sfactors :: SFactors Integer 13+-- SFactors {sfactorsFactors = [(Prime 13,1)]}+sfactors :: forall a m. (Ord a, UniqueFactorisation a, KnownNat m) => SFactors a m+sfactors = if m == 0+  then error "sfactors: modulo must be positive"+  else SFactors (sort (factorise m))+  where+    m = fromIntegral (natVal (Proxy :: Proxy m))++-- | Create a singleton from factors of @m@.+-- Factors must be distinct, as in output of 'factorise'.+--+-- >>> import Math.NumberTheory.Primes+-- >>> someSFactors (factorise 98)+-- SFactors {sfactorsFactors = [(Prime 2,1),(Prime 7,2)]}+someSFactors :: (Ord a, Num a) => [(Prime a, Word)] -> Some (SFactors a)+someSFactors+  = Some+  . SFactors+  -- Just a precaution against ill-formed lists of factors+  . M.assocs+  . M.fromListWith (+)++-- | Convert a singleton to a proof that @m@ is known. Usage example:+--+-- > toModulo :: SFactors Integer m -> Natural+-- > toModulo t = case proofFromSFactors t of Sub Dict -> natVal t+proofFromSFactors :: Integral a => SFactors a m -> (() :- KnownNat m)+proofFromSFactors (SFactors fs) = Sub $ unsafeCoerce (Magic Dict) (fromIntegral (factorBack fs) :: Natural)++-- | This singleton data type establishes a correspondence+-- between a modulo @m@ on type level+-- and a cyclic group of the same order on term level.+data CyclicGroup a (m :: Nat)+  = CG2' -- ^ Residues modulo 2.+  | CG4' -- ^ Residues modulo 4.+  | CGOddPrimePower'       (Prime a) Word+  -- ^ Residues modulo @p@^@k@ for __odd__ prime @p@.+  | CGDoubleOddPrimePower' (Prime a) Word+  -- ^ Residues modulo 2@p@^@k@ for __odd__ prime @p@.+  deriving (Show, Generic)++instance Eq (CyclicGroup a m) where+  _ == _ = True++instance Ord (CyclicGroup a m) where+  _ `compare` _ = EQ++instance NFData a => NFData (CyclicGroup a m)++instance Eq a => Eq (Some (CyclicGroup a)) where+  Some CG2' == Some CG2' = True+  Some CG4' == Some CG4' = True+  Some (CGOddPrimePower' p1 k1) == Some (CGOddPrimePower' p2 k2) =+    p1 == p2 && k1 == k2+  Some (CGDoubleOddPrimePower' p1 k1) == Some (CGDoubleOddPrimePower' p2 k2) =+    p1 == p2 && k1 == k2+  _ == _ = False++instance Ord a => Ord (Some (CyclicGroup a)) where+  compare (Some x) (Some y) = case x of+    CG2' -> case y of+      CG2' -> EQ+      _    -> LT+    CG4' -> case y of+      CG2' -> GT+      CG4' -> EQ+      _    -> LT+    CGOddPrimePower' p1 k1 -> case y of+      CGDoubleOddPrimePower'{} -> LT+      CGOddPrimePower' p2 k2 ->+        p1 `compare` p2 <> k1 `compare` k2+      _ -> GT+    CGDoubleOddPrimePower' p1 k1 -> case y of+      CGDoubleOddPrimePower' p2 k2 ->+        p1 `compare` p2 <> k1 `compare` k2+      _ -> GT++instance Show a => Show (Some (CyclicGroup a)) where+  showsPrec p (Some x) = showsPrec p x++instance NFData a => NFData (Some (CyclicGroup a)) where+  rnf (Some x) = rnf x++-- | Create a singleton from a type-level positive modulo @m@,+-- passed in a constraint.+--+-- >>> :set -XDataKinds+-- >>> import Data.Maybe+-- >>> cyclicGroup :: CyclicGroup Integer 169+-- CGOddPrimePower' (Prime 13) 2+--+-- >>> sfactorsToCyclicGroup (fromModulo 4)+-- Just CG4'+-- >>> sfactorsToCyclicGroup (fromModulo (2 * 13 ^ 3))+-- Just (CGDoubleOddPrimePower' (Prime 13) 3)+-- >>> sfactorsToCyclicGroup (fromModulo (4 * 13))+-- Nothing+cyclicGroup+  :: forall a m.+     (Integral a, UniqueFactorisation a, KnownNat m)+  => Maybe (CyclicGroup a m)+cyclicGroup = fromModuloInternal m+  where+    m = fromIntegral (natVal (Proxy :: Proxy m))++cyclicGroupFromFactors+  :: (Eq a, Num a)+  => [(Prime a, Word)]+  -> Maybe (Some (CyclicGroup a))+cyclicGroupFromFactors = \case+  [(unPrime -> 2, 1)] -> Just $ Some CG2'+  [(unPrime -> 2, 2)] -> Just $ Some CG4'+  [(unPrime -> 2, _)] -> Nothing+  [(p, k)] -> Just $ Some $ CGOddPrimePower' p k+  [(unPrime -> 2, 1), (p, k)] -> Just $ Some $ CGDoubleOddPrimePower' p k+  [(p, k), (unPrime -> 2, 1)] -> Just $ Some $ CGDoubleOddPrimePower' p k+  _ -> Nothing++-- | Similar to 'cyclicGroupFromFactors' . 'factorise',+-- but much faster, because it+-- but performes only one primality test instead of full+-- factorisation.+cyclicGroupFromModulo+  :: (Integral a, UniqueFactorisation a)+  => a+  -> Maybe (Some (CyclicGroup a))+cyclicGroupFromModulo = fmap Some . fromModuloInternal++fromModuloInternal+  :: (Integral a, UniqueFactorisation a)+  => a+  -> Maybe (CyclicGroup a m)+fromModuloInternal = \case+  2 -> Just CG2'+  4 -> Just CG4'+  n+    | even n -> uncurry CGDoubleOddPrimePower' <$> isOddPrimePower (n `div` 2)+    | otherwise -> uncurry CGOddPrimePower' <$> isOddPrimePower n++isOddPrimePower+  :: (Integral a, UniqueFactorisation a)+  => a+  -> Maybe (Prime a, Word)+isOddPrimePower n+  | even n    = Nothing+  | otherwise = (, k) <$> isPrime p+  where+    (p, k) = highestPower n++-- | Convert a cyclic group to a proof that @m@ is known. Usage example:+--+-- > toModulo :: CyclicGroup Integer m -> Natural+-- > toModulo t = case proofFromCyclicGroup t of Sub Dict -> natVal t+proofFromCyclicGroup :: Integral a => CyclicGroup a m -> (() :- KnownNat m)+proofFromCyclicGroup = proofFromSFactors . cyclicGroupToSFactors++-- | Check whether a multiplicative group of residues,+-- characterized by its modulo, is cyclic and, if yes, return its form.+--+-- >>> sfactorsToCyclicGroup (fromModulo 4)+-- Just CG4'+-- >>> sfactorsToCyclicGroup (fromModulo (2 * 13 ^ 3))+-- Just (CGDoubleOddPrimePower' (Prime 13) 3)+-- >>> sfactorsToCyclicGroup (fromModulo (4 * 13))+-- Nothing+sfactorsToCyclicGroup :: (Eq a, Num a) => SFactors a m -> Maybe (CyclicGroup a m)+sfactorsToCyclicGroup (SFactors fs) = case fs of+  [(unPrime -> 2, 1)]         -> Just CG2'+  [(unPrime -> 2, 2)]         -> Just CG4'+  [(unPrime -> 2, _)]         -> Nothing+  [(p, k)]                    -> Just $ CGOddPrimePower' p k+  [(p, k), (unPrime -> 2, 1)] -> Just $ CGDoubleOddPrimePower' p k+  [(unPrime -> 2, 1), (p, k)] -> Just $ CGDoubleOddPrimePower' p k+  _ -> Nothing++-- | Invert 'sfactorsToCyclicGroup'.+--+-- >>> import Data.Maybe+-- >>> cyclicGroupToSFactors (fromJust (sfactorsToCyclicGroup (fromModulo 4)))+-- SFactors {sfactorsModulo = 4, sfactorsFactors = [(Prime 2,2)]}+cyclicGroupToSFactors :: Num a => CyclicGroup a m -> SFactors a m+cyclicGroupToSFactors = SFactors . \case+  CG2' -> [(Prime 2, 1)]+  CG4' -> [(Prime 2, 2)]+  CGOddPrimePower' p k -> [(p, k)]+  CGDoubleOddPrimePower' p k -> [(Prime 2, 1), (p, k)]++-- | Unidirectional pattern for residues modulo 2.+pattern CG2 :: CyclicGroup a m+pattern CG2 <- CG2'++-- | Unidirectional pattern for residues modulo 4.+pattern CG4 :: CyclicGroup a m+pattern CG4 <- CG4'++-- | Unidirectional pattern for residues modulo @p@^@k@ for __odd__ prime @p@.+pattern CGOddPrimePower :: Prime a -> Word -> CyclicGroup a m+pattern CGOddPrimePower p k <- CGOddPrimePower' p k++-- | Unidirectional pattern for residues modulo 2@p@^@k@ for __odd__ prime @p@.+pattern CGDoubleOddPrimePower :: Prime a -> Word -> CyclicGroup a m+pattern CGDoubleOddPrimePower p k <- CGDoubleOddPrimePower' p k++#if __GLASGOW_HASKELL__ > 801+{-# COMPLETE CG2, CG4, CGOddPrimePower, CGDoubleOddPrimePower #-}+#endif

@@ -17,39 +17,29 @@   , sqrtsModFactorisation   , sqrtsModPrimePower   , sqrtsModPrime-    -- * Old interface-  , Old.sqrtModP-  , Old.sqrtModPList-  , Old.sqrtModP'-  , Old.tonelliShanks-  , Old.sqrtModPP-  , Old.sqrtModPPList-  , Old.sqrtModF-  , Old.sqrtModFList   ) where  import Control.Monad (liftM2) import Data.Bits+import Data.Constraint  import Math.NumberTheory.Moduli.Chinese-import Math.NumberTheory.Moduli.Class (Mod, getVal, getMod, KnownNat)+import Math.NumberTheory.Moduli.Class hiding (powMod) import Math.NumberTheory.Moduli.Jacobi+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.Powers.Modular (powMod)-import Math.NumberTheory.Primes.Types-import Math.NumberTheory.Primes.Sieve (sieveFrom)-import Math.NumberTheory.Primes (Prime, factorise)+import Math.NumberTheory.Primes import Math.NumberTheory.Utils (shiftToOddCount, splitOff, recipMod) import Math.NumberTheory.Utils.FromIntegral -import qualified Math.NumberTheory.Moduli.SqrtOld as Old- -- | List all modular square roots. -- -- >>> :set -XDataKinds--- >>> sqrtsMod (1 :: Mod 60)+-- >>> sqrtsMod sfactors (1 :: Mod 60) -- [(1 `modulo` 60),(49 `modulo` 60),(41 `modulo` 60),(29 `modulo` 60),(31 `modulo` 60),(19 `modulo` 60),(11 `modulo` 60),(59 `modulo` 60)]-sqrtsMod :: KnownNat m => Mod m -> [Mod m]-sqrtsMod a = map fromInteger $ sqrtsModFactorisation (getVal a) (factorise (getMod a))+sqrtsMod :: SFactors Integer m -> Mod m -> [Mod m]+sqrtsMod sm a = case proofFromSFactors sm of+  Sub Dict -> map fromInteger $ sqrtsModFactorisation (getVal a) (unSFactors sm)  -- | List all square roots modulo a number, the factorisation of which is -- passed as a second argument.@@ -61,7 +51,7 @@ sqrtsModFactorisation n pps = map fst $ foldl1 (liftM2 comb) cs   where     ms :: [Integer]-    ms = map (\(Prime p, pow) -> p ^ pow) pps+    ms = map (\(p, pow) -> unPrime p ^ pow) pps      rs :: [[Integer]]     rs = map (\(p, pow) -> sqrtsModPrimePower n p pow) pps@@ -81,7 +71,7 @@ -- [3,12,21,24,6,15] sqrtsModPrimePower :: Integer -> Prime Integer -> Word -> [Integer] sqrtsModPrimePower nn p 1 = sqrtsModPrime nn p-sqrtsModPrimePower nn (Prime prime) expo = let primeExpo = prime ^ expo in+sqrtsModPrimePower nn (unPrime -> prime) expo = let primeExpo = prime ^ expo in   case splitOff prime (nn `mod` primeExpo) of     (_, 0) -> [0, prime ^ ((expo + 1) `quot` 2) .. primeExpo - 1]     (kk, n)@@ -115,8 +105,8 @@ -- >>> sqrtsModPrime 2 (fromJust (isPrime 5)) -- [] sqrtsModPrime :: Integer -> Prime Integer -> [Integer]-sqrtsModPrime n (Prime 2) = [n `mod` 2]-sqrtsModPrime n (Prime prime) = case jacobi n prime of+sqrtsModPrime n (unPrime -> 2) = [n `mod` 2]+sqrtsModPrime n (unPrime -> prime) = case jacobi n prime of   MinusOne -> []   Zero     -> [0]   One      -> let r = sqrtModP' (n `mod` prime) prime in [r, prime - r]@@ -176,9 +166,10 @@  -- | prime must be odd, n must be coprime with prime sqrtModPP' :: Integer -> Integer -> Word -> Maybe Integer-sqrtModPP' n prime expo = case sqrtsModPrime n (Prime prime) of-                            []    -> Nothing-                            r : _ -> fixup r+sqrtModPP' n prime expo = case jacobi n prime of+  MinusOne -> Nothing+  Zero     -> Nothing+  One      -> fixup $ sqrtModP' (n `mod` prime) prime   where     fixup r = let diff' = r*r-n               in if diff' == 0@@ -235,11 +226,14 @@  findNonSquare :: Integer -> Integer findNonSquare n-    | rem8 n == 5 || rem8 n == 3  = 2-    | otherwise = search primelist+    | rem8 n == 5 || rem8 n == 3 = 2+    | otherwise = search candidates       where-        primelist = [3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67]-                        ++ map unPrime (sieveFrom (68 + n `rem` 4)) -- prevent sharing+        -- It is enough to consider only prime candidates, but+        -- the probability that the smallest non-residue is > 67+        -- is small and 'jacobi' test is fast,+        -- so we use [71..n] instead of filter isPrime [71..n].+        candidates = 3:5:7:11:13:17:19:23:29:31:37:41:43:47:53:59:61:67:[71..n]         search (p:ps) = case jacobi p n of           MinusOne -> p           _        -> search ps

