NumberTheory (empty) → 0.1.0.0
raw patch · 5 files changed
+1605/−0 lines, 5 filesdep +HUnitdep +basedep +containerssetup-changed
Dependencies added: HUnit, base, containers, primes
Files
- LICENSE +675/−0
- NumberTheory.cabal +26/−0
- NumberTheory.hs +677/−0
- NumberTheory_Tests.hs +225/−0
- Setup.hs +2/−0
+ LICENSE view
@@ -0,0 +1,675 @@+ GNU GENERAL PUBLIC LICENSE+ Version 3, 29 June 2007++ Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>+ Everyone is permitted to copy and distribute verbatim copies+ of this license document, but changing it is not allowed.++ Preamble++ The GNU General Public License is a free, copyleft license for+software and other kinds of works.++ The licenses for most software and other practical works are designed+to take away your freedom to share and change the works. By contrast,+the GNU General Public License is intended to guarantee your freedom to+share and change all versions of a program--to make sure it remains free+software for all its users. We, the Free Software Foundation, use the+GNU General Public License for most of our software; it applies also to+any other work released this way by its authors. You can apply it to+your programs, too.++ When we speak of free software, we are referring to freedom, not+price. 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+ NumberTheory.cabal view
@@ -0,0 +1,26 @@+-- Initial NumberTheory.cabal generated by cabal init. For further +-- documentation, see http://haskell.org/cabal/users-guide/++name: NumberTheory+version: 0.1.0.0+synopsis: A library for number theoretic computations, written in Haskell.+-- description: +license: GPL-3+license-file: LICENSE+author: Chris Fredrickson+maintainer: chris.p.fredrickson@gmail.com+-- copyright: +category: Math+build-type: Simple+cabal-version: >=1.8++library+ exposed-modules: NumberTheory+ -- other-modules: + build-depends: base ==4.*, containers ==0.5.*, primes ==0.2.*+ ghc-options: -Wall++Test-Suite NumberTheory_Tests+ type: exitcode-stdio-1.0+ Main-Is: NumberTheory_Tests.hs+ build-depends: base == 4.*, containers == 0.5.*, HUnit ==1.3.1.*, primes == 0.2.*
+ NumberTheory.hs view
@@ -0,0 +1,677 @@+{-# LANGUAGE ScopedTypeVariables #-}+-- |A library for doing number-theoretic computations. This includes computations+-- in Z mod m (henceforth also written Zm), Z, Z x Zi (the Gaussian integers),+-- and some computations with continued fractions.+module NumberTheory (+ -- pythagorean triples+ pythSide,+ pythLeg,+ pythHyp,+ primPythHyp,+ primPythLeg,+ -- Functions in Z mod m+ canon,+ evalPoly,+ polyCong,+ exponentiate,+ rsaGenKeys,+ rsaEval,+ units,+ nilpotents,+ idempotents,+ roots,+ almostRoots,+ order,+ orders,+ expressAsRoots,+ powerCong,+ ilogBM,+ -- functions in Z+ divisors,+ factorize,+ nonUnitFactorize,+ primes,+ isPrime,+ areCoprime,+ legendre,+ kronecker,+ -- arithmetic functions+ totient,+ tau,+ sigma,+ mobius,+ littleOmega,+ bigOmega,+ -- Gaussian Integer functions+ GaussInt((:+)),+ real,+ imag,+ conjugate,+ magnitude,+ (.+),+ (.-),+ (.*),+ (./),+ (.%),+ gIsPrime,+ gPrimes,+ gGCD,+ gFindPrime,+ gExponentiate,+ gFactorize,+ -- assorted combinatorics+ factorial,+ fibonacci,+ permute,+ choose,+ enumerate,+ asSumOfSquares,+ -- Continued fraction functions+ ContinuedFraction(Finite, Infinite),+ continuedFractionFromDouble,+ continuedFractionFromRational,+ continuedFractionFromQuadratic,+ continuedFractionToRational,+ continuedFractionToFractional+) where++import Data.List ((\\), elemIndex, genericLength, nub, sort)+import qualified Data.Map as Map (fromListWith, toList)+import Data.Monoid+import qualified Data.Numbers.Primes as Primes (primes)+import Data.Ratio ((%), denominator, numerator, Ratio)+import qualified Data.Set as Set (fromList, Set, size, toList)++-- |The canonical representation of x in Z mod m.