@@ -1,232 +0,0 @@--- |--- Module:      Math.NumberTheory.Moduli.Sqrt--- Copyright:   (c) 2011 Daniel Fischer--- Licence:     MIT--- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>------ Modular square roots.-----{-# LANGUAGE BangPatterns #-}-{-# LANGUAGE CPP          #-}-{-# LANGUAGE ViewPatterns #-}--module Math.NumberTheory.Moduli.SqrtOld-  ( sqrtModP-  , sqrtModPList-  , sqrtModP'-  , tonelliShanks-  , sqrtModPP-  , sqrtModPPList-  , sqrtModF-  , sqrtModFList-  ) where--import Control.Monad (liftM2)-import Data.Bits-import Data.List (nub)-import GHC.Integer.GMP.Internals--import Math.NumberTheory.Moduli.Chinese-import Math.NumberTheory.Moduli.Jacobi-import Math.NumberTheory.Primes.Sieve (sieveFrom)-import Math.NumberTheory.Primes.Types (unPrime)-import Math.NumberTheory.Utils (shiftToOddCount, splitOff)-import Math.NumberTheory.Utils.FromIntegral--{-# DEPRECATED sqrtModP, sqrtModP', sqrtModPList, tonelliShanks "Use 'Math.NumberTheory.Moduli.Sqrt.sqrtsModPrime' instead" #-}-{-# DEPRECATED sqrtModPP, sqrtModPPList "Use 'Math.NumberTheory.Moduli.Sqrt.sqrtsModPrimePower' instead" #-}-{-# DEPRECATED sqrtModF, sqrtModFList "Use 'Math.NumberTheory.Moduli.Sqrt.sqrtsModFactorisation' or 'Math.NumberTheory.Moduli.Sqrt.sqrtsMod' instead" #-}----- | @sqrtModP n prime@ calculates a modular square root of @n@ modulo @prime@---   if that exists. The second argument /must/ be a (positive) prime, otherwise---   the computation may not terminate and if it does, may yield a wrong result.---   The precondition is /not/ checked.------   If @prime@ is a prime and @n@ a quadratic residue modulo @prime@, the result---   is @Just r@ where @r^2 ≡ n (mod prime)@, if @n@ is a quadratic nonresidue,---   the result is @Nothing@.-sqrtModP :: Integer -> Integer -> Maybe Integer-sqrtModP n 2 = Just (n `mod` 2)-sqrtModP n prime = case jacobi n prime of-                     MinusOne -> Nothing-                     Zero     -> Just 0-                     One      -> Just (sqrtModP' (n `mod` prime) prime)---- | @sqrtModPList n prime@ computes the list of all square roots of @n@---   modulo @prime@. @prime@ /must/ be a (positive) prime.---   The precondition is /not/ checked.-sqrtModPList :: Integer -> Integer -> [Integer]-sqrtModPList n prime-    | prime == 2    = [n `mod` 2]-    | otherwise     = case sqrtModP n prime of-                        Just 0 -> [0]-                        Just r -> [r,prime-r] -- The group of units in Z/(p) is cyclic-                        _      -> []---- | @sqrtModP' square prime@ finds a square root of @square@ modulo---   prime. @prime@ /must/ be a (positive) prime, and @square@ /must/ be a positive---   quadratic residue modulo @prime@, i.e. @'jacobi square prime == 1@.---   The precondition is /not/ checked.-sqrtModP' :: Integer -> Integer -> Integer-sqrtModP' square prime-    | prime == 2    = square-    | rem4 prime == 3 = powModInteger square ((prime + 1) `quot` 4) prime-    | otherwise     = tonelliShanks square prime---- | @tonelliShanks square prime@ calculates a square root of @square@---   modulo @prime@, where @prime@ is a prime of the form @4*k + 1@ and---   @square@ is a positive quadratic residue modulo @prime@, using the---   Tonelli-Shanks algorithm.---   No checks on the input are performed.-tonelliShanks :: Integer -> Integer -> Integer-tonelliShanks square prime = loop rc t1 generator log2-  where-    (log2,q) = shiftToOddCount (prime-1)-    nonSquare = findNonSquare prime-    generator = powModInteger nonSquare q prime-    rc = powModInteger square ((q+1) `quot` 2) prime-    t1 = powModInteger square q prime-    msqr x = (x*x) `rem` prime-    msquare 0 x = x-    msquare k x = msquare (k-1) (msqr x)-    findPeriod per 1 = per-    findPeriod per x = findPeriod (per+1) (msqr x)-    loop !r t c m-        | t == 1    = r-        | otherwise = loop nextR nextT nextC nextM-          where-            nextM = findPeriod 0 t-            b     = msquare (m - 1 - nextM) c-            nextR = (r*b) `rem` prime-            nextC = msqr b-            nextT = (t*nextC) `rem` prime---- | @sqrtModPP n (prime,expo)@ calculates a square root of @n@---   modulo @prime^expo@ if one exists. @prime@ /must/ be a---   (positive) prime. @expo@ must be positive, @n@ must be coprime---   to @prime@-sqrtModPP :: Integer -> (Integer,Int) -> Maybe Integer-sqrtModPP n (2,e) = sqM2P n e-sqrtModPP n (prime,expo) = case sqrtModP n prime of-                             Just r -> fixup r-                             _      -> Nothing-  where-    fixup r = let diff' = r*r-n-              in if diff' == 0-                   then Just r-                   else case splitOff prime diff' of-                          (wordToInt -> e,q) | expo <= e -> Just r-                                | otherwise -> fmap (\inv -> hoist inv r (q `mod` prime) (prime^e)) (recipMod (2*r) prime)--    hoist inv root elim pp-        | diff' == 0    = root'-        | expo <= wordToInt ex    = root'-        | otherwise     = hoist inv root' (nelim `mod` prime) (prime^ex)-          where-            root' = (root + (inv*(prime-elim))*pp) `mod` (prime*pp)-            diff' = root'*root' - n-            (ex, nelim) = splitOff prime diff'---- dirty, dirty-sqM2P :: Integer -> Int -> Maybe Integer-sqM2P n e-    | e < 2     = Just (n `mod` 2)-    | n' == 0   = Just 0-    | odd k     = Nothing-    | otherwise = fmap ((`mod` mdl) . (`shiftL` k2)) $ solve s e2-      where-        mdl = 1 `shiftL` e-        n' = n `mod` mdl-        (wordToInt -> k,s) = shiftToOddCount n'-        k2 = k `quot` 2-        e2 = e-k-        solve _ 1 = Just 1-        solve 1 _ = Just 1-        solve r _-            | rem4 r == 3   = Nothing  -- otherwise r ≡ 1 (mod 4)-            | rem8 r == 5   = Nothing  -- otherwise r ≡ 1 (mod 8)-            | otherwise     = fixup r (wordToInt $ fst $ shiftToOddCount (r-1))-              where-                fixup x pw-                    | pw >= e2  = Just x-                    | otherwise = fixup x' pw'-                      where-                        x' = x + (1 `shiftL` (pw-1))-                        d = x'*x' - r-                        pw' = if d == 0 then e2 else wordToInt (fst (shiftToOddCount d))---- | @sqrtModF n primePowers@ calculates a square root of @n@ modulo---   @product [p^k | (p,k) <- primePowers]@ if one exists and all primes---   are distinct.---   The list must be non-empty, @n@ must be coprime with all primes.-sqrtModF :: Integer -> [(Integer,Int)] -> Maybe Integer-sqrtModF _ []  = Nothing-sqrtModF n pps = do roots <- mapM (sqrtModPP n) pps-                    chineseRemainder $ zip roots (map (uncurry (^)) pps)---- | @sqrtModFList n primePowers@ calculates all square roots of @n@ modulo---   @product [p^k | (p,k) <- primePowers]@ if all primes are distinct.---   The list must be non-empty, @n@ must be coprime with all primes.-sqrtModFList :: Integer -> [(Integer,Int)] -> [Integer]-sqrtModFList _ []  = []-sqrtModFList n pps = map fst $ foldl1 (liftM2 comb) cs-  where-    ms :: [Integer]-    ms = map (uncurry (^)) pps-    rs :: [[Integer]]-    rs = map (sqrtModPPList n) pps-    cs :: [[(Integer,Integer)]]-    cs = zipWith (\l m -> map (\x -> (x,m)) l) rs ms-    comb t1@(_,m1) t2@(_,m2) = (chineseRemainder2 t1 t2,m1*m2)---- | @sqrtModPPList n (prime,expo)@ calculates the list of all---   square roots of @n@ modulo @prime^expo@. The same restriction---   as in 'sqrtModPP' applies to the arguments.-sqrtModPPList :: Integer -> (Integer,Int) -> [Integer]-sqrtModPPList n (2,1) = [n `mod` 2]-sqrtModPPList n (2,expo)-    = case sqM2P n expo of-        Just r -> let m = 1 `shiftL` (expo-1)-                  in nub [r, (r+m) `mod` (2*m), (m-r) `mod` (2*m), 2*m-r]-        _ -> []-sqrtModPPList n pe@(prime,expo)-    = case sqrtModPP n pe of-        Just 0 -> [0]-        Just r -> [prime^expo - r, r] -- The group of units in Z/(p^e) is cyclic-        _      -> []----- Utilities--{-# SPECIALISE rem4 :: Integer -> Int,-                       Int -> Int,-                       Word -> Int-  #-}-rem4 :: Integral a => a -> Int-rem4 n = fromIntegral n .&. 3--{-# SPECIALISE rem8 :: Integer -> Int,-                       Int -> Int,-                       Word -> Int-  #-}-rem8 :: Integral a => a -> Int-rem8 n = fromIntegral n .&. 7--findNonSquare :: Integer -> Integer-findNonSquare n-    | rem8 n == 5 || rem8 n == 3  = 2-    | otherwise = search primelist-      where-        primelist = [3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67]-                        ++ map unPrime (sieveFrom (68 + n `rem` 4)) -- prevent sharing-        search (p:ps) = case jacobi p n of-          MinusOne -> p-          _        -> search ps-        search _ = error "Should never have happened, prime list exhausted."--recipMod :: Integer -> Integer -> Maybe Integer-recipMod x m = case recipModInteger x m of-  0 -> Nothing-  y -> Just y

@@ -17,20 +17,32 @@  import Control.Monad import Control.Monad.ST-import qualified Data.Vector.Mutable as MV+import Data.Proxy+import qualified Data.Vector.Generic as G+import qualified Data.Vector.Generic.Mutable as MG  import Math.NumberTheory.Powers.Squares+import Math.NumberTheory.Utils.FromIntegral  -- | @totientSum n@ is, for @n > 0@, the sum of @[totient k | k <- [1 .. n]]@, --   computed via generalised Möbius inversion. --   See <http://mathworld.wolfram.com/TotientSummatoryFunction.html> for the --   formula used for @totientSum@.-totientSum :: Int -> Integer-totientSum n-  | n < 1 = 0-  | otherwise = generalInversion (triangle . fromIntegral) n+--+-- >>> import Data.Proxy+-- >>> totientSum (Proxy :: Proxy Data.Vector.Unboxed.Vector) 100 :: Int+-- 3044+-- >>> totientSum (Proxy :: Proxy Data.Vector.Vector) 100 :: Integer+-- 3044+totientSum+    :: (Integral t, G.Vector v t)+    => Proxy v+    -> Word+    -> t+totientSum _ 0 = 0+totientSum proxy n = generalInversion proxy (triangle . fromIntegral) n   where-    triangle k = (k*(k+1)) `quot` 2+    triangle k = (k * (k + 1)) `quot` 2  -- | @generalInversion g n@ evaluates the generalised Möbius inversion of @g@ --   at the argument @n@.@@ -76,28 +88,36 @@ --   The value @f n@ is then computed by @generalInversion g n@. Note that when --   many values of @f@ are needed, there are far more efficient methods, this --   method is only appropriate to compute isolated values of @f@.-generalInversion :: (Int -> Integer) -> Int -> Integer-generalInversion fun n-    | n < 1     = error "Möbius inversion only defined on positive domain"-    | n == 1    = fun 1-    | n == 2    = fun 2 - fun 1-    | n == 3    = fun 3 - 2*fun 1-    | otherwise = fastInvert fun n--fastInvert :: (Int -> Integer) -> Int -> Integer-fastInvert fun n = runST (fastInvertST fun n)+generalInversion+    :: (Num t, G.Vector v t)+    => Proxy v+    -> (Word -> t)+    -> Word+    -> t+generalInversion proxy fun n = case n of+    0 ->error "Möbius inversion only defined on positive domain"+    1 -> fun 1+    2 -> fun 2 - fun 1+    3 -> fun 3 - 2*fun 1+    _ -> runST (fastInvertST proxy (fun . intToWord) (wordToInt n)) -fastInvertST :: forall s. (Int -> Integer) -> Int -> ST s Integer-fastInvertST fun n = do+fastInvertST+    :: forall s t v.+       (Num t, G.Vector v t)+    => Proxy v+    -> (Int -> t)+    -> Int+    -> ST s t+fastInvertST _ fun n = do     let !k0 = integerSquareRoot (n `quot` 2)         !mk0 = n `quot` (2*k0+1)         kmax a m = (a `quot` m - 1) `quot` 2 -    small <- MV.unsafeNew (mk0 + 1) :: ST s (MV.MVector s Integer)-    MV.unsafeWrite small 0 0-    MV.unsafeWrite small 1 $! (fun 1)+    small <- MG.unsafeNew (mk0 + 1) :: ST s (G.Mutable v s t)+    MG.unsafeWrite small 0 0+    MG.unsafeWrite small 1 $! (fun 1)     when (mk0 >= 2) $-        MV.unsafeWrite small 2 $! (fun 2 - fun 1)+        MG.unsafeWrite small 2 $! (fun 2 - fun 1)      let calcit :: Int -> Int -> Int -> ST s (Int, Int)         calcit switch change i@@ -107,22 +127,22 @@                 let mloop !acc k !m                         | k < switch    = kloop acc k                         | otherwise     = do-                            val <- MV.unsafeRead small m+                            val <- MG.unsafeRead small m                             let nxtk = kmax i (m+1)                             mloop (acc - fromIntegral (k-nxtk)*val) nxtk (m+1)                     kloop !acc k                         | k == 0    = do-                            MV.unsafeWrite small i $! acc+                            MG.unsafeWrite small i $! acc                             calcit switch change (i+1)                         | otherwise = do-                            val <- MV.unsafeRead small (i `quot` (2*k+1))+                            val <- MG.unsafeRead small (i `quot` (2*k+1))                             kloop (acc-val) (k-1)                 mloop (fun i - fun (i `quot` 2)) ((i-1) `quot` 2) 1      (sw, ch) <- calcit 1 8 3-    large <- MV.unsafeNew k0 :: ST s (MV.MVector s Integer)+    large <- MG.unsafeNew k0 :: ST s (G.Mutable v s t) -    let calcbig :: Int -> Int -> Int -> ST s (MV.MVector s Integer)+    let calcbig :: Int -> Int -> Int -> ST s (G.Mutable v s t)         calcbig switch change j             | j == 0    = return large             | (2*j-1)*change <= n   = calcbig (switch+1) (change + 4*switch+6) j@@ -131,21 +151,21 @@                     mloop !acc k m                         | k < switch    = kloop acc k                         | otherwise     = do-                            val <- MV.unsafeRead small m+                            val <- MG.unsafeRead small m                             let nxtk = kmax i (m+1)                             mloop (acc - fromIntegral (k-nxtk)*val) nxtk (m+1)                     kloop !acc k                         | k == 0    = do-                            MV.unsafeWrite large (j-1) $! acc+                            MG.unsafeWrite large (j-1) $! acc                             calcbig switch change (j-1)                         | otherwise = do                             let m = i `quot` (2*k+1)                             val <- if m <= mk0-                                     then MV.unsafeRead small m-                                     else MV.unsafeRead large (k*(2*j-1)+j-1)+                                     then MG.unsafeRead small m+                                     else MG.unsafeRead large (k*(2*j-1)+j-1)                             kloop (acc-val) (k-1)                 mloop (fun i - fun (i `quot` 2)) ((i-1) `quot` 2) 1      mvec <- calcbig sw ch k0-    MV.unsafeRead mvec 0+    MG.unsafeRead mvec 0 

@@ -10,7 +10,9 @@ {-# LANGUAGE FlexibleContexts    #-} {-# LANGUAGE ScopedTypeVariables #-} -module Math.NumberTheory.MoebiusInversion.Int+{-# OPTIONS_HADDOCK hide #-}++module Math.NumberTheory.MoebiusInversion.Int {-# DEPRECATED "Use Math.NumberTheory.MoebiusInversion" #-}     ( generalInversion     , totientSum     ) where

@@ -28,6 +28,7 @@ import Data.Bits import Data.List (foldl') import qualified Data.Set as Set+import Data.Vector.Unboxed (toList)  import Numeric.Natural @@ -38,6 +39,7 @@ import qualified Math.NumberTheory.Powers.Squares as P2 import qualified Math.NumberTheory.Powers.Cubes as P3 import qualified Math.NumberTheory.Powers.Fourth as P4+import Math.NumberTheory.Primes.Small import Math.NumberTheory.Utils.FromIntegral (intToWord, wordToInt)  -- | Calculate an integer root, @'integerRoot' k n@ computes the (floor of) the @k@-th@@ -256,20 +258,18 @@       (k,r) -> findHighPower (gcd k e) ((p,k):pws) r (P2.integerSquareRoot r) ps findHighPower e pws m _ [] = finishPower e pws m +smallOddPrimes :: [Integer]+smallOddPrimes+  = takeWhile (< spBound)+  $ map fromIntegral+  $ tail+  $ toList smallPrimes+ spBEx :: Word spBEx = 14  spBound :: Integer spBound = 2^spBEx--smallOddPrimes :: [Integer]-smallOddPrimes = 3:5:primes'-  where-    primes' = 7:11:13:17:19:23:29:filter isPrime (takeWhile (< spBound) $ scanl (+) 31 (cycle [6,4,2,4,2,4,6,2]))-    isPrime n = go primes'-      where-        go (p:ps) = (p*p > n) || (n `rem` p /= 0 && go ps)-        go []     = True  -- n large, has no prime divisors < spBound finishPower :: Word -> [(Integer, Word)] -> Integer -> (Integer, Word)

@@ -18,9 +18,12 @@   , fromFactors   ) where +import Prelude hiding ((^), gcd) import Control.Arrow- import Data.Semigroup+import Data.Semiring (Semiring(..), Mul(..), (^))+import qualified Data.Semiring as Semiring+import Unsafe.Coerce  import Math.NumberTheory.Euclidean import Math.NumberTheory.Euclidean.Coprimes@@ -84,7 +87,7 @@ -- -- >>> fromValue 123 -- Prefactored {prefValue = 123, prefFactors = Coprimes {unCoprimes = [(123,1)]}}-fromValue :: (Eq a, Num a) => a -> Prefactored a+fromValue :: (Eq a, GcdDomain a) => a -> Prefactored a fromValue a = Prefactored a (singleton a 1)  -- | Create 'Prefactored' from a given list of pairwise coprime@@ -94,10 +97,10 @@ -- Prefactored {prefValue = 23100, prefFactors = Coprimes {unCoprimes = [(28,1),(33,1),(5,2)]}} -- >>> fromFactors (splitIntoCoprimes [(140, 2), (165, 3)]) -- Prefactored {prefValue = 88045650000, prefFactors = Coprimes {unCoprimes = [(28,2),(33,3),(5,5)]}}-fromFactors :: Num a => Coprimes a Word -> Prefactored a-fromFactors as = Prefactored (product (map (uncurry (^)) (unCoprimes as))) as+fromFactors :: Semiring a => Coprimes a Word -> Prefactored a+fromFactors as = Prefactored (getMul $ foldMap (\(a, k) -> Mul $ a ^ k) (unCoprimes as)) as -instance Euclidean a => Num (Prefactored a) where+instance (Eq a, Num a, GcdDomain a) => Num (Prefactored a) where   Prefactored v1 _ + Prefactored v2 _     = fromValue (v1 + v2)   Prefactored v1 _ - Prefactored v2 _@@ -109,7 +112,7 @@   signum (Prefactored v _) = Prefactored (signum v) mempty   fromInteger n = fromValue (fromInteger n) -instance (Euclidean a, UniqueFactorisation a) => UniqueFactorisation (Prefactored a) where+instance (Eq a, GcdDomain a, UniqueFactorisation a) => UniqueFactorisation (Prefactored a) where   factorise (Prefactored _ f)     = concatMap (\(x, xm) -> map (\(p, k) -> (Prime $ fromValue $ unPrime p, k * xm)) (factorise x)) (unCoprimes f)   isPrime (Prefactored _ f) = case unCoprimes f of

@@ -10,6 +10,7 @@ {-# LANGUAGE LambdaCase        #-}  {-# OPTIONS_GHC -fno-warn-orphans #-}+{-# OPTIONS_GHC -fno-warn-deprecations #-}  module Math.NumberTheory.Primes     ( Prime@@ -17,6 +18,7 @@     , nextPrime     , precPrime     , UniqueFactorisation(..)+    , factorBack     , -- * Old interface       primes     ) where@@ -97,6 +99,9 @@   factorise = map (coerce integerToNatural *** id) . F.factorise . naturalToInteger   isPrime n = if T.isPrime (toInteger n) then Just (Prime n) else Nothing +factorBack :: Num a => [(Prime a, Word)] -> a+factorBack = product . map (\(p, k) -> unPrime p ^ k)+ -- | Smallest prime, greater or equal to argument. -- -- > nextPrime (-100) ==    2@@ -192,21 +197,21 @@  enumFromThenGeneric :: (Bits a, Integral a, UniqueFactorisation a) => Prime a -> Prime a -> [Prime a] enumFromThenGeneric p@(Prime p') (Prime q') = case p' `compare` q' of-  LT -> filter (\(Prime r') -> (r' - p') `mod` delta == 0) $ enumFromGeneric p+  LT -> filter (\(Prime r') -> (r' - p') `rem` delta == 0) $ enumFromGeneric p     where       delta = q' - p'   EQ -> repeat p-  GT -> filter (\(Prime r') -> (p' - r') `mod` delta == 0) $ reverse $ enumFromToGeneric (Prime 2) p+  GT -> filter (\(Prime r') -> (p' - r') `rem` delta == 0) $ reverse $ enumFromToGeneric (Prime 2) p     where       delta = p' - q'  enumFromThenToGeneric :: (Bits a, Integral a, UniqueFactorisation a) => Prime a -> Prime a -> Prime a -> [Prime a] enumFromThenToGeneric p@(Prime p') (Prime q') r@(Prime r') = case p' `compare` q' of-  LT -> filter (\(Prime t') -> (t' - p') `mod` delta == 0) $ enumFromToGeneric p r+  LT -> filter (\(Prime t') -> (t' - p') `rem` delta == 0) $ enumFromToGeneric p r     where       delta = q' - p'   EQ -> if p' <= r' then repeat p else []-  GT -> filter (\(Prime t') -> (p' - t') `mod` delta == 0) $ reverse $ enumFromToGeneric r p+  GT -> filter (\(Prime t') -> (p' - t') `rem` delta == 0) $ reverse $ enumFromToGeneric r p     where       delta = p' - q' 

@@ -23,8 +23,9 @@ #include "MachDeps.h"  import Math.NumberTheory.Primes.Sieve.Eratosthenes-import Math.NumberTheory.Primes.Sieve.Indexing-import Math.NumberTheory.Primes.Counting.Approximate+    (PrimeSieve(..), primeList, primeSieve, psieveFrom, sieveTo, sieveBits, sieveRange, countFromTo, countToNth, countAll, nthPrimeCt)+import Math.NumberTheory.Primes.Sieve.Indexing (toPrim, idxPr)+import Math.NumberTheory.Primes.Counting.Approximate (nthPrimeApprox, approxPrimeCount) import Math.NumberTheory.Primes.Types import Math.NumberTheory.Powers.Squares import Math.NumberTheory.Powers.Cubes

@@ -1,5 +1,6 @@ -- | -- Module:      Math.NumberTheory.Primes.Factorisation+-- Description: Deprecated -- Copyright:   (c) 2011 Daniel Fischer -- Licence:     MIT -- Maintainer:  Daniel Fischer <daniel.is.fischer@googlemail.com>@@ -12,7 +13,7 @@ -- and in the case of the Carmichael function that the list of prime factors -- with their multiplicities is ascending. -module Math.NumberTheory.Primes.Factorisation+module Math.NumberTheory.Primes.Factorisation {-# DEPRECATED "Use 'Math.NumberTheory.Primes.factorise' instead" #-}     ( -- * Factorisation functions       -- $algorithm       -- ** Complete factorisation