+canon :: Integral a => a -> a -> a+canon x m+ | x < 0 = canon (x + m) m+ | otherwise = x `mod` m++-- square root of an integral value, for brevity elsewhere+sqrti :: Integral a => a -> Double+sqrti = (sqrt :: Double -> Double) . fromIntegral++isIntegral :: Double -> Bool+isIntegral x = (floor :: Double -> Integer) x == ceiling x++-- |List all pythagorean triples that include a given length (either as a leg+-- or hypotenuse).+pythSide :: Integral a => a -> [(a, a, a)]+pythSide s = sort $ pythLeg s ++ pythHyp s++-- |List all pythagorean triples that include a given leg length.+pythLeg :: Integral a => a -> [(a, a, a)]+pythLeg leg = sort [ (k * a, k * b, k * c)+ | k <- divisors leg+ , (a, b, c) <- primPythLeg $ leg `quot` k+ ]++-- |List all primitive pythagorean triples that include a given leg length.+primPythLeg :: Integral a => a -> [(a, a, a)]+primPythLeg leg = -- (a, b, c) = (m^2-n^2, 2mn, m^2+n^2) for some integers m, n+ sort [ (a, b, c)+ | (m, n) <- findMN+ , let (a, b, c) = generatePythTriple m n+ ]+ where+ findMN+ | leg `mod` 2 == 0 =+ [ (m, n)+ | n <- divisors leg+ , let m = quot leg (2 * n)+ , areLegalParametersForPythTriple m n+ ]+ | otherwise =+ [ (m, n)+ | n <- [1 .. leg]+ , let x = sqrti $ leg + n * n+ , isIntegral x+ , let m = floor x+ , areLegalParametersForPythTriple m n+ ]++-- |List all pythagorean triples with a given hypotenuse.+pythHyp :: Integral a => a -> [(a, a, a)]+pythHyp hypotenuse = sort [ (k * a, k * b, k * c)+ | k <- divisors hypotenuse+ , (a, b, c) <- primPythHyp $ hypotenuse `quot` k+ ]++-- |List all primitive pythagorean triples with a given hypotenuse.+primPythHyp :: Integral a => a -> [(a, a, a)]+primPythHyp hypotenuse =+ sort [ (a, b, c)+ | n <- [1 .. floor $ sqrti hypotenuse]+ , let x = sqrti (hypotenuse - n * n)+ , isIntegral x+ , let m = floor x+ , areLegalParametersForPythTriple m n+ , let (a, b, c) = generatePythTriple m n+ ]++generatePythTriple :: Integral a => a -> a -> (a, a, a)+generatePythTriple m n =+ let a = m * m - n * n+ b = 2 * m * n+ c = m * m + n * n+ in (a, b, c)++areLegalParametersForPythTriple :: Integral a => a -> a -> Bool+areLegalParametersForPythTriple m n =+ 0 < n &&+ n < m &&+ gcd m n == 1 && -- m and n must be coprime+ even (m*n) -- exactly 1 of m and n is divisible by 2 (this test is sufficient since they are coprime)++-- |List all divisors of n (not just proper divisors).+divisors :: Integral a => a -> [a]+divisors n+ | n == 0 = []+ | n == 1 = [1]+ | n < 0 = divisors (-n)+ | otherwise = let divisorPairs = [nub [x, quot n x] | x <- [2 .. limit], n `mod` x == 0]+ limit = floor $ sqrti n+ in sort . ([1, n] ++) $ concat divisorPairs++-- |List the prime factors of n, and their multiplicities.+factorize :: (Integral a) => a -> [(a, a)]+factorize n+ | n == 0 = []+ | n < 0 = (-1, 1) : nonUnitFactorize (-n)+ | otherwise = (1, 1) : nonUnitFactorize n++nonUnitFactorize :: (Integral a) => a -> [(a, a)]+nonUnitFactorize n+ | n == 0 = []+ | n < 0 = nonUnitFactorize (-n)+ | otherwise = let findFactors :: Integral a => a -> [a] -> [a]+ findFactors 1 acc = sort acc+ findFactors k acc =+ let d = head . tail $ divisors k+ in findFactors (quot k d) (d : acc)+ in collapseMultiplicities $ findFactors n []++-- collapse a list of elements to (elt, multiplicity) pairs+collapseMultiplicities :: (Ord a, Num b) => [a] -> [(a, b)]+collapseMultiplicities list = Map.toList (Map.fromListWith (+) [(x, 1)| x <- list])++-- | Compute Euler's phi. This is equal to the number of integers <= n that are+-- relatively prime to n.+totient :: Integral a => a -> a+totient n+ | n < 0 = totient (-n)+ | n == 0 = 0+ | n == 1 = 1+ | otherwise = let primeList = primes n+ offset = n ^ (genericLength primeList - (1 :: Integer))+ diffList = map ((`subtract` n) . quot n) primeList+ in product diffList `quot` offset++-- |List the unique prime factors of n.+primes :: Integral a => a -> [a]+primes = map fst . nonUnitFactorize+++-- |Compute if n is prime.+isPrime :: Integral a => a -> Bool+isPrime n = n `elem` takeWhile (<= n) (dropWhile (< n) Primes.primes)++-- |Compute whether two integers are relatively prime to each other. That is, if+-- their GCD == 1.+areCoprime :: Integral a => a -> a -> Bool+areCoprime = ((1 ==) .) . gcd++-- |Evaluate a polynomial (in Zm) with given coefficients at a given point+-- using Horner's method.+evalPoly :: forall a. Integral a => a -> a -> [a] -> a+evalPoly m x cs = evalPolyHelper . reverse $ map (`canon` m) cs+ where+ evalPolyHelper :: [a] -> a+ evalPolyHelper [] = 0+ evalPolyHelper [c] = c `mod` m+ evalPolyHelper (c : ct) = let val = evalPolyHelper ct+ in (val * x + c) `mod` m++-- |Find the zeros to a given polynomial in Zm, where the coefficients are+-- given in order of descending degree.+polyCong :: Integral a => a -> [a] -> [a]+polyCong m cs = filter (\x -> evalPoly m x cs == 0) [0 .. m - 1]++-- |Raise a to the power of e in Zm.+exponentiate :: (Integral a) => a -> a -> a -> a+exponentiate a e m+ | e < 0 && a `elem` us = exponentiate a (canon (ul + e) m) m+ | e < 0 = error "a is not invertible in Z mod m"+ | e == 0 = 1+ | even e = canon (s * s) m+ | otherwise = canon (q * a) m+ where+ s = exponentiate a (quot e 2) m+ q = exponentiate a (e - 1) m+ us = units m+ ul = genericLength us++-- |A type to represent a public or private key.+type Key a = (a, a)+-- |Given primes p and q, generate all pairs of public/private keys derived+-- from those values.+rsaGenKeys :: Integral a => a -> a -> [(Key a, Key a)]+rsaGenKeys p q+ | not (isPrime p && isPrime q) = error "p and q must both be prime"+ | otherwise =+ [ ((e, n), (d, n))+ | let n = p * q+ phi = totient n+ , e <- filter (areCoprime phi) [2 .. phi - 1]+ , d <- polyCong phi [e, -1]+ ]++-- |Use the given key to encode/decode the message or ciphertext.+rsaEval :: (Integral a) => Key a -> a -> a+rsaEval (k, n) text = exponentiate text k n++-- |Compute the group of units of Zm.+units :: Integral a => a -> [a]+units n = filter (areCoprime n) [1 .. n - 1]++-- |Compute the nilpotent elements of Zm.+nilpotents :: (Integral a) => a -> [a]+nilpotents m+ | r == 0 = []+ | otherwise = [ n+ | n <- [0 .. m - 1]+ , let powers = map (\e -> exponentiate n e m) [1 .. r]+ , 0 `elem` powers+ ]+ where r = genericLength $ units m++-- |Compute the idempotent elements of Zm.+idempotents :: Integral a => a -> [a]+idempotents = flip polyCong [1, -1, 0]++-- |Compute the primitive roots of Zm.+roots :: (Integral a) => a -> [a]+roots m+ | null us = []+ | otherwise = [ u | u <- us, order u m == genericLength us]+ where us = units m++-- |Compute the "almost roots" of Zm. An almost root is a unit, is not a+-- primitive root, and generates the whole group of units when exponentiated.+almostRoots :: forall a. (Integral a) => a -> [a]+almostRoots m = let unitCount = genericLength $ units m+ expList = [1 .. unitCount + 1]+ generateUnits :: a -> Set.Set a+ generateUnits u = Set.fromList $ concat+ [ [k, canon (-k) m]+ | e <- expList+ , let k = exponentiate u e m+ ]+ in sort [ u+ | u <- units m \\ roots m+ , unitCount == (fromIntegral . Set.size $ generateUnits u)+ ]++-- |Compute the order of x in Zm.