@@ -1,5 +1,6 @@ -- | -- Module:      Math.NumberTheory.Primes.Factorisation.Certified+-- Description: Deprecated -- Copyright:   (c) 2011 Daniel Fischer -- Licence:     MIT -- Maintainer:  Daniel Fischer <daniel.is.fischer@googlemail.com>@@ -9,7 +10,7 @@ -- For large numbers, this will be very slow in general. -- Use only if you're paranoid or must be /really/ sure. {-# LANGUAGE BangPatterns, CPP #-}-module Math.NumberTheory.Primes.Factorisation.Certified+module Math.NumberTheory.Primes.Factorisation.Certified {-# DEPRECATED "This module will be removed in the next release" #-}   ( certifiedFactorisation   , certificateFactorisation   , provenFactorisation@@ -31,27 +32,24 @@ certifiedFactorisation :: Integer -> [(Integer, Word)] certifiedFactorisation = map fst . certificateFactorisation --- | @'certificateFactorisation' n@ produces a 'provenFactorisation'---   with a default bound of @100000@.+-- | @'certificateFactorisation' n@ produces a 'provenFactorisation'. certificateFactorisation :: Integer -> [((Integer, Word),PrimalityProof)]-certificateFactorisation n = provenFactorisation 100000 n+certificateFactorisation n = provenFactorisation n --- | @'provenFactorisation' bound n@ constructs a the prime factorisation of @n@+-- | @'provenFactorisation' n@ constructs a the prime factorisation of @n@ --   (which must be positive) together with proofs of primality of the factors,---   using trial division up to @bound@ (which is arbitrarily replaced by @2000@---   if the supplied value is smaller) and elliptic curve factorisation for the+--   using trial division up to 2^16 and elliptic curve factorisation for the --   remaining factors if necessary. -- --   Construction of primality proofs can take a /very/ long time, so this --   will usually be slow (but should be faster than using 'factorise' and --   proving the primality of the factors from scratch).-provenFactorisation :: Integer -> Integer -> [((Integer, Word),PrimalityProof)]-provenFactorisation _ 1 = []-provenFactorisation bd n+provenFactorisation :: Integer -> [((Integer, Word),PrimalityProof)]+provenFactorisation 1 = []+provenFactorisation n     | n < 2     = error "provenFactorisation: argument not positive"-    | bd < 2000 = provenFactorisation 2000 n-    | otherwise = test $-      case smallFactors bd n of+    | otherwise = let bd = 65536 in test $+      case smallFactors n of         (sfs,mb) -> map (\t@(p,_) -> (t, smallCert p)) sfs             ++ case mb of                  Nothing -> []

@@ -59,6 +59,7 @@ import Data.Semigroup #endif import Data.Traversable+import Data.Vector.Unboxed (toList)  import GHC.TypeNats.Compat @@ -67,10 +68,10 @@ import Math.NumberTheory.Moduli.Class import Math.NumberTheory.Powers.General     (highestPower, largePFPower) import Math.NumberTheory.Powers.Squares     (integerSquareRoot')-import Math.NumberTheory.Primes.Sieve.Eratosthenes-import Math.NumberTheory.Primes.Sieve.Indexing+import Math.NumberTheory.Primes.Sieve.Eratosthenes (PrimeSieve(..), psieveFrom)+import Math.NumberTheory.Primes.Sieve.Indexing (toPrim)+import Math.NumberTheory.Primes.Small import Math.NumberTheory.Primes.Testing.Probabilistic-import Math.NumberTheory.Primes.Types (unPrime) import Math.NumberTheory.Unsafe import Math.NumberTheory.Utils @@ -100,10 +101,10 @@ --   seem to be slower than the 'StdGen' based variant. stepFactorisation :: Integer -> [(Integer, Word)] stepFactorisation n-    = let (sfs,mb) = smallFactors 100000 n+    = let (sfs,mb) = smallFactors n       in sfs ++ case mb of                   Nothing -> []-                  Just r  -> curveFactorisation (Just 10000000000) bailliePSW+                  Just r  -> curveFactorisation (Just $ 65536 * 65536) bailliePSW                                                 (\m k -> (if k < (m-1) then k else error "Curves exhausted",k+1)) 6 Nothing r  -- | @'defaultStdGenFactorisation'@ first strips off all small prime factors and then,@@ -122,10 +123,10 @@ --   @n@ must be larger than @1@. defaultStdGenFactorisation' :: StdGen -> Integer -> [(Integer, Word)] defaultStdGenFactorisation' sg n-    = let (sfs,mb) = smallFactors 100000 n+    = let (sfs,mb) = smallFactors n       in sfs ++ case mb of                   Nothing -> []-                  Just m  -> stdGenFactorisation (Just 10000000000) sg Nothing m+                  Just m  -> stdGenFactorisation (Just $ 65536 * 65536) sg Nothing m  ---------------------------------------------------------------------------------------------------- --                                    Factorisation wrappers                                      --@@ -155,7 +156,7 @@ --   make a huge difference. So, if the default takes too long, try another one; or you can improve your --   chances for a quick result by running several instances in parallel. -----   'curveFactorisation' @n@ requires that small (< 100000) prime factors of @n@+--   'curveFactorisation' @n@ requires that small (< 65536) prime factors of @n@ --   have been stripped before. Otherwise it is likely to cycle forever. When in doubt, --   use 'defaultStdGenFactorisation'. --@@ -288,8 +289,9 @@       g -> Just g   where     n = getMod s-    smallPrimes = takeWhile (<= b1) (2 : 3 : 5 : list primeStore)-    smallPowers = map findPower smallPrimes+    smallPowers+      = map findPower+      $ takeWhile (<= b1) (2 : 3 : 5 : list primeStore)     findPower p = go p       where         go acc@@ -346,23 +348,24 @@ list sieves = concat [[off + toPrim i | i <- [0 .. li], unsafeAt bs i]                                 | PS vO bs <- sieves, let { (_,li) = bounds bs; off = fromInteger vO; }] --- | @'smallFactors' bound n@ finds all prime divisors of @n > 1@ up to @bound@ by trial division and returns the+-- | @'smallFactors' n@ finds all prime divisors of @n > 1@ up to 2^16 by trial division and returns the --   list of these together with their multiplicities, and a possible remaining factor which may be composite.-smallFactors :: Integer -> Integer -> ([(Integer, Word)], Maybe Integer)-smallFactors bd n = case shiftToOddCount n of+smallFactors :: Integer -> ([(Integer, Word)], Maybe Integer)+smallFactors n = case shiftToOddCount n of                       (0,m) -> go m prms                       (k,m) -> (2,k) <: if m == 1 then ([],Nothing) else go m prms   where-    prms = map unPrime $ tail (primeStore >>= primeList)+    prms = map fromIntegral $ toList smallPrimes     x <: ~(l,b) = (x:l,b)+    go m []+      | m < 65536 * 65536 = ([(m, 1)], Nothing)+      | otherwise         = ([], Just m)     go m (p:ps)-        | m < p*p   = ([(m,1)], Nothing)-        | bd < p    = ([], Just m)-        | otherwise = case splitOff p m of-                        (0,_) -> go m ps-                        (k,r) | r == 1 -> ([(p,k)], Nothing)-                              | otherwise -> (p,k) <: go r ps-    go m [] = ([(m,1)], Nothing)+      | m < p*p   = ([(m,1)], Nothing)+      | otherwise = case splitOff p m of+                      (0,_) -> go m ps+                      (k,r) | r == 1 -> ([(p,k)], Nothing)+                            | otherwise -> (p,k) <: go r ps  -- | For a given estimated decimal length of the smallest prime factor -- ("tier") return parameters B1, B2 and the number of curves to try

@@ -17,7 +17,7 @@     , trialDivisionPrimeTo     ) where -import Math.NumberTheory.Primes.Sieve.Eratosthenes+import Math.NumberTheory.Primes.Sieve.Eratosthenes (primeList, primeSieve, psieveList) import Math.NumberTheory.Powers.Squares import Math.NumberTheory.Primes.Types import Math.NumberTheory.Utils

@@ -1,5 +1,6 @@ -- | -- Module:      Math.NumberTheory.Primes.Sieve+-- Description: Deprecated -- Copyright:   (c) 2011 Daniel Fischer -- Licence:     MIT -- Maintainer:  Daniel Fischer <daniel.is.fischer@googlemail.com>@@ -13,7 +14,8 @@ -- However, that means we must store data for primes up to the square root of -- where sieving is done, thus sieving primes up to @n@ requires -- @/O/(sqrt n/log n)@ space.-module Math.NumberTheory.Primes.Sieve++module Math.NumberTheory.Primes.Sieve {-# DEPRECATED "Use 'Enum' instance of 'Math.NumberTheory.Primes.Prime' instead" #-}     ( -- * Limitations       -- $limits 

@@ -0,0 +1,20 @@+-- |+-- Module:      Math.NumberTheory.Primes.Small+-- Copyright:   (c) 2019 Andrew Lelechenko+-- Licence:     MIT+-- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>+--+-- This is an internal module,+-- defining an array of precalculated primes < 2^16.+--++module Math.NumberTheory.Primes.Small+  ( smallPrimes+  ) where++import Data.Vector.Unboxed (Vector, fromList)+import Data.Word++smallPrimes :: Vector Word16+smallPrimes = fromList+  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@@ -1,11 +1,12 @@ -- | -- Module:      Math.NumberTheory.Primes.Testing.Certificates+-- Description: Deprecated -- Copyright:   (c) 2011 Daniel Fischer -- Licence:     MIT -- Maintainer:  Daniel Fischer <daniel.is.fischer@googlemail.com> -- -- Certificates for primality or compositeness.-module Math.NumberTheory.Primes.Testing.Certificates+module Math.NumberTheory.Primes.Testing.Certificates {-# DEPRECATED "This module will be removed in the next release" #-}     ( -- * Certificates       Certificate(..)     , argueCertificate

@@ -33,13 +33,13 @@ import GHC.Integer.GMP.Internals  import Math.NumberTheory.Moduli.Class-import Math.NumberTheory.Utils+import Math.NumberTheory.Powers.Squares+import Math.NumberTheory.Primes (unPrime) import Math.NumberTheory.Primes.Factorisation.TrialDivision import Math.NumberTheory.Primes.Factorisation.Montgomery import Math.NumberTheory.Primes.Testing.Probabilistic-import Math.NumberTheory.Primes.Sieve.Eratosthenes-import Math.NumberTheory.Primes.Types (unPrime)-import Math.NumberTheory.Powers.Squares+import Math.NumberTheory.Primes.Sieve.Eratosthenes (primeList, primeSieve)+import Math.NumberTheory.Utils  -- | A certificate of either compositeness or primality of an --   'Integer'. Only numbers @> 1@ can be certified, trying to@@ -248,7 +248,7 @@                     ((p,_):_) | p < n       -> Composite (Factors n p (n `quot` p))                               | otherwise   -> Prime (TrialDivision n r2)                     _ -> error "Impossible"-    | otherwise = case smallFactors 100000 n of+    | otherwise = case smallFactors n of                     ([], Just _) | not (isStrongFermatPP n 2) -> Composite (StrongFermat n 2)                                  | not (lucasTest n) -> Composite (LucasSelfridge n)                                  | otherwise -> Prime (certifyBPSW n)       -- if it isn't we error and ask for a report.

@@ -29,16 +29,15 @@ import Control.DeepSeq import Data.Coerce import Data.List                                       (mapAccumL, partition)-import Data.Maybe                                      (fromMaybe)+import Data.Maybe import Data.Ord                                        (comparing)+import qualified Data.Semiring as S import GHC.Generics                                    (Generic)  import qualified Math.NumberTheory.Euclidean            as ED import Math.NumberTheory.Moduli.Sqrt-import qualified Math.NumberTheory.Primes.Sieve         as Sieve-import qualified Math.NumberTheory.Primes.Testing       as Testing import Math.NumberTheory.Primes.Types-import qualified Math.NumberTheory.Primes  as U+import qualified Math.NumberTheory.Primes as U import Math.NumberTheory.Utils                          (mergeBy) import Math.NumberTheory.Utils.FromIntegral @@ -75,6 +74,16 @@     fromInteger n = n :+ 0     signum = snd . absSignum +instance S.Semiring EisensteinInteger where+    plus          = (+)+    times         = (*)+    zero          = 0 :+ 0+    one           = 1 :+ 0+    fromNatural n = fromIntegral n :+ 0++instance S.Ring EisensteinInteger where+    negate = negate+ -- | Returns an @EisensteinInteger@'s sign, and its associate in the first -- sextant. absSignum :: EisensteinInteger -> (EisensteinInteger, EisensteinInteger)@@ -96,25 +105,26 @@ associates :: EisensteinInteger -> [EisensteinInteger] associates e = map (e *) ids +instance ED.GcdDomain EisensteinInteger+ instance ED.Euclidean EisensteinInteger where-  quotRem = divHelper quot-  divMod  = divHelper div+    degree = fromInteger . norm+    quotRem = divHelper  -- | Function that does most of the underlying work for @divMod@ and -- @quotRem@, apart from choosing the specific integer division algorithm. -- This is instead done by the calling function (either @divMod@ which uses -- @div@, or @quotRem@, which uses @quot@.) divHelper-    :: (Integer -> Integer -> Integer)-    -> EisensteinInteger+    :: EisensteinInteger     -> EisensteinInteger     -> (EisensteinInteger, EisensteinInteger)-divHelper divide g h =-    let nr :+ ni = g * conjugate h+divHelper g h = (q, r)+    where+        nr :+ ni = g * conjugate h         denom = norm h-        q = divide nr denom :+ divide ni denom-        p = h * q-    in (q, g - p)+        q = ((nr + signum nr * denom `quot` 2) `quot` denom) :+   ((ni + signum ni * denom `quot` 2) `quot` denom)+        r = g - h * q  -- | Conjugate a Eisenstein integer. conjugate :: EisensteinInteger -> EisensteinInteger@@ -133,8 +143,8 @@           -- Special case, @1 - ω@ is the only Eisenstein prime with norm @3@,           --  and @abs (1 - ω) = 2 + ω@.           | a' == 2 && b' == 1         = True-          | b' == 0 && a' `mod` 3 == 2 = Testing.isPrime a'-          | nE `mod` 3 == 1            = Testing.isPrime nE+          | b' == 0 && a' `mod` 3 == 2 = isJust $ U.isPrime a'+          | nE `mod` 3 == 1            = isJust $ U.isPrime nE           | otherwise = False   where nE       = norm e         a' :+ b' = abs e@@ -179,7 +189,7 @@ findPrime :: Prime Integer -> U.Prime EisensteinInteger findPrime p = case sqrtsModPrime (9*k*k - 1) p of     []    -> error "findPrime: argument must be prime p = 6k + 1"-    z : _ -> Prime $ ED.gcd (unPrime p :+ 0) ((z - 3 * k) :+ 1)+    z : _ -> Prime $ abs $ ED.gcd (unPrime p :+ 0) ((z - 3 * k) :+ 1)     where         k :: Integer         k = unPrime p `div` 6@@ -194,7 +204,7 @@ primes = coerce $ (2 :+ 1) : mergeBy (comparing norm) l r   where     leftPrimes, rightPrimes :: [Prime Integer]-    (leftPrimes, rightPrimes) = partition (\p -> unPrime p `mod` 3 == 2) Sieve.primes+    (leftPrimes, rightPrimes) = partition (\p -> unPrime p `mod` 3 == 2) [U.nextPrime 2 ..]     rightPrimes' = filter (\prime -> unPrime prime `mod` 3 == 1) $ tail rightPrimes     l = [unPrime p :+ 0 | p <- leftPrimes]     r = [g | p <- rightPrimes', let x :+ y = unPrime (findPrime p), g <- [x :+ y, x :+ (x - y)]]@@ -255,7 +265,7 @@       | unPrime p `mod` 3 == 2       = let e' = e `quot` 2 in (z `quotI` (unPrime p ^ e'), [(Prime (unPrime p :+ 0), e')]) -      -- The @`mod` 3 == 0@ case need not be verified because the+      -- The @`rem` 3 == 0@ case need not be verified because the       -- only Eisenstein primes whose norm are a multiple of 3       -- are @1 - ω@ and its associates, which have already been       -- removed by the above @go z (3, e)@ pattern match.

@@ -24,18 +24,16 @@ import Control.DeepSeq (NFData) import Data.Coerce import Data.List (mapAccumL, partition)-import Data.Maybe (fromMaybe)+import Data.Maybe import Data.Ord (comparing)+import qualified Data.Semiring as S import GHC.Generics - import qualified Math.NumberTheory.Euclidean as ED import Math.NumberTheory.Moduli.Sqrt import Math.NumberTheory.Powers (integerSquareRoot) import Math.NumberTheory.Primes.Types-import qualified Math.NumberTheory.Primes.Sieve as Sieve-import qualified Math.NumberTheory.Primes.Testing as Testing-import qualified Math.NumberTheory.Primes  as U+import qualified Math.NumberTheory.Primes as U import Math.NumberTheory.Utils              (mergeBy) import Math.NumberTheory.Utils.FromIntegral @@ -70,6 +68,16 @@     fromInteger n = n :+ 0     signum = snd . absSignum +instance S.Semiring GaussianInteger where+    plus          = (+)+    times         = (*)+    zero          = 0 :+ 0+    one           = 1 :+ 0+    fromNatural n = fromIntegral n :+ 0++instance S.Ring GaussianInteger where+    negate = negate+ absSignum :: GaussianInteger -> (GaussianInteger, GaussianInteger) absSignum z@(a :+ b)     | a == 0 && b == 0 =   (z, 0)              -- origin@@ -78,21 +86,22 @@     | a <  0 && b <= 0 = ((-a) :+ (-b), -1)    -- third quadrant: (-inf, 0) x (-inf, 0]i     | otherwise        = ((-b) :+   a, -ι)     -- fourth quadrant: [0, inf) x (-inf, 0)i +instance ED.GcdDomain GaussianInteger+ instance ED.Euclidean GaussianInteger where-    quotRem = divHelper quot-    divMod  = divHelper div+    degree = fromInteger . norm+    quotRem = divHelper  divHelper-    :: (Integer -> Integer -> Integer)-    -> GaussianInteger+    :: GaussianInteger     -> GaussianInteger     -> (GaussianInteger, GaussianInteger)-divHelper divide g h =-    let nr :+ ni = g * conjugate h+divHelper g h = (q, r)+    where+        nr :+ ni = g * conjugate h         denom = norm h-        q = divide nr denom :+ divide ni denom-        p = h * q-    in (q, g - p)+        q = ((nr + signum nr * denom `quot` 2) `quot` denom) :+ ((ni + signum ni * denom `quot` 2) `quot` denom)+        r = g - h * q  -- |Conjugate a Gaussian integer. conjugate :: GaussianInteger -> GaussianInteger@@ -105,9 +114,9 @@ -- |Compute whether a given Gaussian integer is prime. isPrime :: GaussianInteger -> Bool isPrime g@(x :+ y)-    | x == 0 && y /= 0 = abs y `mod` 4 == 3 && Testing.isPrime y-    | y == 0 && x /= 0 = abs x `mod` 4 == 3 && Testing.isPrime x-    | otherwise        = Testing.isPrime $ norm g+    | x == 0 && y /= 0 = abs y `mod` 4 == 3 && isJust (U.isPrime y)+    | y == 0 && x /= 0 = abs x `mod` 4 == 3 && isJust (U.isPrime x)+    | otherwise        = isJust $ U.isPrime $ norm g  -- |An infinite list of the Gaussian primes. Uses primes in Z to exhaustively -- generate all Gaussian primes (up to associates), in order of ascending@@ -116,7 +125,7 @@ primes = coerce $ (1 :+ 1) : mergeBy (comparing norm) l r   where     leftPrimes, rightPrimes :: [Prime Integer]-    (leftPrimes, rightPrimes) = partition (\p -> unPrime p `mod` 4 == 3) (tail Sieve.primes)+    (leftPrimes, rightPrimes) = partition (\p -> unPrime p `mod` 4 == 3) [U.nextPrime 3 ..]     l = [unPrime p :+ 0 | p <- leftPrimes]     r = [g | p <- rightPrimes, let Prime (x :+ y) = findPrime p, g <- [x :+ y, y :+ x]] 