+order :: (Integral a) => a -> a -> a+order x m = head [ ord+ | ord <- [1 .. genericLength $ units m]+ , exponentiate (canon x m) ord m == 1+ ]++-- |Computes the orders of all units in Zm.+orders :: (Integral a) => a -> [a]+orders m = map (`order` m) $ units m++rootsOrAlmostRoots :: (Integral a) => a -> [a]+rootsOrAlmostRoots m =+ case roots m of+ [] -> almostRoots m+ rs -> rs++-- |Find powers of all the primitive roots of Zm that are equal to x.+-- Equivalently, express x as powers of roots (almost or primitive) in Zm.+expressAsRoots :: (Integral a) => a -> a -> [(a, a)]+expressAsRoots x m =+ let rs = rootsOrAlmostRoots m+ in [ (r', e)+ | r <- rs+ , e <- [1 .. order r m]+ , let k = exponentiate r e m+ , r' <- [ r | k == x ]+ ++ [ -r | canon (-k) m == x ]+ ]++-- |Solve the power congruence for x, given e, k, m: x^e = k (mod m)+powerCong :: (Integral a) => a -> a -> a -> [a]+powerCong e k m = [ x+ | x <- [1 .. m]+ , exponentiate x e m == canon k m+ ]++-- |Compute the integer log base B of k in Zm.+-- Equivalently, given 2 elements of Zm, find what powers of b produce k, if any.+ilogBM :: (Integral a) => a -> a -> a -> [a]+ilogBM b k m = let bc = canon b m+ kc = canon k m+ in [ e+ | e <- [1 .. order bc m]+ , exponentiate bc e m == kc+ ]++-- |Compute the Legendre symbol of p and q.+legendre :: (Integral a) => a -> a -> a+legendre q p+ | not $ isPrime p = error "p is not prime"+ | p == 2 = error "p must be odd"+ | otherwise = let r = exponentiate q (quot (p - 1) 2) p+ in if r > 1 then (-1) else r++-- |Compute the Kronecker symbol (a|n).+kronecker :: (Integral a) => a -> a -> a+kronecker a n+ | n == (-1) && a < 0 = -1+ | n == (-1) = 1+ | n == 0 && abs a == 1 = 1+ | n == 0 = 0+ | n == 1 = 1+ | n == 2 && even a = 0+ | n == 2 && abs (a `mod` 8) == 1 = 1+ | n == 2 = -1+ | isPrime n = legendre a n+ | otherwise = kronecker a u * product [ kronecker a p ^ e | (p, e) <- nonUnitFactorize n]+ where u = if a < 0 then -1 else 1+++-- |Compute tau(n), the number of divisors of n.+tau :: Integral a => a -> a+tau = genericLength . divisors++-- |Compute sigma(n), the sum of powers of divisors of n.+sigma :: Integral a => a -> a -> a+sigma k = sum . map (^ k) . divisors++-- |Compute mobius(n): (-1)^littleOmega(n) if n is square-free, 0 otherwise.+mobius :: (Integral a) => a -> a+mobius n+ | isSquareFree n = (-1) ^ littleOmega n+ | otherwise = 0+ where+ isSquareFree :: Integral a => a -> Bool+ isSquareFree = all (odd . snd) . nonUnitFactorize++-- |Compute littleOmega(n), the number of unique prime factors.+littleOmega :: Integral a => a -> a+littleOmega = genericLength . nonUnitFactorize++-- |Compute bigOmega(n), the number of prime factors of n (including multiplicities).+bigOmega :: Integral a => a -> a+bigOmega = sum . map snd . nonUnitFactorize++---------------------------------------------------------------------------------+infix 6 :++-- |A Gaussian integer is a+bi, where a and b are both integers.+data GaussInt a = a :+ a deriving (Ord, Eq)++instance (Show a, Ord a, Num a) => Show (GaussInt a) where+ show (a :+ b) = show a ++ op ++ b' ++ "i"+ where op = if b > 0 then "+" else "-"+ b' = if abs b /= 1 then show (abs b) else ""++instance (Monoid a) => Monoid (GaussInt a) where+ mempty = (mempty :: a) :+ (mempty :: a)+ (c :+ d) `mappend` (e :+ f) = (c `mappend` e) :+ (d `mappend` f)++-- |The real part of a Gaussian integer.+real :: GaussInt a -> a+real (x :+ _) = x++-- |The imaginary part of a Gaussian integer.+imag :: GaussInt a -> a+imag (_ :+ y) = y++-- |Conjugate a Gaussian integer.+conjugate :: Num a => GaussInt a -> GaussInt a+conjugate (r :+ i) = r :+ (-i)++-- |The square of the magnitude of a Gaussian integer.+magnitude :: Num a => GaussInt a -> a+magnitude (x :+ y) = x * x + y * y++-- |Add two Gaussian integers together.+(.+) :: Num a => GaussInt a -> GaussInt a -> GaussInt a+(gr :+ gi) .+ (hr :+ hi) = (gr + hr) :+ (gi + hi)++-- |Subtract one Gaussian integer from another.+(.-) :: Num a => GaussInt a -> GaussInt a -> GaussInt a+(gr :+ gi) .- (hr :+ hi) = (gr - hr) :+ (gi - hi)++-- |Multiply two Gaussian integers.+(.*) :: Num a => GaussInt a -> GaussInt a -> GaussInt a+(gr :+ gi) .* (hr :+ hi) = (gr * hr - hi * gi) :+ (gr * hi + gi * hr)++-- "div" truncates toward -infinity, "quot" truncates toward 0, but we need+-- something that truncates toward the nearest integer. I.e., we want to+-- truncate with "round".+divToNearest :: (Integral a, Integral b) => a -> a -> b+divToNearest x y = round ((x % 1) / (y % 1))++-- |Divide one Gaussian integer by another.+(./) :: Integral a => GaussInt a -> GaussInt a -> GaussInt a+g ./ h =+ let nr :+ ni = g .* conjugate h+ denom = magnitude h+ in divToNearest nr denom :+ divToNearest ni denom++-- |Compute the remainder when dividing one Gaussian integer by another.+(.%) :: Integral a => GaussInt a -> GaussInt a -> GaussInt a+g .% m =+ let q = g ./ m+ p = m .* q+ in g .- p++-- |Compute whether a given Gaussian integer is prime.+gIsPrime :: Integral a => GaussInt a -> Bool+gIsPrime = isPrime . magnitude++-- |An infinte list of the Gaussian primes. This list is in order of ascending magnitude.+gPrimes :: Integral a => [GaussInt a]+gPrimes = [ a' :+ b'+ | mag <- Primes.primes+ , let radius = floor $ sqrti mag+ , a <- [0 .. radius]+ , let y = sqrti $ mag - a*a+ , isIntegral y+ , let b = floor y+ , a' <- [-a, a]+ , b' <- [-b, b]+ ]++-- |Compute the GCD of two Gaussian integers.+gGCD :: Integral a => GaussInt a -> GaussInt a -> GaussInt a+gGCD g h+ | h == 0 :+ 0 = g --done recursing+ | otherwise = gGCD h (g .% h)++-- |Find a Gaussian integer whose magnitude squared is the given prime number.+gFindPrime :: (Integral a) => a -> [GaussInt a]+gFindPrime 2 = [1 :+ 1]+gFindPrime p+ | p `mod` 4 == 1 && isPrime p =+ let r = head $ roots p+ z = exponentiate r (quot (p - 1) 4) p+ in [gGCD (p :+ 0) (z :+ 1)]+ | otherwise = []++-- |Raise a Gaussian integer to a given power.+gExponentiate :: (Integral a) => GaussInt a -> a -> GaussInt a+gExponentiate a e+ | e < 0 = error "Cannot exponentiate Gaussian Int to negative power"+ | e == 0 = 1 :+ 0+ | even e = s .* s+ | otherwise = a .* m+ where+ s = gExponentiate a (quot e 2)+ m = gExponentiate a (e - 1)++-- |Compute the prime factorization of a Gaussian integer. This is unique up to units (+/- 1, +/- i).+gFactorize :: forall a. (Integral a) => GaussInt a -> [(GaussInt a, a)]+gFactorize g+ | g == 0 :+ 0 = [(0 :+ 0, 1)]+ | otherwise =+ let nonUnits = concatMap processPrime . nonUnitFactorize $ magnitude g+ nonUnitProduct = foldr ((.*) . uncurry gExponentiate) (1 :+ 0) nonUnits+ remainderUnit = (g ./ nonUnitProduct, 1)+ in remainderUnit : nonUnits+ where+ processPrime :: (a, a) -> [(GaussInt a, a)]+ processPrime (p, e)+ --deal with primes congruent to 3 (mod 4)+ | p `mod` 4 == 3 = [(p :+ 0, quot e 2)]+ --deal with all other primes+ | otherwise = collapseMultiplicities $ processGaussPrime g []+ where+ processGaussPrime :: GaussInt a -> [GaussInt a] -> [GaussInt a]+ processGaussPrime g' acc = do+ gp <- gFindPrime p -- find a GaussInt whose magnitude is p+ let fs = filter (\f -> g' .