@@ -1,17 +0,0 @@--- |--- Module:      Math.NumberTheory.Recurrencies--- Description: Deprecated--- Copyright:   (c) 2018 Alexandre Rodrigues Baldé--- Licence:     MIT--- Maintainer:  Alexandre Rodrigues Baldé <alexandrer_b@outlook.com>-----module Math.NumberTheory.Recurrencies {-# DEPRECATED "Use `Math.NumberTheory.Recurrences` instead." #-}-    ( module Math.NumberTheory.Recurrences.Linear-    , module Math.NumberTheory.Recurrences.Bilinear-    , module Math.NumberTheory.Recurrences.Pentagonal-    ) where--import Math.NumberTheory.Recurrences.Bilinear-import Math.NumberTheory.Recurrences.Linear-import Math.NumberTheory.Recurrences.Pentagonal (partition)

@@ -1,38 +0,0 @@--- |--- Module:      Math.NumberTheory.Recurrencies.Bilinear--- Description: Deprecated--- Copyright:   (c) 2016 Andrew Lelechenko--- Licence:     MIT--- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>------ Bilinear recurrent sequences and Bernoulli numbers,--- roughly covering Ch. 5-6 of /Concrete Mathematics/--- by R. L. Graham, D. E. Knuth and O. Patashnik.------ #memory# __Note on memory leaks and memoization.__--- Top-level definitions in this module are polymorphic, so the results of computations are not retained in memory.--- Make them monomorphic to take advantages of memoization. Compare------ >>> :set +s--- >>> binomial !! 1000 !! 1000 :: Integer--- 1--- (0.01 secs, 1,385,512 bytes)--- >>> binomial !! 1000 !! 1000 :: Integer--- 1--- (0.01 secs, 1,381,616 bytes)------ against------ >>> let binomial' = binomial :: [[Integer]]--- >>> binomial' !! 1000 !! 1000 :: Integer--- 1--- (0.01 secs, 1,381,696 bytes)--- >>> binomial' !! 1000 !! 1000 :: Integer--- 1--- (0.01 secs, 391,152 bytes)--module Math.NumberTheory.Recurrencies.Bilinear {-# DEPRECATED "Use `Math.NumberTheory.Recurrences.Bilinear` instead." #-}-    ( module Math.NumberTheory.Recurrences.Bilinear-    ) where--import Math.NumberTheory.Recurrences.Bilinear

@@ -1,14 +0,0 @@--- |--- Module:      Math.NumberTheory.Recurrencies.Linear--- Description: Deprecated--- Copyright:   (c) 2011 Daniel Fischer--- Licence:     MIT--- Maintainer:  Daniel Fischer <daniel.is.fischer@googlemail.com>------ Efficient calculation of linear recurrent sequences, including Fibonacci and Lucas sequences.--module Math.NumberTheory.Recurrencies.Linear {-# DEPRECATED "Use `Math.NumberTheory.Recurrences.Linear` instead." #-}-    ( module Math.NumberTheory.Recurrences.Linear-    ) where--import Math.NumberTheory.Recurrences.Linear

@@ -10,7 +10,9 @@ -- over a set {3, 4}, and 24 is not. -- +{-# LANGUAGE FlexibleContexts    #-} {-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE TypeApplications    #-}  module Math.NumberTheory.SmoothNumbers   ( -- * Create a smooth basis@@ -29,12 +31,14 @@   ) where  import Prelude hiding (div, mod, gcd)+import Data.Bits (Bits) import Data.Coerce import Data.List (nub)+import Data.Semiring (isZero) import qualified Data.Set as S+ import qualified Math.NumberTheory.Euclidean as E-import Math.NumberTheory.Primes (unPrime)-import Math.NumberTheory.Primes.Sieve (primes)+import Math.NumberTheory.Primes  -- | An abstract representation of a smooth basis. -- It consists of a set of numbers ≥2.@@ -49,8 +53,9 @@ -- Just (SmoothBasis {unSmoothBasis = [2,4]}) -- >>> fromSet (Set.fromList [1, 3]) -- should be >= 2 -- Nothing-fromSet :: E.Euclidean a => S.Set a -> Maybe (SmoothBasis a)+fromSet :: (Eq a, E.GcdDomain a) => S.Set a -> Maybe (SmoothBasis a) fromSet s = if isValid l then Just (SmoothBasis l) else Nothing where l = S.elems s+{-# DEPRECATED fromSet "Use 'fromList' instead " #-}  -- | Build a 'SmoothBasis' from a list of numbers ≥2. --@@ -62,7 +67,7 @@ -- Just (SmoothBasis {unSmoothBasis = [2,4]}) -- >>> fromList [1, 3] -- should be >= 2 -- Nothing-fromList :: E.Euclidean a => [a] -> Maybe (SmoothBasis a)+fromList :: (Eq a, E.GcdDomain a) => [a] -> Maybe (SmoothBasis a) fromList l = if isValid l' then Just (SmoothBasis l') else Nothing   where     l' = nub l@@ -73,10 +78,14 @@ -- Just (SmoothBasis {unSmoothBasis = [2,3,5,7]}) -- >>> fromSmoothUpperBound 1 -- Nothing-fromSmoothUpperBound :: Integral a => a -> Maybe (SmoothBasis a)-fromSmoothUpperBound n = if (n < 2)-                         then Nothing-                         else Just $ SmoothBasis $ takeWhile (<= n) $ map unPrime primes+fromSmoothUpperBound+  :: (Integral a, Enum (Prime a), Bits a, UniqueFactorisation a)+  => a+  -> Maybe (SmoothBasis a)+fromSmoothUpperBound n+  | n < 2     = Nothing+  | otherwise = Just $ SmoothBasis $ map unPrime [nextPrime 2 .. precPrime n]+{-# DEPRECATED fromSmoothUpperBound "Use 'fromList' with an appropriate list of primes instead " #-}  -- | Helper used by @smoothOver@ (@Integral@ constraint) and @smoothOver'@ -- (@Euclidean@ constraint) Since the typeclass constraint is just@@ -85,10 +94,14 @@ -- This function relies on the fact that for any element of a smooth basis @p@ -- and any @a@ it is true that @norm (a * p) > norm a@. -- This condition is not checked.-smoothOver' :: forall a b . (Eq a, Num a, Ord b) => (a -> b) -> SmoothBasis a -> [a]+smoothOver'+  :: forall a b. (Eq a, Num a, Ord b)+  => (a -> b)+  -> SmoothBasis a+  -> [a] smoothOver' norm pl =     foldr-    (\p l -> mergeListLists $ iterate (map $ abs . (p*)) l)+    (\p l -> mergeListLists $ iterate (map (* p)) l)     [1]     (nub $ unSmoothBasis pl)   where@@ -104,9 +117,9 @@         go2 a [] = a         go2 [] b = b         go2 a@(ah:at) b@(bh:bt)-          | norm bh < norm ah   = bh : (go2 a bt)-          | ah == bh    = ah : (go2 at bt)-          | otherwise = ah : (go2 at b)+          | norm bh < norm ah = bh : (go2 a bt)+          | abs ah == abs bh  = ah : (go2 at bt)+          | otherwise         = ah : (go2 at b)  -- | Generate an infinite ascending list of -- <https://en.wikipedia.org/wiki/Smooth_number smooth numbers>@@ -115,7 +128,7 @@ -- >>> import Data.Maybe -- >>> take 10 (smoothOver (fromJust (fromList [2, 5]))) -- [1,2,4,5,8,10,16,20,25,32]-smoothOver :: Integral a => SmoothBasis a -> [a]+smoothOver :: (Ord a, Num a) => SmoothBasis a -> [a] smoothOver = smoothOver' abs  -- | Generate an ascending list of@@ -129,12 +142,12 @@ -- >>> import Data.Maybe -- >>> smoothOverInRange (fromJust (fromList [2, 5])) 100 200 -- [100,125,128,160,200]-smoothOverInRange :: forall a. Integral a => SmoothBasis a -> a -> a -> [a]+smoothOverInRange :: (Ord a, Num a) => SmoothBasis a -> a -> a -> [a] smoothOverInRange s lo hi   = takeWhile (<= hi)   $ dropWhile (< lo)-  $ coerce-  $ smoothOver (coerce s :: SmoothBasis (E.WrappedIntegral a))+  $ smoothOver s+{-# DEPRECATED smoothOverInRange "Use 'smoothOver' instead" #-}  -- | Generate an ascending list of -- <https://en.wikipedia.org/wiki/Smooth_number smooth numbers>@@ -150,7 +163,7 @@ -- >>> smoothOverInRangeBF (fromJust (fromList [2, 5])) 100 200 -- [100,125,128,160,200] smoothOverInRangeBF-  :: forall a. (Enum a, E.Euclidean a)+  :: (Eq a, Enum a, E.GcdDomain a)   => SmoothBasis a   -> a   -> a@@ -159,23 +172,19 @@   = coerce   $ filter (isSmooth prs)   $ coerce [lo..hi]+{-# DEPRECATED smoothOverInRangeBF "Use filtering by 'isSmooth' instead" #-} --- | isValid assumes that the list is sorted and unique and then checks if the list is suitable to be a SmoothBasis.-isValid :: (Eq a, Num a) => [a] -> Bool-isValid pl = length pl /= 0 && v' pl-  where-    v' :: (Eq a, Num a) => [a] -> Bool-    v' []     = True-    v' (x:xs) = x /= 0 && abs x /= 1 && abs x == x && v' xs+isValid :: (Eq a, E.GcdDomain a) => [a] -> Bool+isValid [] = False+isValid xs = all (\x -> not (isZero x) && not (E.isUnit x)) xs  -- | @isSmooth@ checks if a given number is smooth under a certain @SmoothBasis@. -- Does not check if the @SmoothBasis@ is valid.-isSmooth :: forall a . E.Euclidean a => SmoothBasis a -> a -> Bool-isSmooth prs x = mf (unSmoothBasis prs) x+isSmooth :: (Eq a, E.GcdDomain a) => SmoothBasis a -> a -> Bool+isSmooth prs x = not (isZero x) && go (unSmoothBasis prs) x   where-    mf :: [a] -> a -> Bool-    mf _         0 = False-    mf []        n = abs n == 1 -- mf means manually factor-    mf pl@(p:ps) n = if E.mod n p == 0-                     then mf pl (E.div n p)-                     else mf ps n+    go :: (Eq a, E.GcdDomain a) => [a] -> a -> Bool+    go [] n = E.isUnit n+    go pps@(p:ps) n = case n `E.divide` p of+      Nothing -> go ps n+      Just q  -> go pps q || go ps n

@@ -1,13 +0,0 @@--- |--- Module:      Math.NumberTheory.Recurrencies--- Description: Deprecated--- Copyright:   (c) 2019 Andrew Lelechenko--- Licence:     MIT--- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>-----module Math.NumberTheory.UniqueFactorisation {-# DEPRECATED "Use `Math.NumberTheory.Primes` instead." #-}-    ( module Math.NumberTheory.Primes-    ) where--import Math.NumberTheory.Primes

@@ -39,6 +39,7 @@ import GHC.Natural  import Data.Bits+import Data.Semiring (Semiring(..), isZero) import Math.NumberTheory.Euclidean  uncheckedShiftR :: Word -> Int -> Word@@ -160,12 +161,13 @@ bitCountInt :: Int -> Int bitCountInt = popCount -splitOff :: Euclidean a => a -> a -> (Word, a)-splitOff _ 0 = (0, 0) -- prevent infinite loop-splitOff p n = go 0 n+splitOff :: (Eq a, GcdDomain a) => a -> a -> (Word, a)+splitOff p n+  | isZero n  = (0, zero) -- prevent infinite loop+  | otherwise = go 0 n   where-    go !k m = case m `quotRem` p of-      (q, 0) -> go (k + 1) q+    go !k m = case m `divide` p of+      Just q -> go (k + 1) q       _      -> (k, m) {-# INLINABLE splitOff #-} @@ -194,7 +196,7 @@  -- | Work around https://ghc.haskell.org/trac/ghc/ticket/14085 recipMod :: Integer -> Integer -> Maybe Integer-recipMod x m = case recipModInteger (x `mod` m) m of+recipMod x m = case recipModInteger (x `P.mod` m) m of   0 -> Nothing   y -> Just y 

@@ -27,6 +27,7 @@ import Data.Coerce import Data.Map (Map) import qualified Data.Map.Strict as M+import Data.Maybe import Data.Semiring (Semiring(..)) import Numeric.Natural @@ -65,7 +66,7 @@ -- and all a_i and b_i are divisors of n. Return Dirichlet series cs, -- which contains all terms as * bs = sum_i m_i/c_i^s such that c_i divides n. timesAndCrop-  :: (Euclidean a, Ord a, Semiring b)+  :: (Num a, Euclidean a, Ord a, Semiring b)   => a   -> DirichletSeries a b   -> DirichletSeries a b@@ -78,7 +79,7 @@   | (b, fb) <- M.assocs bs   , let nb = n `quot` b   , (a, fa) <- takeWhile ((<= nb) . fst) (M.assocs as)-  , nb `rem` a == 0+  , isJust (nb `divide` a)   ] {-# SPECIALISE timesAndCrop :: Semiring b => Int -> DirichletSeries Int b -> DirichletSeries Int b -> DirichletSeries Int b #-} {-# SPECIALISE timesAndCrop :: Semiring b => Word -> DirichletSeries Word b -> DirichletSeries Word b -> DirichletSeries Word b #-}

@@ -1,21 +1,24 @@ -- | -- Module:      Math.NumberTheory.Zeta--- Copyright:   (c) 2018 Andrew Lelechenko+-- Copyright:   (c) 2018 Alexandre Rodrigues Baldé, Andrew Lelechenko -- Licence:     MIT -- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com> ----- Interface to work with Riemann zeta-function and Dirichlet beta-function.+-- Numeric evaluation of various zeta-functions.  {-# LANGUAGE ScopedTypeVariables #-}  module Math.NumberTheory.Zeta-  ( module Math.NumberTheory.Zeta.Dirichlet-  , module Math.NumberTheory.Zeta.Hurwitz-  , module Math.NumberTheory.Zeta.Riemann-  , module Math.NumberTheory.Zeta.Utils+  ( -- * Riemann zeta-function+    zetas+  , zetasEven+    -- * Dirichlet beta-function+  , betas+  , betasOdd+    -- * Hurwitz zeta-functions+  , zetaHurwitz   ) where  import Math.NumberTheory.Zeta.Dirichlet import Math.NumberTheory.Zeta.Hurwitz import Math.NumberTheory.Zeta.Riemann-import Math.NumberTheory.Zeta.Utils

@@ -8,6 +8,8 @@  {-# LANGUAGE ScopedTypeVariables #-} +{-# OPTIONS_HADDOCK hide #-}+ module Math.NumberTheory.Zeta.Dirichlet   ( betas   , betasEven@@ -52,11 +54,6 @@  -- | Infinite sequence of approximate (up to given precision) -- values of Dirichlet beta-function at integer arguments, starting with @β(0)@.------ The algorithm previously used to compute @β@ for even arguments was derived--- from <https://arxiv.org/pdf/0910.5004.pdf An Euler-type formula for β(2n) and closed-form expressions for a class of zeta series>--- by F. M. S. Lima, formula (12), but is now based on the--- 'Math.NumberTheory.Zeta.Hurwitz.zetaHurwitz' recurrence. -- -- >>> take 5 (betas 1e-14) :: [Double] -- [0.5,0.7853981633974483,0.9159655941772189,0.9689461462593694,0.9889445517411051]

@@ -8,6 +8,8 @@  {-# LANGUAGE ScopedTypeVariables #-} +{-# OPTIONS_HADDOCK hide #-}+ module Math.NumberTheory.Zeta.Hurwitz   ( zetaHurwitz   ) where@@ -15,13 +17,11 @@ import Math.NumberTheory.Recurrences (bernoulli, factorial) import Math.NumberTheory.Zeta.Utils  (skipEvens, skipOdds) --- | Values of Hurwitz zeta function evaluated at @ζ(s, a)@ with--- @forall t1 . (Floating t1, Ord t1) => a ∈ t1@, and @s ∈ [0, 1 ..]@.+-- | Values of Hurwitz zeta function evaluated at @ζ(s, a)@ for @s ∈ [0, 1 ..]@. -- -- The algorithm used was based on the Euler-Maclaurin formula and was derived -- from <http://fredrikj.net/thesis/thesis.pdf Fast and Rigorous Computation of Special Functions to High Precision> -- by F. Johansson, chapter 4.8, formula 4.8.5.--- -- The error for each value in this recurrence is given in formula 4.8.9 as an --  indefinite integral, and in formula 4.8.12 as a closed form formula. --@@ -29,8 +29,8 @@ -- the type chosen. -- -- For instance, when using @Double@s, it does not make sense--- to provide a number @ε >= 1e-53@ as the desired precision. For @Float@s,--- providing an @ε >= 1e-24@ also does not make sense.+-- to provide a number @ε < 1e-53@ as the desired precision. For @Float@s,+-- providing an @ε < 1e-24@ also does not make sense. -- Example of how to call the function: -- -- >>> zetaHurwitz 1e-15 0.25 !! 5

@@ -8,6 +8,8 @@  {-# LANGUAGE ScopedTypeVariables #-} +{-# OPTIONS_HADDOCK hide #-}+ module Math.NumberTheory.Zeta.Riemann   ( zetas   , zetasEven@@ -45,11 +47,6 @@  -- | Infinite sequence of approximate (up to given precision) -- values of Riemann zeta-function at integer arguments, starting with @ζ(0)@.------ Computations for odd arguments were formerly performed in accordance to--- <https://cr.yp.to/bib/2000/borwein.pdf Computational strategies for the Riemann zeta function>--- by J. M. Borwein, D. M. Bradley, R. E. Crandall, formula (57), but now use--- the 'Math.NumberTheory.Zeta.Hurwitz.zetaHurwitz' recurrence. -- -- >>> take 5 (zetas 1e-14) :: [Double] -- [-0.5,Infinity,1.6449340668482264,1.2020569031595942,1.0823232337111381]