% f == 0 :+ 0) [gp, conjugate gp] --find the list of even divisors+ case fs of+ [] -> acc -- Couldn't find a factor, so stop recursing+ f : _ -> processGaussPrime (g' ./ f) (f : acc) -- add this factor to the list, and keep looking++---------------------------------------------------------------------------------+--Combinatorics and other fun things++-- |Compute the factorial of a given integer.+factorial :: Integral a => a -> a+factorial n = product [1 .. n]++-- |The Fibonacci sequence.+fibonacci :: Num a => [a]+fibonacci = 0 : 1 : zipWith (+) fibonacci (tail fibonacci)++-- |Given a set of n elements, compute the number of ways to arrange k elements of it.+permute :: Integral a => a -> a -> a+permute n k = factorial n `quot` factorial (n - k)++-- |Given a set of n elements, compute the number of ways to choose r elements of it.+choose :: Integral a => a -> a -> a+choose n r = (n `permute` r) `quot` factorial r++-- |Given a list of spots, where each spot is a list of its possible values,+-- enumerate all possible assignments of values to spots.+enumerate :: [[a]] -> [[a]]+enumerate [] = [[]]+enumerate (c:cs) = [ a : as+ | a <- c+ , as <- enumerate cs+ ]++-- |Given an integer n, find all ways of expressing n as the sum of two squares.+asSumOfSquares :: Integral a => a -> [(a, a)]+asSumOfSquares n = Set.toList . Set.fromList $+ [ (x', y')+ | x <- [1 .. floor $ sqrti n]+ , let d = n - x * x+ , d > 0+ , let sd = sqrti d+ , isIntegral sd+ , let y = floor sd+ [x', y'] = sort [x, y]+ ]++---------------------------------------------------------------------------------+-- Continued fractions++-- |A (simple) continued fraction can be represented as a list of coefficients.+-- This list is either finite (in the case of rational numbers), or infinite (in+-- the case of irrational numbers. If the fraction represents a quadratic number+-- (that is, a number that can be the root of some quadratic polynomial), then+-- the infinite list of coefficients consists of a finite sequence of coefficients+-- followed by a (finite) sequence of coefficients that repeats indefinitely.+data ContinuedFraction a = Finite [a] | Infinite ([a], [a])++instance (Show a) => Show (ContinuedFraction a) where+ show (Finite as) = "Finite " ++ show as+ show (Infinite (as, ps)) = "Infinite " ++ show as ++ show ps ++ "..."++-- |Convert a Double to a (finite) continued fraction. This is inherently lossy.+continuedFractionFromDouble :: forall a. (Integral a) => Double -> a -> ContinuedFraction a+continuedFractionFromDouble x precision+ | precision < 1 = Finite []+ | otherwise =+ let ts = getTs (fractionalPart x) precision+ in Finite $ integralPart x : map (integralPart . recip) (filter (/= 0) ts)+ where+ integralPart :: Double -> a+ integralPart n = fst $ (properFraction :: Double -> (a, Double)) n+ fractionalPart :: Double -> Double+ fractionalPart 0 = 0+ fractionalPart n = snd $ (properFraction :: Double -> (Integer, Double)) n+ getTs :: Integral a => Double -> a -> [Double]+ getTs y n = reverse $ tRunner [y] n+ where+ tRunner [] _ = error "improper call of tRunner. This should never happen."+ tRunner ts 0 = ts+ tRunner ts@(t : _) m+ | tn == 0 = ts+ | otherwise = tRunner (tn : ts) (m - 1)+ where tn = fractionalPart $ recip t++-- |Convert the quadratic number (m0 + sqrt(d)) / q0 to its continued fraction+-- representation.+continuedFractionFromQuadratic :: forall a. (Integral a) => a -> a -> a -> ContinuedFraction a+continuedFractionFromQuadratic m0 d q0+ | q0 == 0 = error "Cannot divide by 0"+ | isIntegral $ sqrti d = continuedFractionFromRational ((m0 + (floor . sqrti $ d)) % q0)+ | not . isIntegral $ getNextQ m0 q0 = continuedFractionFromQuadratic (m0 * q0) (d * q0 * q0) (q0 * q0)+ | otherwise =+ let a0 = truncate $ (fromIntegral m0 + sqrti d) / fromIntegral q0+ in helper [(m0, q0, a0)]+ where+ helper :: [(a, a, a)] -> ContinuedFraction a+ helper [] = error "improper call to helper function. This will never happen."+ helper ts@((mp, qp, ap) : _) =+ let mn = ap * qp - mp+ qn = (truncate :: Double -> a) $ getNextQ mn qp+ an = truncate ((fromIntegral mn + sqrti d) / fromIntegral qn)+ ts' = reverse ts+ as' = map third ts'+ in case elemIndex (mn, qn, an) ts' of+ -- We've hit the first repetition of the period+ Just idx -> Infinite (take idx as', drop idx as')+ -- Haven't hit the end of the period yet, keep going as usual+ Nothing -> helper $ (mn, qn, an) : ts+ getNextQ :: a -> a -> Double+ getNextQ mp qp = fromIntegral (d - mp * mp) / fromIntegral qp+ third :: (a, b, c) -> c+ third (_, _, x) = x++-- |Convert a continued fraction to a rational number. If the fraction is finite,+-- then this is an exact conversion. If the fraction is infinite, this conversion+-- is necessarily lossy, since the fraction does not represent a rational number.+continuedFractionToRational :: (Integral a) => ContinuedFraction a -> Ratio a+continuedFractionToRational frac =+ let list = case frac of+ Finite as -> as+ Infinite (as, periods) -> as ++ take 35 (cycle periods)+ in foldr (\ai rat -> (ai % 1) + (1 / rat)) (last list % 1) (init list)++-- |Convert a rational number to a continued fraction. This is an exact conversion.+continuedFractionFromRational :: Integral a => Ratio a -> ContinuedFraction a+continuedFractionFromRational rat+ | denominator rat == 1 = Finite [numerator rat]+ | numerator fracPart == 1 = Finite [intPart, denominator fracPart]+ | otherwise =+ let Finite trail = continuedFractionFromRational (1 / fracPart)+ in Finite (intPart : trail)+ where+ intPart = numerator rat `div` denominator rat+ fracPart = rat - (intPart % 1)++-- |Convert a continued fraction to a Fractional type. This is lossy due to+-- precision in the Fractional type, and due to conversion of irrational continued+-- fractions to rational types.+continuedFractionToFractional :: (Fractional a) => ContinuedFraction Integer -> a+continuedFractionToFractional = fromRational . continuedFractionToRational
+ NumberTheory_Tests.hs view
@@ -0,0 +1,225 @@+module Main where++import Data.List+import qualified Data.Numbers.Primes as Primes+import NumberTheory+import Test.HUnit++main :: IO Counts+main = runTestTT tests++tests :: Test+tests = TestList+ [ TestLabel "Continued Fraction Tests" continuedFractionTests+ , TestLabel "Pythagorean Triples Tests" pythTests+ , TestLabel "Z mod M Tests" zModMTests+ , TestLabel "Z Tests" zTests+ , TestLabel "Arithmetic Functions tests" arithmeticFnsTests+ , TestLabel "Gaussian Integer Tests" gaussianIntTests+ ]++limit :: [a] -> [a]+limit = take 20000+--limit = id++pythTests :: Test+pythTests = TestList+ [ TestCase $ assertEqual "test pythSide" [(35, 12, 37),(37, 684, 685)] (pythSide (37 :: Int))+ , TestCase $ assertEqual "test pythLeg" [(15, 8, 17),(15, 20, 25),(15, 36, 39),(15, 112, 113)] (pythLeg (15 :: Int))+ , TestCase $ assertEqual "test pythHyp" [(7, 24, 25),(15, 20, 25)] (pythHyp (25 :: Int))+ ]++-- Note: don't use any functions from NumberTheory to define these (e.g. isPrime).+sampleMixed :: [Integer]+sampleMixed = [1 .. 