@@ -1,15 +1,15 @@ name:          arithmoi-version:       0.9.0.0+version:       0.10.0.0 cabal-version: >=1.10 build-type:    Simple license:       MIT license-file:  LICENSE-copyright:     (c) 2011 Daniel Fischer, 2016-2018 Andrew Lelechenko, Carter Schonwald-maintainer:    Carter Schonwald  carter at wellposed dot com,-               Andrew Lelechenko andrew dot lelechenko at gmail dot com+copyright:     (c) 2016-2019 Andrew Lelechenko, Carter Schonwald, 2011 Daniel Fischer+maintainer:    Andrew Lelechenko andrew dot lelechenko at gmail dot com,+               Carter Schonwald  carter at wellposed dot com stability:     Provisional-homepage:      https://github.com/cartazio/arithmoi-bug-reports:   https://github.com/cartazio/arithmoi/issues+homepage:      https://github.com/Bodigrim/arithmoi+bug-reports:   https://github.com/Bodigrim/arithmoi/issues synopsis:      Efficient basic number-theoretic functions. description:   A library of basic functionality needed for@@ -18,14 +18,14 @@   Primes and related things (totients, factorisation),   powers (integer roots and tests, modular exponentiation). category:      Math, Algorithms, Number Theory-author:        Daniel Fischer+author:        Andrew Lelechenko, Daniel Fischer tested-with:   GHC ==8.0.2 GHC ==8.2.2 GHC ==8.4.4 GHC ==8.6.5 GHC ==8.8.1 extra-source-files:   Changes  source-repository head   type: git-  location: https://github.com/cartazio/arithmoi+  location: https://github.com/Bodigrim/arithmoi  flag check-bounds   description:@@ -38,6 +38,7 @@     base >=4.9 && <5,     array >=0.5 && <0.6,     containers >=0.5 && <0.7,+    constraints,     deepseq,     exact-pi >=0.5,     ghc-prim <0.6,@@ -45,7 +46,7 @@     integer-logarithms >=1.0,     random >=1.0 && <1.2,     transformers >=0.4 && <0.6,-    semirings >= 0.2,+    semirings >= 0.4.2,     vector >= 0.12   exposed-modules:     GHC.TypeNats.Compat@@ -65,6 +66,7 @@     Math.NumberTheory.Moduli.Equations     Math.NumberTheory.Moduli.Jacobi     Math.NumberTheory.Moduli.PrimitiveRoot+    Math.NumberTheory.Moduli.Singleton     Math.NumberTheory.Moduli.Sqrt     Math.NumberTheory.MoebiusInversion     Math.NumberTheory.MoebiusInversion.Int@@ -81,27 +83,22 @@     Math.NumberTheory.Primes.Factorisation     Math.NumberTheory.Primes.Factorisation.Certified     Math.NumberTheory.Primes.Sieve+    Math.NumberTheory.Primes.Small     Math.NumberTheory.Primes.Testing     Math.NumberTheory.Primes.Testing.Certificates     Math.NumberTheory.Quadratic.GaussianIntegers     Math.NumberTheory.Quadratic.EisensteinIntegers     Math.NumberTheory.Recurrences-    Math.NumberTheory.Recurrencies     Math.NumberTheory.Recurrences.Bilinear-    Math.NumberTheory.Recurrencies.Bilinear     Math.NumberTheory.Recurrences.Linear-    Math.NumberTheory.Recurrencies.Linear     Math.NumberTheory.SmoothNumbers-    Math.NumberTheory.UniqueFactorisation     Math.NumberTheory.Zeta     Math.NumberTheory.Zeta.Dirichlet     Math.NumberTheory.Zeta.Hurwitz     Math.NumberTheory.Zeta.Riemann   other-modules:     Math.NumberTheory.ArithmeticFunctions.Class-    Math.NumberTheory.ArithmeticFunctions.SieveBlock.Unboxed     Math.NumberTheory.ArithmeticFunctions.Standard-    Math.NumberTheory.Moduli.SqrtOld     Math.NumberTheory.Primes.Counting.Approximate     Math.NumberTheory.Primes.Counting.Impl     Math.NumberTheory.Primes.Factorisation.Montgomery@@ -155,8 +152,8 @@     Math.NumberTheory.Moduli.EquationsTests     Math.NumberTheory.Moduli.JacobiTests     Math.NumberTheory.Moduli.PrimitiveRootTests+    Math.NumberTheory.Moduli.SingletonTests     Math.NumberTheory.Moduli.SqrtTests-    Math.NumberTheory.MoebiusInversion.IntTests     Math.NumberTheory.MoebiusInversionTests     Math.NumberTheory.Powers.CubesTests     Math.NumberTheory.Powers.FourthTests@@ -191,6 +188,7 @@     base,     arithmoi,     array,+    constraints,     containers,     deepseq,     gauge,

@@ -1,6 +1,7 @@ {-# LANGUAGE ExistentialQuantification #-}-{-# LANGUAGE RankNTypes #-}-{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE RankNTypes                #-}+{-# LANGUAGE ScopedTypeVariables       #-}+{-# LANGUAGE TypeApplications          #-}  {-# OPTIONS_GHC -fno-warn-type-defaults #-} @@ -11,6 +12,7 @@   ) where  import Gauge.Main+import Control.Monad import Data.Maybe import GHC.TypeNats.Compat import Data.Proxy@@ -18,7 +20,8 @@  import Math.NumberTheory.Moduli.Class (isMultElement, KnownNat, MultMod, multElement, getVal,Mod) import Math.NumberTheory.Moduli.DiscreteLogarithm (discreteLogarithm)-import Math.NumberTheory.Moduli.PrimitiveRoot (PrimitiveRoot, isPrimitiveRoot, unPrimitiveRoot, cyclicGroupFromModulo)+import Math.NumberTheory.Moduli.PrimitiveRoot+import Math.NumberTheory.Moduli.Singleton  data Case = forall m. KnownNat m => Case (PrimitiveRoot m) (MultMod m) String @@ -31,7 +34,7 @@ makeCase (a,b,n,s) =   case someNatVal n of     SomeNat (_ :: Proxy m) ->-      Case <$> isPrimitiveRoot a' <*> isMultElement b' <*> pure s+      Case <$> join (isPrimitiveRoot @Integer <$> cyclicGroup <*> pure a') <*> isMultElement b' <*> pure s         where a' = fromInteger a :: Mod m               b' = fromInteger b @@ -45,15 +48,16 @@ rangeCases :: Natural -> Int -> [Case] rangeCases start num = take num $ do   n <- [start..]-  _cg <- maybeToList $ cyclicGroupFromModulo n   case someNatVal n of-    SomeNat (_ :: Proxy m) -> do-      a <- take 1 $ mapMaybe isPrimitiveRoot [2 :: Mod m .. maxBound]-      b <- take 1 $ filter (/= unPrimitiveRoot a) $ mapMaybe isMultElement [2 .. maxBound]-      return $ Case a b (show n)+    SomeNat (_ :: Proxy m) -> case cyclicGroup :: Maybe (CyclicGroup Integer m) of+      Nothing -> []+      Just cg -> do+        a <- take 1 $ mapMaybe (isPrimitiveRoot cg) [2 :: Mod m .. maxBound]+        b <- take 1 $ filter (/= unPrimitiveRoot a) $ mapMaybe isMultElement [2 .. maxBound]+        return $ Case a b (show n)  discreteLogarithm' :: Case -> Natural-discreteLogarithm' (Case a b _) = discreteLogarithm a b+discreteLogarithm' (Case a b _) = discreteLogarithm (fromJust cyclicGroup) a b  benchSuite :: Benchmark benchSuite = bgroup "Discrete logarithm"

@@ -1,3 +1,4 @@+{-# LANGUAGE FlexibleContexts      #-} {-# LANGUAGE TypeApplications      #-}  {-# OPTIONS_GHC -fno-warn-type-defaults #-}@@ -7,6 +8,7 @@   ) where  import Gauge.Main+import Data.Bits (Bits) import Numeric.Natural  import Math.NumberTheory.ArithmeticFunctions.Inverse@@ -22,7 +24,7 @@ countInverseTotient :: (Ord a, Euclidean a, UniqueFactorisation a) => a -> Word countInverseTotient = inverseTotient (const 1) -countInverseSigma :: (Integral a, Euclidean a, UniqueFactorisation a) => a -> Word+countInverseSigma :: (Integral a, Euclidean a, UniqueFactorisation a, Enum (Prime a), Bits a) => a -> Word countInverseSigma = inverseSigma (const 1)  benchSuite :: Benchmark

@@ -8,7 +8,7 @@ import System.Random  import Math.NumberTheory.Logarithms (integerLog2)-import Math.NumberTheory.Primes.Factorisation+import Math.NumberTheory.Primes (factorise) import Math.NumberTheory.Primes.Testing  genInteger :: Int -> Int -> Integer

@@ -1,3 +1,5 @@+{-# LANGUAGE RankNTypes #-}+ {-# OPTIONS_GHC -fno-warn-type-defaults #-}  module Math.NumberTheory.PrimitiveRootsBench@@ -5,20 +7,29 @@   ) where  import Gauge.Main+import Data.Constraint import Data.Maybe  import Math.NumberTheory.Moduli.PrimitiveRoot+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.Primes  primRootWrap :: Integer -> Word -> Integer -> Bool-primRootWrap p k g = isPrimitiveRoot' (CGOddPrimePower p' k) g-  where p' = fromJust $ isPrime p+primRootWrap p k g = case fromJust $ cyclicGroupFromFactors [(p', k)] of+  Some cg -> case proofFromCyclicGroup cg of+    Sub Dict -> isJust $ isPrimitiveRoot cg (fromInteger g)+  where+    p' = fromJust $ isPrime p  primRootWrap2 :: Integer -> Word -> Integer -> Bool-primRootWrap2 p k g = isPrimitiveRoot' (CGDoubleOddPrimePower p' k) g-  where p' = fromJust $ isPrime p+primRootWrap2 p k g = case fromJust $ cyclicGroupFromFactors [(two, 1), (p', k)] of+  Some cg -> case proofFromCyclicGroup cg of+    Sub Dict -> isJust $ isPrimitiveRoot cg (fromInteger g)+  where+    two = fromJust $ isPrime 2+    p'  = fromJust $ isPrime p -cyclicWrap :: Integer -> Maybe (CyclicGroup Integer)+cyclicWrap :: Integer -> Maybe (Some (CyclicGroup Integer)) cyclicWrap = cyclicGroupFromModulo  benchSuite :: Benchmark

@@ -1,4 +1,5 @@ {-# OPTIONS_GHC -fno-warn-type-defaults #-}+{-# OPTIONS_GHC -fno-warn-deprecations #-}  module Math.NumberTheory.SequenceBench   ( benchSuite@@ -11,8 +12,8 @@ import Data.Bits  import Math.NumberTheory.Primes (Prime(..))-import Math.NumberTheory.Primes.Sieve as P-import Math.NumberTheory.Primes.Testing as P+import Math.NumberTheory.Primes.Sieve+import Math.NumberTheory.Primes.Testing  filterIsPrime :: (Integer, Integer) -> Integer filterIsPrime (p, q) = sum $ takeWhile (<= q) $ dropWhile (< p) $ filter isPrime (map toPrim [toIdx p .. toIdx q])

@@ -4,19 +4,18 @@   ( benchSuite   ) where -import Data.List (genericTake) import Data.Maybe import Gauge.Main -import Math.NumberTheory.Euclidean (Euclidean)+import Math.NumberTheory.Primes import Math.NumberTheory.SmoothNumbers -doBench :: (Euclidean a, Integral a) => a -> a-doBench lim = sum $ genericTake lim $ smoothOver $ fromJust $ fromSmoothUpperBound lim+doBench :: Int -> Int+doBench lim = sum $ take lim $ smoothOver $ fromJust $ fromList $ map unPrime [nextPrime 2 .. precPrime lim]  benchSuite :: Benchmark benchSuite = bgroup "SmoothNumbers"-  [ bench "100"      $ nf doBench    (100 :: Int)-  , bench "1000"     $ nf doBench   (1000 :: Int)-  , bench "10000"    $ nf doBench  (10000 :: Int)+  [ bench "100"      $ nf doBench   100+  , bench "1000"     $ nf doBench  1000+  , bench "10000"    $ nf doBench 10000   ]

@@ -20,6 +20,7 @@ import Test.Tasty import Test.Tasty.HUnit +import Data.Bits (Bits) import qualified Data.Set as S  import Math.NumberTheory.ArithmeticFunctions@@ -134,10 +135,10 @@ ------------------------------------------------------------------------------- -- Sigma -sigmaProperty1 :: forall a. (Euclidean a, UniqueFactorisation a, Integral a) => Positive a -> Bool+sigmaProperty1 :: forall a. (Euclidean a, UniqueFactorisation a, Integral a, Enum (Prime a), Bits a) => Positive a -> Bool sigmaProperty1 (Positive x) = x `S.member` asSetOfPreimages inverseSigma (sigma 1 x) -sigmaProperty2 :: (Euclidean a, UniqueFactorisation a, Integral a) => Positive a -> Bool+sigmaProperty2 :: (Euclidean a, UniqueFactorisation a, Integral a, Enum (Prime a), Bits a) => Positive a -> Bool sigmaProperty2 (Positive x) = all (== x) (S.map (sigma 1) (asSetOfPreimages inverseSigma x))  -- | http://oeis.org/A055486

@@ -23,22 +23,15 @@ import Data.Semigroup #endif import qualified Data.Vector as V-import qualified Data.Vector.Unboxed as U  import Math.NumberTheory.ArithmeticFunctions import Math.NumberTheory.ArithmeticFunctions.SieveBlock-import Math.NumberTheory.Primes (unPrime)  pointwiseTest :: (Eq a, Show a) => ArithmeticFunction Word a -> Word -> Word -> IO () pointwiseTest f lowIndex len = assertEqual "pointwise"     (runFunctionOverBlock f lowIndex len)     (V.generate (fromIntegral len) (runFunction f . (+ lowIndex) . fromIntegral)) -unboxedTest :: (Eq a, U.Unbox a, Show a) => SieveBlockConfig a -> IO ()-unboxedTest config = assertEqual "unboxed"-    (sieveBlock config 1 1000)-    (U.convert $ sieveBlockUnboxed config 1 1000)- moebiusTest :: Word -> Word -> Bool moebiusTest m n   = m == 0@@ -68,13 +61,6 @@     pairToTest :: Word -> Word -> TestTree     pairToTest m n = testCase (show m ++ "," ++ show n) $ assertBool "should be equal" $ moebiusTest m n -multiplicativeConfig :: (Word -> Word -> Word) -> SieveBlockConfig Word-multiplicativeConfig f = SieveBlockConfig-  { sbcEmpty                = 1-  , sbcAppend               = (*)-  , sbcFunctionOnPrimePower = f . unPrime-  }- moebiusConfig :: SieveBlockConfig Moebius moebiusConfig = SieveBlockConfig   { sbcEmpty = MoebiusP@@ -95,12 +81,6 @@     , testCase "smallOmega" $ pointwiseTest smallOmegaA 1 1000     , testCase "bigOmega"   $ pointwiseTest bigOmegaA   1 1000     , testCase "carmichael" $ pointwiseTest carmichaelA 1 1000-    ]-  , testGroup "unboxed"-    [ testCase "id"      $ unboxedTest $ multiplicativeConfig (^)-    , testCase "tau"     $ unboxedTest $ multiplicativeConfig (\_ a -> succ a )-    , testCase "moebius" $ unboxedTest moebiusConfig-    , testCase "totient" $ unboxedTest $ multiplicativeConfig (\p a -> (p - 1) * p ^ (a - 1))     ]   , testGroup "special moebius" moebiusSpecialCases   ]

@@ -46,7 +46,7 @@  -- | All divisors of n truly divides n. divisorsProperty3 :: NonZero Natural -> Bool-divisorsProperty3 (NonZero n) = all (\d -> n `mod` d == 0) (runFunction divisorsA n)+divisorsProperty3 (NonZero n) = all (\d -> n `rem` d == 0) (runFunction divisorsA n)  -- | 'divisorsA' matches 'divisorsSmallA' divisorsProperty4 :: NonZero Int -> Bool@@ -129,10 +129,10 @@ -- | congruences 1,2,3,4 from https://en.wikipedia.org/wiki/Ramanujan_tau_function ramanujanCongruence1 :: NonZero Natural -> Bool ramanujanCongruence1 (NonZero n)-  | k == 1 = (ramanujan n' - sigma 11 n') `mod` (2^11) == 0-  | k == 3 = (ramanujan n' - 1217 * sigma 11 n') `mod` (2^13) == 0-  | k == 5 = (ramanujan n' - 1537 * sigma 11 n') `mod` (2^12) == 0-  | k == 7 = (ramanujan n' - 705 * sigma 11 n') `mod` (2^14) == 0+  | k == 1 = (ramanujan n' - sigma 11 n') `rem` (2^11) == 0+  | k == 3 = (ramanujan n' - 1217 * sigma 11 n') `rem` (2^13) == 0+  | k == 5 = (ramanujan n' - 1537 * sigma 11 n') `rem` (2^12) == 0+  | k == 7 = (ramanujan n' - 705 * sigma 11 n') `rem` (2^14) == 0   | otherwise = True   where k = n `mod` 8         n' = fromIntegral n :: Integer@@ -140,8 +140,8 @@ -- | congruences 8,9 from https://en.wikipedia.org/wiki/Ramanujan_tau_function ramanujanCongruence2 :: NonZero Natural -> Bool ramanujanCongruence2 (NonZero n)-  | (n `mod` 7) `elem` [0,1,2,4] = m `mod` 7 == 0-  | otherwise                    = m `mod` 49 == 0+  | (n `mod` 7) `elem` [0,1,2,4] = m `rem` 7 == 0+  | otherwise                    = m `rem` 49 == 0   where m = ramanujan n' - n' * sigma 9 n'         n' = fromIntegral n :: Integer @@ -228,7 +228,7 @@  -- | carmichaeil divides totient carmichaelProperty1 :: NonZero Natural -> Bool-carmichaelProperty1 (NonZero n) = runFunction totientA n `mod` runFunction carmichaelA n == 0+carmichaelProperty1 (NonZero n) = runFunction totientA n `rem` runFunction carmichaelA n == 0  -- | carmichael matches baseline from OEIS. carmichaelOeis :: Assertion

@@ -17,11 +17,11 @@ import Test.Tasty                                     (TestTree, testGroup) import Test.Tasty.HUnit                               (Assertion, assertEqual,                                                       testCase)+import Test.Tasty.QuickCheck as QC hiding (Positive(..))  import qualified Math.NumberTheory.Euclidean as ED import qualified Math.NumberTheory.Quadratic.EisensteinIntegers as E import Math.NumberTheory.Primes-import Math.NumberTheory.Primes.Sieve (primes) import Math.NumberTheory.TestUtils                    (Positive (..),                                                        testSmallAndQuick) @@ -43,22 +43,10 @@     inFirstSextant = x' > y' && y' >= 0     isAssociate = z' `elem` map (\e -> z * (1 E.:+ 1) ^ e) [0 .. 5] --- | Verify that @div@ and @mod@ are what `divMod` produces.-divModProperty1 :: E.EisensteinInteger -> E.EisensteinInteger -> Bool-divModProperty1 x y = y == 0 || (q == q' && r == r')-  where-    (q, r) = ED.divMod x y-    q'     = ED.div x y-    r'     = ED.mod x y---- | Verify that @divModE` produces the right quotient and remainder.-divModProperty2 :: E.EisensteinInteger -> E.EisensteinInteger -> Bool-divModProperty2 x y = (y == 0) || (x `ED.div` y) * y + (x `ED.mod` y) == x---- | Verify that @divModE@ produces a remainder smaller than the divisor with+-- | Verify that @rem@ produces a remainder smaller than the divisor with -- regards to the Euclidean domain's function.-modProperty1 :: E.EisensteinInteger -> E.EisensteinInteger -> Bool-modProperty1 x y = (y == 0) || (E.norm $ x `ED.mod` y) < (E.norm y)+remProperty1 :: E.EisensteinInteger -> E.EisensteinInteger -> Bool+remProperty1 x y = (y == 0) || (E.norm $ x `ED.rem` y) < (E.norm y)  -- | Verify that @quot@ and @rem@ are what `quotRem` produces. quotRemProperty1 :: E.EisensteinInteger -> E.EisensteinInteger -> Bool@@ -76,7 +64,7 @@ gcdEProperty1 :: E.EisensteinInteger -> E.EisensteinInteger -> Bool gcdEProperty1 z1 z2   = z1 == 0 && z2 == 0-  || z1 `ED.rem` z == 0 && z2 `ED.rem` z == 0 && z == abs z+  || z1 `ED.rem` z == 0 && z2 `ED.rem` z == 0   where     z = ED.gcd z1 z2 @@ -91,7 +79,7 @@  -- | A special case that tests rounding/truncating in GCD. gcdESpecialCase1 :: Assertion-gcdESpecialCase1 = assertEqual "gcd" 1 $ ED.gcd (12 E.:+ 23) (23 E.:+ 34)+gcdESpecialCase1 = assertEqual "gcd" (1 E.:+ 1) $ ED.gcd (12 E.:+ 23) (23 E.:+ 34)  findPrimesProperty1 :: Positive Int -> Bool findPrimesProperty1 (Positive index) =@@ -152,13 +140,10 @@ testSuite :: TestTree testSuite = testGroup "EisensteinIntegers" $   [ testSmallAndQuick "forall z . z == signum z * abs z" signumAbsProperty-  , testSmallAndQuick "abs z always returns an @EisensteinInteger@ in the\-                      \ first sextant of the complex plane" absProperty+  , testSmallAndQuick "abs z rotates to the first sextant" absProperty   , testGroup "Division"-    [ testSmallAndQuick "divE and modE work properly" divModProperty1-    , testSmallAndQuick "divModE works properly" divModProperty2-    , testSmallAndQuick "The remainder's norm is smaller than the divisor's"-                        modProperty1+    [ testSmallAndQuick "The remainder's norm is smaller than the divisor's"+                        remProperty1      , testSmallAndQuick "quotE and remE work properly" quotRemProperty1     , testSmallAndQuick "quotRemE works properly" quotRemProperty2@@ -167,24 +152,21 @@   , testGroup "g.c.d."     [ testSmallAndQuick "The g.c.d. of two Eisenstein integers divides them"                         gcdEProperty1-    , testSmallAndQuick "A common divisor of two Eisenstein integers always\-                        \ divides the g.c.d. of those two integers"+    -- smallcheck takes too long+    , QC.testProperty "Common divisor divides gcd"                         gcdEProperty2     , testCase          "g.c.d. (12 :+ 23) (23 :+ 34)" gcdESpecialCase1     ]   , testSmallAndQuick "The Eisenstein norm function is multiplicative"                     euclideanDomainProperty1   , testGroup "Primality"-    [ testSmallAndQuick "Eisenstein primes found by the norm search used in\-                        \ findPrime are really prime"+    [ testSmallAndQuick "findPrime returns prime"                         findPrimesProperty1-    , testSmallAndQuick "Eisenstein primes generated by `primes` are actually\-                        \ primes"+    , testSmallAndQuick "primes are actually prime"                         primesProperty1-    , testSmallAndQuick "The infinite list of Eisenstein primes produced by\-                        \ `primes` is ordered. "+    , testSmallAndQuick "primes is ordered"                         primesProperty2-    , testSmallAndQuick "All generated primes are in the first sextant"+    , testSmallAndQuick "primes are in the first sextant"                         primesProperty3     ] 

@@ -9,6 +9,7 @@  {-# LANGUAGE CPP                 #-} {-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE TypeApplications    #-}  {-# OPTIONS_GHC -fno-warn-type-defaults  #-} {-# OPTIONS_GHC -fno-warn-unused-imports #-}@@ -21,6 +22,7 @@ import Prelude hiding (gcd) import Test.Tasty import Test.Tasty.HUnit+import Test.Tasty.QuickCheck as QC hiding (Positive(..))  import Control.Arrow import Data.Bits@@ -31,10 +33,11 @@  import Math.NumberTheory.Euclidean import Math.NumberTheory.Euclidean.Coprimes+import Math.NumberTheory.Quadratic.GaussianIntegers import Math.NumberTheory.TestUtils  -- | Check that 'extendedGCD' is consistent with documentation.-extendedGCDProperty :: forall a. (Bits a, Euclidean a, Ord a) => AnySign a -> AnySign a -> Bool+extendedGCDProperty :: forall a. (Bits a, Num a, GcdDomain a, Euclidean a, Ord a) => AnySign a -> AnySign a -> Bool extendedGCDProperty (AnySign a) (AnySign b)   | isNatural a = True -- extendedGCD does not make sense for Natural   | otherwise =@@ -50,43 +53,62 @@ isNatural a = isNothing (bitSizeMaybe a) && not (isSigned a)  -- | Check that numbers are coprime iff their gcd equals to 1.-coprimeProperty :: (Euclidean a) => AnySign a -> AnySign a -> Bool+coprimeProperty :: (Eq a, Num a, GcdDomain a, Euclidean a) => AnySign a -> AnySign a -> Bool coprimeProperty (AnySign a) (AnySign b) = coprime a b == (gcd a b == 1) -splitIntoCoprimesProperty1 :: [(Positive Natural, Power Word)] -> Bool+splitIntoCoprimesProperty1+  :: (Eq a, Num a, GcdDomain a)+  => [(a, Power Word)]+  -> Bool splitIntoCoprimesProperty1 fs' = factorback fs == factorback (unCoprimes $ splitIntoCoprimes fs)   where-    fs = map (getPositive *** getPower) fs'-    factorback = product . map (uncurry (^))+    fs = map (id *** getPower) fs'+    factorback = abs . product . map (uncurry (^)) -splitIntoCoprimesProperty2 :: [(Positive Natural, Power Word)] -> Bool+splitIntoCoprimesProperty2+  :: (Eq a, Num a, GcdDomain a)+  => [(NonZero a, Power Word)]+  -> Bool splitIntoCoprimesProperty2 fs' = multiplicities fs <= multiplicities (unCoprimes $ splitIntoCoprimes fs)   where-    fs = map (getPositive *** getPower) fs'-    multiplicities = sum . map snd . filter ((/= 1) . fst)+    fs = map (getNonZero *** getPower) fs'+    multiplicities = sum . map snd . filter ((/= 1) . abs . fst) -splitIntoCoprimesProperty3 :: [(Positive Natural, Power Word)] -> Bool+splitIntoCoprimesProperty3+  :: (Eq a, Num a, GcdDomain a)+  => [(a, Power Word)]+  -> Bool splitIntoCoprimesProperty3 fs' = and [ coprime x y | (x : xs) <- tails fs, y <- xs ]   where-    fs = map fst $ unCoprimes $ splitIntoCoprimes $ map (getPositive *** getPower) fs'+    fs = map fst $ unCoprimes $ splitIntoCoprimes $ map (id *** getPower) fs'  -- | Check that evaluation never freezes.-splitIntoCoprimesProperty4 :: [(Integer, Word)] -> Bool+splitIntoCoprimesProperty4+  :: (Eq a, Num a, GcdDomain a)+  => [(a, Word)]+  -> Bool splitIntoCoprimesProperty4 fs' = fs == fs   where     fs = splitIntoCoprimes fs' +splitIntoCoprimesProperty5+  :: (Eq a, Num a, GcdDomain a)+  => [(a, Word)]+  -> Bool+splitIntoCoprimesProperty5 =+  all ((/= 1) . abs . fst) . unCoprimes . splitIntoCoprimes+ -- | This is an undefined behaviour, but at least it should not -- throw exceptions or loop forever. splitIntoCoprimesSpecialCase1 :: Assertion splitIntoCoprimesSpecialCase1 =-  assertBool "should not fail" $ splitIntoCoprimesProperty4 [(0, 0), (0, 0)]+  assertBool "should not fail" $ splitIntoCoprimesProperty4 @Integer [(0, 0), (0, 0)]  -- | This is an undefined behaviour, but at least it should not -- throw exceptions or loop forever. splitIntoCoprimesSpecialCase2 :: Assertion splitIntoCoprimesSpecialCase2 =-  assertBool "should not fail" $ splitIntoCoprimesProperty4 [(0, 1), (-2, 0)]+  assertBool "should not fail" $ splitIntoCoprimesProperty4 @Integer [(0, 1), (-2, 0)]  toListReturnsCorrectValues :: Assertion toListReturnsCorrectValues = assertEqual@@ -117,32 +139,62 @@       expected = [(2,10), (5,2), (7,1)]   in assertEqual "should be equal" expected actual -unionProperty1 :: [(Positive Natural, Power Word)] -> [(Positive Natural, Power Word)] -> Bool+unionProperty1+  :: (Ord a, GcdDomain a)+  => [(a, Power Word)]+  -> [(a, Power Word)]+  -> Bool unionProperty1 xs ys   =  sort (unCoprimes (splitIntoCoprimes (xs' <> ys')))   == sort (unCoprimes (splitIntoCoprimes xs' <> splitIntoCoprimes ys'))   where-    xs' = map (getPositive *** getPower) xs-    ys' = map (getPositive *** getPower) ys+    xs' = map (id *** getPower) xs+    ys' = map (id *** getPower) ys  testSuite :: TestTree testSuite = testGroup "Euclidean"   [ testSameIntegralProperty "extendedGCD" extendedGCDProperty   , testSameIntegralProperty "coprime"     coprimeProperty   , testGroup "splitIntoCoprimes"-    [ testSmallAndQuick "preserves product of factors"        splitIntoCoprimesProperty1-    , testSmallAndQuick "number of factors is non-decreasing" splitIntoCoprimesProperty2-    , testSmallAndQuick "output factors are coprime"          splitIntoCoprimesProperty3--    , testCase          "does not freeze 1"                   splitIntoCoprimesSpecialCase1-    , testCase          "does not freeze 2"                   splitIntoCoprimesSpecialCase2-    , testSmallAndQuick "does not freeze random"              splitIntoCoprimesProperty4+    [ testGroup "preserves product of factors"+      [ testSmallAndQuick "Natural" (splitIntoCoprimesProperty1 @Natural)+      , testSmallAndQuick "Integer" (splitIntoCoprimesProperty1 @Integer)+      , testSmallAndQuick "Gaussian" (splitIntoCoprimesProperty1 @GaussianInteger)+      ]+    , testGroup "number of factors is non-decreasing"+      [ testSmallAndQuick "Natural" (splitIntoCoprimesProperty2 @Natural)+      , testSmallAndQuick "Integer" (splitIntoCoprimesProperty2 @Integer)+      , testSmallAndQuick "Gaussian" (splitIntoCoprimesProperty2 @GaussianInteger)+      ]+    , testGroup "output factors are coprime"+      [ testSmallAndQuick "Natural" (splitIntoCoprimesProperty3 @Natural)+      , testSmallAndQuick "Integer" (splitIntoCoprimesProperty3 @Integer)+      , testSmallAndQuick "Gaussian" (splitIntoCoprimesProperty3 @GaussianInteger)+      ]+    , testGroup "does not freeze"+      [ testCase          "case 1"                   splitIntoCoprimesSpecialCase1+      , testCase          "case 2"                   splitIntoCoprimesSpecialCase2+      , testSmallAndQuick "Natural" (splitIntoCoprimesProperty4 @Natural)+      -- smallcheck for Integer and GaussianInteger takes too long+      , QC.testProperty "Integer" (splitIntoCoprimesProperty4 @Integer)+      , QC.testProperty "Gaussian" (splitIntoCoprimesProperty4 @GaussianInteger)+      ]+    , testGroup "output factors are non-unit"+      [ testSmallAndQuick "Natural" (splitIntoCoprimesProperty5 @Natural)+      -- smallcheck for Integer and GaussianInteger takes too long+      , QC.testProperty "Integer" (splitIntoCoprimesProperty5 @Integer)+      , QC.testProperty "Gaussian" (splitIntoCoprimesProperty5 @GaussianInteger)+      ]     ]   , testGroup "Coprimes"     [  testCase         "test equality"                       toListReturnsCorrectValues     ,  testCase         "test union"                          unionReturnsCorrectValues     ,  testCase         "test insert with coprime base"       insertReturnsCorrectValuesWhenCoprimeBase     ,  testCase         "test insert with non-coprime base"   insertReturnsCorrectValuesWhenNotCoprimeBase-    ,  testSmallAndQuick "property union"                     unionProperty1+    ,  testGroup "property union"+      [ testSmallAndQuick "Natural" (unionProperty1 @Natural)+      -- smallcheck for Integer takes too long+      , QC.testProperty "Integer" (unionProperty1 @Integer)+      ]     ]   ]

@@ -18,6 +18,7 @@ import Data.Maybe (fromJust, mapMaybe) import Test.Tasty import Test.Tasty.HUnit+import Test.Tasty.QuickCheck as QC hiding (NonNegative(..), Positive(..))  import qualified Math.NumberTheory.Euclidean as ED import Math.NumberTheory.Quadratic.GaussianIntegers@@ -112,8 +113,8 @@  numberOfPrimes :: Assertion numberOfPrimes = assertEqual "counting primes: OEIS A091100"-  [16,100,668,4928,38404,313752,2658344]-  [4 * (length $ takeWhile ((<= 10^n) . norm . unPrime) primes) | n <- [1..7]]+  [16,100,668,4928,38404,313752]+  [4 * (length $ takeWhile ((<= 10^n) . norm . unPrime) primes) | n <- [1..6]]  -- | signum and abs should satisfy: z == signum z * abs z signumAbsProperty :: GaussianInteger -> Bool@@ -128,10 +129,15 @@     inFirstQuadrant = x' > 0 && y' >= 0     -- first quadrant includes the positive real axis, but not the origin or the positive imaginary axis     isAssociate = z' `elem` map (\e -> z * (0 :+ 1) ^ e) [0 .. 3] +-- | Verify that @rem@ produces a remainder smaller than the divisor with+-- regards to the Euclidean domain's function.+remProperty :: GaussianInteger -> GaussianInteger -> Bool+remProperty x y = (y == 0) || (norm $ x `ED.rem` y) < (norm y)+ gcdGProperty1 :: GaussianInteger -> GaussianInteger -> Bool gcdGProperty1 z1 z2   = z1 == 0 && z2 == 0-  || z1 `ED.rem` z == 0 && z2 `ED.rem` z == 0 && z == abs z+  || z1 `ED.rem` z == 0 && z2 `ED.rem` z == 0   where     z = ED.gcd z1 z2 @@ -145,8 +151,11 @@  -- | a special case that tests rounding/truncating in GCD. gcdGSpecialCase1 :: Assertion-gcdGSpecialCase1 = assertEqual "gcdG" 1 $ ED.gcd (12 :+ 23) (23 :+ 34)+gcdGSpecialCase1 = assertEqual "gcdG" (-1) $ ED.gcd (12 :+ 23) (23 :+ 34) +gcdGSpecialCase2 :: Assertion+gcdGSpecialCase2 = assertEqual "gcdG" (0 :+ (-1)) $ ED.gcd (0 :+ 3) (2 :+ 2)+ testSuite :: TestTree testSuite = testGroup "GaussianIntegers" $   [ testGroup "factorise" (@@ -167,9 +176,12 @@   , testCase          "counting primes"          numberOfPrimes   , testSmallAndQuick "signumAbsProperty"        signumAbsProperty   , testSmallAndQuick "absProperty"              absProperty+  , testSmallAndQuick "remProperty"              remProperty   , testGroup "gcd"     [ testSmallAndQuick "is divisor"            gcdGProperty1-    , testSmallAndQuick "is greatest"           gcdGProperty2+    -- smallcheck takes too long+    , QC.testProperty   "is greatest"           gcdGProperty2     , testCase          "(12 :+ 23) (23 :+ 34)" gcdGSpecialCase1+    , testCase          "(0 :+ 3) (2 :+ 2)"     gcdGSpecialCase2     ]   ]

@@ -37,14 +37,17 @@  -- | Check that 'chineseRemainder' matches 'chineseRemainder2'. chineseRemainder2Property :: Integer -> Positive Integer -> Integer -> Positive Integer -> Bool-chineseRemainder2Property r1 (Positive m1) r2 (Positive m2) = gcd m1 m2 /= 1-  || Just (chineseRemainder2 (r1, m1) (r2, m2)) == chineseRemainder [(r1, m1), (r2, m2)]+chineseRemainder2Property r1 (Positive m1) r2 (Positive m2)+  | gcd m1 m2 /= 1 = True+  | otherwise      = case chineseRemainder [(r1, m1), (r2, m2)] of+    Nothing -> False+    Just ch -> (ch - chineseRemainder2 (r1, m1) (r2, m2)) `rem` (m1 * m2) == 0  chineseCoprimeProperty :: Integer -> Positive Integer -> Integer -> Positive Integer -> Bool chineseCoprimeProperty n1 (Positive m1) n2 (Positive m2) = case gcd m1 m2 of   1 -> case chineseCoprime (n1, m1) (n2, m2) of     Nothing -> False-    Just n  -> n `mod` m1 == n1 `mod` m1 && n `mod` m2 == n2 `mod` m2+    Just n  -> (n - n1) `rem` m1 == 0 && (n - n2) `rem` m2 == 0   _ -> case chineseCoprime (n1, m1) (n2, m2) of     Nothing -> True     Just{}  -> False@@ -53,13 +56,13 @@ chineseProperty n1 (Positive m1) n2 (Positive m2) = if compatible   then case chinese (n1, m1) (n2, m2) of     Nothing -> False-    Just n  -> n `mod` m1 == n1 `mod` m1 && n `mod` m2 == n2 `mod` m2+    Just n  -> (n - n1) `rem` m1 == 0 && (n - n2) `rem` m2 == 0   else case chineseCoprime (n1, m1) (n2, m2) of     Nothing -> True     Just{}  -> False   where     g = gcd m1 m2-    compatible = n1 `mod` g == n2 `mod` g+    compatible = (n1 - n2) `rem` g == 0   testSuite :: TestTree

@@ -34,7 +34,7 @@ -- | Check that 'invertMod' inverts numbers modulo. invertModProperty :: AnySign Integer -> Positive Integer -> Bool invertModProperty (AnySign k) (Positive m) = case invertMod k m of-  Nothing            -> k `mod` m == 0 || gcd k m > 1+  Nothing            -> k `rem` m == 0 || gcd k m > 1   Just InfMod{}      -> False   Just (SomeMod inv) -> gcd k m == 1 && k * getVal inv `mod` m == 1 

@@ -13,10 +13,11 @@ import Data.Proxy import GHC.TypeNats.Compat +import Math.NumberTheory.ArithmeticFunctions (totient) import Math.NumberTheory.Moduli.Class-import Math.NumberTheory.Moduli.PrimitiveRoot import Math.NumberTheory.Moduli.DiscreteLogarithm-import Math.NumberTheory.ArithmeticFunctions (totient)+import Math.NumberTheory.Moduli.PrimitiveRoot+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.TestUtils  -- | Ensure 'discreteLogarithm' returns in the appropriate range.@@ -24,27 +25,30 @@ discreteLogRange (Positive m) a b =   case someNatVal m of     SomeNat (_ :: Proxy m) -> fromMaybe True $ do-      a' <- isPrimitiveRoot (fromInteger a :: Mod m)+      cg <- cyclicGroup :: Maybe (CyclicGroup Integer m)+      a' <- isPrimitiveRoot cg (fromInteger a)       b' <- isMultElement (fromInteger b)-      return $ discreteLogarithm a' b' < totient m+      return $ discreteLogarithm cg a' b' < totient m  -- | Check that 'discreteLogarithm' inverts exponentiation. discreteLogarithmProperty :: Positive Natural -> Integer -> Integer -> Bool discreteLogarithmProperty (Positive m) a b =   case someNatVal m of     SomeNat (_ :: Proxy m) -> fromMaybe True $ do-      a' <- isPrimitiveRoot (fromInteger a :: Mod m)+      cg <- cyclicGroup :: Maybe (CyclicGroup Integer m)+      a' <- isPrimitiveRoot cg (fromInteger a)       b' <- isMultElement (fromInteger b)-      return $ discreteLogarithm a' b' `stimes` unPrimitiveRoot a' == b'+      return $ discreteLogarithm cg a' b' `stimes` unPrimitiveRoot a' == b'  -- | Check that 'discreteLogarithm' inverts exponentiation in the other direction. discreteLogarithmProperty' :: Positive Natural -> Integer -> Natural -> Bool discreteLogarithmProperty' (Positive m) a k =   case someNatVal m of     SomeNat (_ :: Proxy m) -> fromMaybe True $ do-      a'' <- isPrimitiveRoot (fromInteger a :: Mod m)+      cg <- cyclicGroup :: Maybe (CyclicGroup Integer m)+      a'' <- isPrimitiveRoot cg (fromInteger a)       let a' = unPrimitiveRoot a''-      return $ discreteLogarithm a'' (k `stimes` a') == k `mod` totient m+      return $ discreteLogarithm cg a'' (k `stimes` a') == k `mod` totient m  testSuite :: TestTree testSuite = testGroup "Discrete logarithm"

@@ -20,6 +20,7 @@  import Math.NumberTheory.Moduli.Class import Math.NumberTheory.Moduli.Equations+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.TestUtils  solveLinearProp :: KnownNat m => Mod m -> Mod m -> Bool@@ -31,7 +32,7 @@   SomeNat (_ :: Proxy t) -> solveLinearProp (fromInteger a :: Mod t) (fromInteger b)  solveQuadraticProp :: KnownNat m => Mod m -> Mod m -> Mod m -> Bool-solveQuadraticProp a b c = sort (solveQuadratic a b c) ==+solveQuadraticProp a b c = sort (solveQuadratic sfactors a b c) ==   filter (\x -> a * x * x + b * x + c == 0) [minBound .. maxBound]  solveQuadraticProperty1 :: Positive Natural -> Integer -> Integer -> Integer -> Bool

@@ -18,28 +18,27 @@  import Prelude hiding (gcd) import Test.Tasty+import Test.Tasty.HUnit  import qualified Data.Set as S import Data.List (genericTake, genericLength) import Data.Maybe (isJust, isNothing, mapMaybe)-import Control.Arrow (first) import Numeric.Natural import Data.Proxy import GHC.TypeNats.Compat  import Math.NumberTheory.ArithmeticFunctions (totient) import Math.NumberTheory.Euclidean-import Math.NumberTheory.Euclidean.Coprimes-import Math.NumberTheory.Moduli.Class (Mod, SomeMod(..), modulo)+import Math.NumberTheory.Moduli.Class import Math.NumberTheory.Moduli.PrimitiveRoot-import Math.NumberTheory.Prefactored (fromFactors, prefFactors, prefValue, Prefactored)+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.Primes import Math.NumberTheory.TestUtils -cyclicGroupProperty1 :: (Euclidean a, Integral a, UniqueFactorisation a) => AnySign a -> Bool-cyclicGroupProperty1 (AnySign n) = case cyclicGroupFromModulo n of+cyclicGroupProperty1 :: (Euclidean a, Integral a, UniqueFactorisation a) => Positive a -> Bool+cyclicGroupProperty1 (Positive n) = case cyclicGroupFromModulo n of   Nothing -> True-  Just cg -> prefValue (cyclicGroupToModulo cg) == n+  Just (Some cg) -> factorBack (unSFactors (cyclicGroupToSFactors cg)) == n  -- | Multiplicative groups modulo primes are always cyclic. cyclicGroupProperty2 :: (Integral a, UniqueFactorisation a) => Positive a -> Bool@@ -54,6 +53,9 @@   Just _  -> 2 * n < n {- overflow check -}           || isJust (cyclicGroupFromModulo n) +cyclicGroupSpecialCase1 :: Assertion+cyclicGroupSpecialCase1 = assertBool "should be non-cyclic" $ isNothing $ cyclicGroupFromModulo (8 :: Integer)+ allUnique :: Ord a => [a] -> Bool allUnique = go S.empty   where@@ -61,68 +63,64 @@     go acc (x : xs) = if x `S.member` acc then False else go (S.insert x acc) xs  isPrimitiveRoot'Property1-  :: (Euclidean a, Integral a, UniqueFactorisation a)-  => AnySign a -> CyclicGroup a -> Bool-isPrimitiveRoot'Property1 (AnySign n) cg-  = gcd (toInteger n) (prefValue (castPrefactored (cyclicGroupToModulo cg))) == 1-  || not (isPrimitiveRoot' cg n)--castPrefactored :: Integral a => Prefactored a -> Prefactored Integer-castPrefactored = fromFactors . splitIntoCoprimes . map (first toInteger) . unCoprimes . prefFactors+  :: forall a. (Euclidean a, Integral a, UniqueFactorisation a)+  => AnySign a+  -> Positive Natural+  -> Bool+isPrimitiveRoot'Property1 (AnySign n) (Positive m) = case someNatVal m of+  SomeNat (_ :: Proxy m) -> case cyclicGroup :: Maybe (CyclicGroup a m) of+    Nothing -> True+    Just cg -> case isPrimitiveRoot cg (fromIntegral n) of+      Nothing -> True+      Just rt -> gcd (toInteger m) (getVal (multElement (unPrimitiveRoot rt))) == 1  isPrimitiveRootProperty1 :: AnySign Integer -> Positive Natural -> Bool-isPrimitiveRootProperty1 (AnySign n) (Positive m)-  = case n `modulo` m of-    SomeMod n' -> gcd n (toInteger m) == 1-               || isNothing (isPrimitiveRoot n')-    InfMod{}   -> False+isPrimitiveRootProperty1 (AnySign n) (Positive m) = case someNatVal m of+  SomeNat (_ :: Proxy m) -> case cyclicGroup :: Maybe (CyclicGroup Integer m) of+    Nothing -> True+    Just cg -> gcd n (toInteger m) == 1+            || isNothing (isPrimitiveRoot cg (fromInteger n))  isPrimitiveRootProperty2 :: Positive Natural -> Bool-isPrimitiveRootProperty2 (Positive m)-  = isNothing (cyclicGroupFromModulo m)-  || case someNatVal m of-    SomeNat (_ :: Proxy t) -> any (isJust . isPrimitiveRoot) [(minBound :: Mod t) .. maxBound]+isPrimitiveRootProperty2 (Positive m) = case someNatVal m of+  SomeNat (_ :: Proxy m) -> case cyclicGroup :: Maybe (CyclicGroup Integer m) of+    Nothing -> True+    Just cg -> any (isJust . isPrimitiveRoot cg) [minBound..maxBound]  isPrimitiveRootProperty3 :: AnySign Integer -> Positive Natural -> Bool-isPrimitiveRootProperty3 (AnySign n) (Positive m)-  = case n `modulo` m of-    SomeMod n' -> isNothing (isPrimitiveRoot n')-               || allUnique (genericTake (totient m - 1) (iterate (* n') 1))-    InfMod{}   -> False--isPrimitiveRootProperty4 :: AnySign Integer -> Positive Natural -> Bool-isPrimitiveRootProperty4 (AnySign n) (Positive m)-  = isJust (cyclicGroupFromModulo m)-  || case n `modulo` m of-    SomeMod n' -> isNothing (isPrimitiveRoot n')-    InfMod{}   -> False+isPrimitiveRootProperty3 (AnySign n) (Positive m) = case someNatVal m of+  SomeNat (_ :: Proxy m) -> case cyclicGroup :: Maybe (CyclicGroup Integer m) of+    Nothing -> True+    Just cg -> let n' = fromInteger n+      in isNothing (isPrimitiveRoot cg n')+      || allUnique (genericTake (totient m - 1) (iterate (* n') 1))  isPrimitiveRootProperty5 :: Positive Natural -> Bool-isPrimitiveRootProperty5 (Positive m)-  = isNothing (cyclicGroupFromModulo m)-  || case someNatVal m of-       SomeNat (_ :: Proxy t) -> genericLength (mapMaybe isPrimitiveRoot [(minBound :: Mod t) .. maxBound]) == totient (totient m)+isPrimitiveRootProperty5 (Positive m) = case someNatVal m of+  SomeNat (_ :: Proxy m) -> case cyclicGroup :: Maybe (CyclicGroup Integer m) of+    Nothing -> True+    Just cg -> genericLength (mapMaybe (isPrimitiveRoot cg) [minBound..maxBound]) == totient (totient m)  testSuite :: TestTree testSuite = testGroup "Primitive root"   [ testGroup "CyclicGroup"-    [ testIntegralProperty "cyclicGroupToModulo . cyclicGroupFromModulo" cyclicGroupProperty1+    [ testIntegralProperty "cyclicGroupFromModulo" cyclicGroupProperty1     , testIntegralProperty "cyclic group mod p" cyclicGroupProperty2     , testIntegralProperty "cyclic group mod 2p" cyclicGroupProperty3+    , testCase "cyclic group mod 8" cyclicGroupSpecialCase1     ]   , testGroup "isPrimitiveRoot'"     [ testGroup "primitive root is coprime with modulo"-      [ testSmallAndQuick "Integer" (isPrimitiveRoot'Property1 :: AnySign Integer -> CyclicGroup Integer -> Bool)-      , testSmallAndQuick "Natural" (isPrimitiveRoot'Property1 :: AnySign Natural -> CyclicGroup Natural -> Bool)-      , testSmallAndQuick "Int"     (isPrimitiveRoot'Property1 :: AnySign Int     -> CyclicGroup Int     -> Bool)-      , testSmallAndQuick "Word"    (isPrimitiveRoot'Property1 :: AnySign Word    -> CyclicGroup Word    -> Bool)+      [ testSmallAndQuick "Integer" (isPrimitiveRoot'Property1 :: AnySign Integer -> Positive Natural -> Bool)+      , testSmallAndQuick "Natural" (isPrimitiveRoot'Property1 :: AnySign Natural -> Positive Natural -> Bool)+      , testSmallAndQuick "Int"     (isPrimitiveRoot'Property1 :: AnySign Int     -> Positive Natural -> Bool)+      , testSmallAndQuick "Word"    (isPrimitiveRoot'Property1 :: AnySign Word    -> Positive Natural -> Bool)       ]     ]   , testGroup "isPrimitiveRoot"     [ testSmallAndQuick "primitive root is coprime with modulo"            isPrimitiveRootProperty1     , testSmallAndQuick "cyclic group has a primitive root"                isPrimitiveRootProperty2     , testSmallAndQuick "primitive root generates cyclic group"            isPrimitiveRootProperty3-    , testSmallAndQuick "no primitive root in non-cyclic group"            isPrimitiveRootProperty4     , testSmallAndQuick "cyclic group has right number of primitive roots" isPrimitiveRootProperty5     ]   ]

@@ -0,0 +1,46 @@+-- |+-- Module:      Math.NumberTheory.Moduli.SingletonTests+-- Copyright:   (c) 2019 Andrew Lelechenko+-- Licence:     MIT+-- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>+--+-- Tests for Math.NumberTheory.Moduli.Singleton+--++{-# LANGUAGE TypeApplications #-}++{-# OPTIONS_GHC -fno-warn-type-defaults #-}++module Math.NumberTheory.Moduli.SingletonTests+  ( testSuite+  ) where++import Test.Tasty++import qualified Data.Map as M++import Math.NumberTheory.Moduli.Singleton+import Math.NumberTheory.Primes+import Math.NumberTheory.TestUtils++someSFactorsProperty1+  :: (Ord a, Num a)+  => [(Prime a, Word)]+  -> Bool+someSFactorsProperty1 xs = case someSFactors xs of+  Some sm -> unSFactors sm == M.assocs (M.fromListWith (+) xs)++cyclicGroupFromModuloProperty1+  :: (Integral a, UniqueFactorisation a)+  => Positive a+  -> Bool+cyclicGroupFromModuloProperty1 (Positive m) = mcg1 == mcg2+  where+    mcg1 = cyclicGroupFromModulo m+    mcg2 = cyclicGroupFromFactors (factorise m)++testSuite :: TestTree+testSuite = testGroup "Singleton"+  [ testSmallAndQuick "unSFactors . someSFactors = id" (someSFactorsProperty1 @Integer)+  , testIntegralPropertyNoLarge "cyclicGroupFromModulo = cyclicGroupFromFactors . factorise" cyclicGroupFromModuloProperty1+  ]

@@ -25,6 +25,7 @@ import Numeric.Natural  import Math.NumberTheory.Moduli hiding (invertMod)+import Math.NumberTheory.Moduli.Singleton import Math.NumberTheory.Primes (unPrime, isPrime, Prime) import Math.NumberTheory.TestUtils @@ -39,10 +40,10 @@ sqrtsModPrimeProperty1 :: AnySign Integer -> Prime Integer -> Bool sqrtsModPrimeProperty1 (AnySign n) p'@(unPrime -> p) = case sqrtsModPrime n p' of   []     -> jacobi n p == MinusOne-  rt : _ -> (p == 2 || jacobi n p /= MinusOne) && rt ^ 2 `mod` p == n `mod` p+  rt : _ -> (p == 2 || jacobi n p /= MinusOne) && (rt ^ 2 - n) `rem` p == 0  sqrtsModPrimeProperty2 :: AnySign Integer -> Prime Integer -> Bool-sqrtsModPrimeProperty2 (AnySign n) p'@(unPrime -> p) = all (\rt -> rt ^ 2 `mod` p == n `mod` p) (sqrtsModPrime n p')+sqrtsModPrimeProperty2 (AnySign n) p'@(unPrime -> p) = all (\rt -> (rt ^ 2 - n) `rem` p == 0) (sqrtsModPrime n p')  sqrtsModPrimeProperty3 :: AnySign Integer -> Prime Integer -> Bool sqrtsModPrimeProperty3 (AnySign n) p'@(unPrime -> p) = nubOrd rts == sort rts@@ -58,7 +59,7 @@     rt : _ = sqrtsModPrime n p'  tonelliShanksProperty2 :: Prime Integer -> Bool-tonelliShanksProperty2 p'@(unPrime -> p) = p `mod` 4 /= 1 || rt ^ 2 `mod` p == n `mod` p+tonelliShanksProperty2 p'@(unPrime -> p) = p `mod` 4 /= 1 || (rt ^ 2 - n) `rem` p == 0   where     n  = head $ filter (\s -> jacobi s p == One) [2..p-1]     rt : _ = sqrtsModPrime n p'@@ -80,7 +81,7 @@  sqrtsModPrimePowerProperty1 :: AnySign Integer -> (Prime Integer, Power Word) -> Bool sqrtsModPrimePowerProperty1 (AnySign n) (p'@(unPrime -> p), Power e) = gcd n p > 1-  || all (\rt -> rt ^ 2 `mod` (p ^ e) == n `mod` (p ^ e)) (sqrtsModPrimePower n p' e)+  || all (\rt -> (rt ^ 2 - n) `rem` (p ^ e) == 0) (sqrtsModPrimePower n p' e)  sqrtsModPrimePowerProperty2 :: AnySign Integer -> Power Word -> Bool sqrtsModPrimePowerProperty2 n e = sqrtsModPrimePowerProperty1 n (fromJust $ isPrime (2 :: Integer), e)@@ -152,7 +153,7 @@ sqrtsModFactorisationProperty1 :: AnySign Integer -> [(Prime Integer, Power Word)] -> Bool sqrtsModFactorisationProperty1 (AnySign n) (take 10 . map unwrapPP -> pes'@(map (first unPrime) -> pes))   = nubOrd ps /= sort ps || all-    (\rt -> all (\(p, e) -> rt ^ 2 `mod` (p ^ e) == n `mod` (p ^ e)) pes)+    (\rt -> all (\(p, e) -> (rt ^ 2 - n) `rem` (p ^ e) == 0) pes)     (take 1000 $ sqrtsModFactorisation n pes')   where     ps = map fst pes@@ -185,7 +186,7 @@  sqrtsModProperty1 :: AnySign Integer -> Positive Natural -> Bool sqrtsModProperty1 (AnySign n) (Positive m) = case n `modulo` m of-  SomeMod x -> sort (sqrtsMod x) == filter (\rt -> rt * rt == x) [minBound .. maxBound]+  SomeMod x -> sort (sqrtsMod sfactors x) == filter (\rt -> rt * rt == x) [minBound .. maxBound]   InfMod{} -> True  testSuite :: TestTree

@@ -1,52 +0,0 @@--- |--- Module:      Math.NumberTheory.MoebiusInversion.IntTests--- Copyright:   (c) 2016 Andrew Lelechenko--- Licence:     MIT--- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>------ Tests for Math.NumberTheory.MoebiusInversion.Int-----{-# OPTIONS_GHC -fno-warn-type-defaults #-}--module Math.NumberTheory.MoebiusInversion.IntTests-  ( testSuite-  ) where--import Test.Tasty-import Test.Tasty.HUnit-import Test.Tasty.QuickCheck as QC hiding (Positive)--import Math.NumberTheory.MoebiusInversion.Int-import Math.NumberTheory.ArithmeticFunctions-import Math.NumberTheory.TestUtils--totientSumProperty :: Positive Int -> Bool-totientSumProperty (Positive n) = toInteger (totientSum n) == sum (map totient [1 .. toInteger n])--totientSumSpecialCase1 :: Assertion-totientSumSpecialCase1 = assertEqual "totientSum" 4496 (totientSum 121)--totientSumSpecialCase2 :: Assertion-totientSumSpecialCase2 = assertEqual "totientSum" 0 (totientSum (-9001))--totientSumZero :: Assertion-totientSumZero = assertEqual "totientSum" 0 (totientSum 0)--generalInversionProperty :: (Int -> Int) -> Positive Int -> Bool-generalInversionProperty g (Positive n)-  =  g n == sum [f (n `quot` k) | k <- [1 .. n]]-  && f n == sum [runMoebius (moebius k) * g (n `quot` k) | k <- [1 .. n]]-  where-    f = generalInversion g--testSuite :: TestTree-testSuite = testGroup "Int"-  [ testGroup "totientSum"-    [ testSmallAndQuick "matches definitions" totientSumProperty-    , testCase          "special case 1"      totientSumSpecialCase1-    , testCase          "special case 2"      totientSumSpecialCase2-    , testCase          "zero"                totientSumZero-    ]-  , QC.testProperty "generalInversion" generalInversionProperty-  ]

@@ -17,35 +17,37 @@ import Test.Tasty.HUnit import Test.Tasty.QuickCheck as QC hiding (Positive) +import Data.Proxy+import Data.Vector.Unboxed (Vector)+ import Math.NumberTheory.MoebiusInversion import Math.NumberTheory.ArithmeticFunctions import Math.NumberTheory.TestUtils -totientSumProperty :: Positive Int -> Bool-totientSumProperty (Positive n) = totientSum n == sum (map totient [1 .. toInteger n])+proxy :: Proxy Vector+proxy = Proxy -totientSumSpecialCase1 :: Assertion-totientSumSpecialCase1 = assertEqual "totientSum" 4496 (totientSum 121)+totientSumProperty :: AnySign Word -> Bool+totientSumProperty (AnySign n) = (totientSum proxy n :: Word) == sum (map totient [1..n]) -totientSumSpecialCase2 :: Assertion-totientSumSpecialCase2 = assertEqual "totientSum" 0 (totientSum (-9001))+totientSumSpecialCase1 :: Assertion+totientSumSpecialCase1 = assertEqual "totientSum" 4496 (totientSum proxy 121 :: Word)  totientSumZero :: Assertion-totientSumZero = assertEqual "totientSum" 0 (totientSum 0)+totientSumZero = assertEqual "totientSum" 0 (totientSum proxy 0 :: Word) -generalInversionProperty :: (Int -> Integer) -> Positive Int -> Bool+generalInversionProperty :: (Word -> Word) -> Positive Word -> Bool generalInversionProperty g (Positive n)   =  g n == sum [f (n `quot` k) | k <- [1 .. n]]   && f n == sum [runMoebius (moebius k) * g (n `quot` k) | k <- [1 .. n]]   where-    f = generalInversion g+    f = generalInversion proxy g  testSuite :: TestTree testSuite = testGroup "MoebiusInversion"   [ testGroup "totientSum"     [ testSmallAndQuick "matches definitions" totientSumProperty     , testCase          "special case 1"      totientSumSpecialCase1-    , testCase          "special case 2"      totientSumSpecialCase2     , testCase          "zero"                totientSumZero     ]   , QC.testProperty "generalInversion" generalInversionProperty

@@ -21,12 +21,12 @@ import Data.List (tails) import Numeric.Natural -import Math.NumberTheory.Euclidean (Euclidean, coprime)+import Math.NumberTheory.Euclidean import Math.NumberTheory.Euclidean.Coprimes import Math.NumberTheory.Prefactored import Math.NumberTheory.TestUtils -isValid :: Euclidean a => Prefactored a -> Bool+isValid :: (Eq a, Num a, GcdDomain a, Euclidean a) => Prefactored a -> Bool isValid pref   = abs n == abs (product (map (uncurry (^)) fs))   && and [ coprime g h | ((g, _) : gs) <- tails fs, (h, _) <- gs ]

@@ -36,7 +36,7 @@   , (10^10,  455052511)   , (10^11,  4118054813)   , (10^12,  37607912018)-  , (10^13,  346065536839)+  -- , (10^13,  346065536839)   -- , (10^14,  3204941750802)   -- , (10^15,  29844570422669)   -- , (10^16,  279238341033925)

@@ -16,11 +16,12 @@ import Test.Tasty import Test.Tasty.HUnit +import Control.Arrow import Control.Monad (zipWithM_) import Data.List (nub, sort)+import Data.Maybe -import Math.NumberTheory.Primes.Factorisation-import Math.NumberTheory.Primes.Testing+import Math.NumberTheory.Primes import Math.NumberTheory.TestUtils  specialCases :: [(Integer, [(Integer, Word)])]@@ -43,7 +44,7 @@   , (16757651897802863152387219654541878160,[(2,4),(5,1),(12323,1),(1424513,1),(6205871923,1),(1922815011093901,1)])   , (16757651897802863152387219654541878162,[(2,1),(29,1),(78173,1),(401529283,1),(1995634649,1),(4612433663779,1)])   , (16757651897802863152387219654541878163,[(11,1),(31,1),(112160981904206269,1),(438144115295608147,1)])-  , (16757651897802863152387219654541878166,[(2,1),(23,1),(277,1),(505353699591289,1),(2602436338718275457,1)])+  -- , (16757651897802863152387219654541878166,[(2,1),(23,1),(277,1),(505353699591289,1),(2602436338718275457,1)])   , ((10 ^ 80 - 1) `div` 9, [(11,1),(17,1),(41,1),(73,1),(101,1),(137,1),(271,1),(3541,1),(9091,1),(27961,1),                              (1676321,1),(5070721,1),(5882353,1),(5964848081,1),(19721061166646717498359681,1)])   ]@@ -58,13 +59,13 @@   ]  factoriseProperty1 :: Assertion-factoriseProperty1 = assertEqual "0" [] (factorise 1)+factoriseProperty1 = assertEqual "0" [] (factorise (1 :: Int))  factoriseProperty2 :: Positive Integer -> Bool factoriseProperty2 (Positive n) = factorise n == factorise (negate n)  factoriseProperty3 :: Positive Integer -> Bool-factoriseProperty3 (Positive n) = all (isPrime . fst) (factorise n)+factoriseProperty3 (Positive n) = all (isJust . isPrime . unPrime . fst) (factorise n)  factoriseProperty4 :: Positive Integer -> Bool factoriseProperty4 (Positive n) = bases == nub (sort bases)@@ -72,13 +73,13 @@     bases = map fst $ factorise n  factoriseProperty5 :: Positive Integer -> Bool-factoriseProperty5 (Positive n) = product (map (uncurry (^)) (factorise n)) == n+factoriseProperty5 (Positive n) = product (map (\(p, k) -> unPrime p ^ k) (factorise n)) == n  factoriseProperty6 :: (Integer, [(Integer, Word)]) -> Assertion-factoriseProperty6 (n, fs) = assertEqual (show n) (sort fs) (sort (factorise n))+factoriseProperty6 (n, fs) = assertEqual (show n) (sort fs) (sort $ map (first unPrime) $ factorise n)  factoriseProperty7 :: (Integer, [(Integer, Word)]) -> Assertion-factoriseProperty7 (n, fs) = zipWithM_ (assertEqual (show n)) fs (factorise n)+factoriseProperty7 (n, fs) = zipWithM_ (assertEqual (show n)) fs (map (first unPrime) $ factorise n)  testSuite :: TestTree testSuite = testGroup "Factorisation"

@@ -139,7 +139,7 @@   = case signum (bernoulli !! m) of     1  -> m == 0 || m `mod` 4 == 2     0  -> m /= 1 && odd m-    -1 -> m == 1 || (m /= 0 && m `mod` 4 == 0)+    -1 -> m == 1 || (m /= 0 && m `rem` 4 == 0)     _  -> False  bernoulliProperty2 :: NonNegative Int -> Bool

@@ -13,27 +13,23 @@   ( testSuite   ) where -import Prelude hiding (mod)+import Prelude hiding (mod, rem) import Test.Tasty import Test.Tasty.HUnit  import Data.Coerce-import Data.List (genericDrop, nub, sort)+import Data.List (nub) import Data.Maybe (fromJust)-import qualified Data.Set as S import Numeric.Natural -import Math.NumberTheory.Euclidean (Euclidean (..), WrappedIntegral (..))+import Math.NumberTheory.Euclidean import Math.NumberTheory.Primes (Prime (..)) import qualified Math.NumberTheory.Quadratic.GaussianIntegers as G import qualified Math.NumberTheory.Quadratic.EisensteinIntegers as E-import Math.NumberTheory.SmoothNumbers+import Math.NumberTheory.SmoothNumbers (SmoothBasis, fromList, isSmooth, smoothOver, smoothOver') import Math.NumberTheory.TestUtils -fromSetListProperty :: (Euclidean a, Ord a) => [a] -> Bool-fromSetListProperty xs = fromSet (S.fromList xs) == fromList (sort xs)--isSmoothPropertyHelper :: Euclidean a => (a -> Integer) -> [a] -> Int -> Int -> Bool+isSmoothPropertyHelper :: (Eq a, Num a, Euclidean a) => (a -> Integer) -> [a] -> Int -> Int -> Bool isSmoothPropertyHelper norm primes' i1 i2 =     let primes = take i1 primes'         basis  = fromJust (fromList primes)@@ -47,19 +43,29 @@ isSmoothProperty2 (Positive i1) (Positive i2) =     isSmoothPropertyHelper E.norm (map unPrime E.primes) i1 i2 -fromSmoothUpperBoundProperty :: Integral a => Positive a -> Bool-fromSmoothUpperBoundProperty (Positive n') = case fromSmoothUpperBound n of-    Nothing -> n < 2-    Just sb -> head (genericDrop (n - 1) (smoothOver (coerce sb))) == n-  where-    n = WrappedIntegral n' `mod` 5000+smoothOverInRange :: (Ord a, Num a) => SmoothBasis a -> a -> a -> [a]+smoothOverInRange s lo hi+  = takeWhile (<= hi)+  $ dropWhile (< lo)+  $ smoothOver s +smoothOverInRangeBF+  :: (Eq a, Enum a, GcdDomain a)+  => SmoothBasis a+  -> a+  -> a+  -> [a]+smoothOverInRangeBF prs lo hi+  = coerce+  $ filter (isSmooth prs)+  $ coerce [lo..hi]+ smoothOverInRangeProperty :: Integral a => SmoothBasis a -> Positive a -> Positive a -> Bool smoothOverInRangeProperty s (Positive lo') (Positive diff')   = xs == ys   where-    lo   = WrappedIntegral lo'   `mod` 2^18-    diff = WrappedIntegral diff' `mod` 2^18+    lo   = WrapIntegral lo'   `rem` 2^18+    diff = WrapIntegral diff' `rem` 2^18     hi   = lo + diff     xs   = smoothOverInRange   (coerce s) lo hi     ys   = smoothOverInRangeBF (coerce s) lo hi@@ -76,17 +82,14 @@     b = fromJust $ fromList [1+3*G.ι,6+8*G.ι]     l = take 10 $ map abs $ smoothOver' G.norm b +isSmoothSpecialCase2 :: Assertion+isSmoothSpecialCase2 = assertBool "should be smooth" $ isSmooth b 6+  where+    b = fromJust $ fromList [4, 3, 6, 10, 7::Int]  testSuite :: TestTree testSuite = testGroup "SmoothNumbers"-  [ testGroup "fromSet == fromList"-    [ testSmallAndQuick "Int"     (fromSetListProperty :: [Int] -> Bool)-    , testSmallAndQuick "Word"    (fromSetListProperty :: [Word] -> Bool)-    , testSmallAndQuick "Integer" (fromSetListProperty :: [Integer] -> Bool)-    , testSmallAndQuick "Natural" (fromSetListProperty :: [Natural] -> Bool)-    ]-  , testIntegralProperty "fromSmoothUpperBound" fromSmoothUpperBoundProperty-  , testGroup "smoothOverInRange == smoothOverInRangeBF"+  [ testGroup "smoothOverInRange == smoothOverInRangeBF"     [ testSmallAndQuick "Int"       (smoothOverInRangeProperty :: SmoothBasis Int -> Positive Int -> Positive Int -> Bool)     , testSmallAndQuick "Word"@@ -103,11 +106,11 @@       (smoothNumbersAreUniqueProperty :: SmoothBasis Natural -> Positive Int -> Bool)     ]   , testGroup "Quadratic rings (Gaussian/Eisenstein)"-    [ testGroup "Check that a list of smooth numbers generated by `smoothOver` \-                \ only contains valid smooth numbers for the generated basis."+    [ testGroup "smoothOver generates valid smooth numbers"       [ testSmallAndQuick "Gaussian" isSmoothProperty1       , testSmallAndQuick "Eisenstein" isSmoothProperty2       ]     , testCase "all distinct for base [1+3*i,6+8*i]" isSmoothSpecialCase1+    , testCase "6 is smooth for base [4,3,6,10,7]" isSmoothSpecialCase2     ]   ]

@@ -59,8 +59,7 @@ import Math.NumberTheory.Euclidean import qualified Math.NumberTheory.Quadratic.EisensteinIntegers as E (EisensteinInteger(..)) import Math.NumberTheory.Quadratic.GaussianIntegers (GaussianInteger(..))-import Math.NumberTheory.Moduli.PrimitiveRoot (CyclicGroup(..))-import Math.NumberTheory.Primes (UniqueFactorisation, Prime, unPrime)+import Math.NumberTheory.Primes (Prime, UniqueFactorisation) import qualified Math.NumberTheory.SmoothNumbers as SN  import Math.NumberTheory.TestUtils.MyCompose@@ -85,43 +84,12 @@   series = cons2 (:+)  ---------------------------------------------------------------------------------- Cyclic group--instance (Eq a, Num a, UniqueFactorisation a, Arbitrary a) => Arbitrary (CyclicGroup a) where-  arbitrary = frequency-    [ (1, pure CG2)-    , (1, pure CG4)-    , (9, CGOddPrimePower-      <$> (arbitrary :: Gen (Prime a)) `suchThatMap` isOddPrime-      <*> (getPower <$> arbitrary))-    , (9, CGDoubleOddPrimePower-      <$> (arbitrary :: Gen (Prime a)) `suchThatMap` isOddPrime-      <*> (getPower <$> arbitrary))-    ]--instance (Monad m, Eq a, Num a, UniqueFactorisation a, Serial m a) => Serial m (CyclicGroup a) where-  series = pure CG2-        \/ pure CG4-        \/ (CGOddPrimePower-           <$> (series :: Series m (Prime a)) `suchThatMapSerial` isOddPrime-           <*> (getPower <$> series))-        \/ (CGDoubleOddPrimePower-           <$> (series :: Series m (Prime a)) `suchThatMapSerial` isOddPrime-           <*> (getPower <$> series))--isOddPrime-  :: forall a. (Eq a, Num a, UniqueFactorisation a)-  => Prime a-  -> Maybe (Prime a)-isOddPrime p = if (unPrime p :: a) == 2 then Nothing else Just p--------------------------------------------------------------------------------- -- SmoothNumbers -instance (Ord a, Euclidean a, Arbitrary a) => Arbitrary (SN.SmoothBasis a) where+instance (Ord a, Num a, Euclidean a, Arbitrary a) => Arbitrary (SN.SmoothBasis a) where   arbitrary = (fmap getPositive <$> arbitrary) `suchThatMap` SN.fromList -instance (Ord a, Euclidean a, Serial m a) => Serial m (SN.SmoothBasis a) where+instance (Ord a, Num a, Euclidean a, Serial m a) => Serial m (SN.SmoothBasis a) where   series = (fmap getPositive <$> series) `suchThatMapSerial` SN.fromList  -------------------------------------------------------------------------------@@ -152,7 +120,7 @@  testIntegralProperty   :: forall wrapper bool. (TestableIntegral wrapper, SC.Testable IO bool, QC.Testable bool)-  => String -> (forall a. (Euclidean a, Semiring a, Integral a, Bits a, UniqueFactorisation a, Show a) => wrapper a -> bool) -> TestTree+  => String -> (forall a. (GcdDomain a, Euclidean a, Semiring a, Integral a, Bits a, UniqueFactorisation a, Show a) => wrapper a -> bool) -> TestTree testIntegralProperty name f = testGroup name   [ SC.testProperty "smallcheck Int"     (f :: wrapper Int     -> bool)   , SC.testProperty "smallcheck Word"    (f :: wrapper Word    -> bool)@@ -170,7 +138,7 @@  testIntegralPropertyNoLarge   :: forall wrapper bool. (TestableIntegral wrapper, SC.Testable IO bool, QC.Testable bool)-  => String -> (forall a. (Euclidean a, Semiring a, Integral a, Bits a, UniqueFactorisation a, Show a) => wrapper a -> bool) -> TestTree+  => String -> (forall a. (Euclidean a, Semiring a, Integral a, Bits a, UniqueFactorisation a, Show a, Enum (Prime a)) => wrapper a -> bool) -> TestTree testIntegralPropertyNoLarge name f = testGroup name   [ SC.testProperty "smallcheck Int"     (f :: wrapper Int     -> bool)   , SC.testProperty "smallcheck Word"    (f :: wrapper Word    -> bool)@@ -184,7 +152,7 @@  testSameIntegralProperty   :: forall wrapper1 wrapper2 bool. (TestableIntegral wrapper1, TestableIntegral wrapper2, SC.Testable IO bool, QC.Testable bool)-  => String -> (forall a. (Euclidean a, Integral a, Bits a, UniqueFactorisation a, Show a) => wrapper1 a -> wrapper2 a -> bool) -> TestTree+  => String -> (forall a. (GcdDomain a, Euclidean a, Integral a, Bits a, UniqueFactorisation a, Show a) => wrapper1 a -> wrapper2 a -> bool) -> TestTree testSameIntegralProperty name f = testGroup name   [ SC.testProperty "smallcheck Int"     (f :: wrapper1 Int     -> wrapper2 Int     -> bool)   , SC.testProperty "smallcheck Word"    (f :: wrapper1 Word    -> wrapper2 Word    -> bool)

@@ -28,18 +28,19 @@ import Control.Applicative import Data.Coerce import Data.Functor.Classes+import Data.Semiring (Semiring)  import Test.Tasty.QuickCheck as QC hiding (Positive, NonNegative, generate, getNonNegative, getPositive) import Test.SmallCheck.Series (Positive(..), NonNegative(..), Serial(..), Series) -import Math.NumberTheory.Euclidean (Euclidean)+import Math.NumberTheory.Euclidean (GcdDomain, Euclidean) import Math.NumberTheory.Primes (Prime, UniqueFactorisation(..))  ------------------------------------------------------------------------------- -- AnySign  newtype AnySign a = AnySign { getAnySign :: a }-  deriving (Eq, Ord, Read, Show, Num, Enum, Bounded, Integral, Real, Functor, Foldable, Traversable, Arbitrary, Euclidean)+  deriving (Eq, Ord, Read, Show, Num, Enum, Bounded, Integral, Real, Functor, Foldable, Traversable, Arbitrary, Semiring, GcdDomain, Euclidean)  instance (Monad m, Serial m a) => Serial m (AnySign a) where   series = AnySign <$> series@@ -57,6 +58,8 @@ -- Positive from smallcheck  deriving instance Functor Positive+deriving instance Semiring a => Semiring (Positive a)+deriving instance GcdDomain a => GcdDomain (Positive a) deriving instance Euclidean a => Euclidean (Positive a)  instance (Num a, Ord a, Arbitrary a) => Arbitrary (Positive a) where@@ -80,6 +83,8 @@ -- NonNegative from smallcheck  deriving instance Functor NonNegative+deriving instance Semiring a => Semiring (NonNegative a)+deriving instance GcdDomain a => GcdDomain (NonNegative a) deriving instance Euclidean a => Euclidean (NonNegative a)  instance (Num a, Ord a, Arbitrary a) => Arbitrary (NonNegative a) where@@ -134,7 +139,7 @@ -- Power  newtype Power a = Power { getPower :: a }-  deriving (Eq, Ord, Read, Show, Num, Enum, Bounded, Integral, Real, Functor, Foldable, Traversable, Euclidean)+  deriving (Eq, Ord, Read, Show, Num, Enum, Bounded, Integral, Real, Functor, Foldable, Traversable, Semiring, GcdDomain, Euclidean)  instance (Monad m, Num a, Ord a, Serial m a) => Serial m (Power a) where   series = Power <$> series `suchThatSerial` (> 0)

@@ -12,10 +12,10 @@ import qualified Math.NumberTheory.Moduli.EquationsTests as ModuliEquations import qualified Math.NumberTheory.Moduli.JacobiTests as ModuliJacobi import qualified Math.NumberTheory.Moduli.PrimitiveRootTests as ModuliPrimitiveRoot+import qualified Math.NumberTheory.Moduli.SingletonTests as ModuliSingleton import qualified Math.NumberTheory.Moduli.SqrtTests as ModuliSqrt  import qualified Math.NumberTheory.MoebiusInversionTests as MoebiusInversion-import qualified Math.NumberTheory.MoebiusInversion.IntTests as MoebiusInversionInt  import qualified Math.NumberTheory.Powers.CubesTests as Cubes import qualified Math.NumberTheory.Powers.FourthTests as Fourth@@ -72,12 +72,10 @@     , ModuliEquations.testSuite     , ModuliJacobi.testSuite     , ModuliPrimitiveRoot.testSuite+    , ModuliSingleton.testSuite     , ModuliSqrt.testSuite     ]-  , testGroup "MoebiusInversion"-    [ MoebiusInversion.testSuite-    , MoebiusInversionInt.testSuite-    ]+  , MoebiusInversion.testSuite   , Prefactored.testSuite   , testGroup "Primes"     [ Primes.testSuite