100]+samplePrimes :: [Integer]+samplePrimes = takeWhile (<= last sampleMixed) Primes.primes+sampleComposites :: [Integer]+sampleComposites = filter (not . flip elem samplePrimes) sampleMixed+sampleMixedGaussInts :: [GaussInt Integer]+sampleMixedGaussInts = [a :+ b | a <- [-25 .. 25], b <- [-25 .. 25]]++zTests :: Test+zTests = TestList+ [ TestList $ limit [ TestCase $ assertEqual "divisors divide evenly" 0 remainder+ | n <- sampleMixed+ , let divs = divisors n+ , d <- divs+ , let remainder = n `mod` d+ ]+ , TestList $ limit [ TestCase $ assertEqual "primes are only divisible by themselves and 1" [1, p] divs+ | p <- samplePrimes+ , let divs = divisors p+ ]+ , TestList $ limit [ TestCase $ assertBool "each divisor has a mate to produce n" found+ | n <- sampleMixed+ , let divs = divisors n+ , d <- divs+ , let found = any (\d' -> d * d' == n) divs+ ]+ , TestList $ limit [ TestCase $ assertEqual "product of factors from factorize is original" n prod+ | n <- sampleMixed+ , let facs = (factorize :: Integer -> [(Integer, Integer)]) n+ , let prod = product $ map (uncurry (^)) facs+ ]+ , TestList $ limit [ TestCase $ assertEqual "test primes on primes" [p] ps+ | p <- samplePrimes+ , let ps = primes p+ ]+ , TestList $ limit [ TestCase $ assertBool "test primes on composites" res+ | n <- sampleMixed+ , let res = all isPrime $ primes n+ ]+ , TestList $ limit [ TestCase $ assertBool "test isPrime on primes" (isPrime p)+ | p <- samplePrimes+ ]+ , TestList $ limit [ TestCase $ assertBool "test isPrime on composites" (not $ isPrime n)+ | n <- sampleComposites+ ]+ , TestList $ limit [ TestCase $ assertBool "test areCoprime on common multiples" res+ | x <- [1 .. 10] :: [Integer]+ , let res = not $ areCoprime 5 (5 * x)+ ]+ , TestList $ limit [ TestCase $ assertBool "test areCoprime on primes" res+ | p <- delete 3 samplePrimes+ , let res = areCoprime 3 p+ ]+ ]++zModMTests :: Test+zModMTests = TestList+ [ TestList $ limit [ TestCase $ assertBool+ ("test canon bounds: " ++ show n ++ " mod " ++ show m)+ (n' >= 0 && n' < m && n `mod` m == n')+ | m <- sampleMixed+ , n <- sampleMixed ++ map negate sampleMixed+ , let n' = canon n m+ ]+ , TestCase $ assertEqual "test evalPoly" 2 (evalPoly 5 3 [4, 5, 6 :: Integer])+ , TestCase $ assertEqual "test polyCong" [1, 4] (polyCong 5 [4, 5, 6 :: Integer])+ , TestCase $ assertEqual "test exponentiate" 3 (exponentiate 9 12 (6 :: Integer))+ , TestCase $ assertEqual "test exponentiate negative" 3 (exponentiate (-9) 12 (6 :: Integer))+ , TestList $ limit [ TestCase $ assertEqual ("test inverses with exponentiation (" ++ show x ++ "^" ++ show e ++ " mod " ++ show n ++ ")") 1 p+ | n <- sampleMixed+ , let us = units n+ , u <- us+ , e <- [1 .. genericLength us]+ , let x = exponentiate u e n+ , let y = exponentiate u (-e) n+ , let p = canon (x * y) n+ ]+ , TestList $ limit [ TestCase $ assertBool "test rsaGenKeys (ed == 1 mod phi(n))" (canon (privk * pubk) (totient n) == (1 :: Integer) && n == n')+ | p <- samplePrimes+ , q <- delete p samplePrimes+ , let keys = rsaGenKeys p q+ , ((pubk, n), (privk, n')) <- keys+ ]+ , TestList $ limit [ TestCase $ assertEqual "test rsaGenKeys (inverses)" text plain+ | text <- sampleMixed+ , p <- samplePrimes+ , q <- delete p samplePrimes+ , let keys = rsaGenKeys p q+ , (pub, priv) <- keys+ , let cipher = rsaEval pub text+ , let plain = rsaEval priv cipher+ ]+ , TestList $ limit [ TestCase $ assertBool+ ("test units invertibility: " ++ show n)+ (all (\u -> any (\u' -> canon (u * u') n == 1) us) us)+ | n <- sampleMixed+ , let us = units n+ ]+ , TestList $ limit [ TestCase $ assertBool+ ("test nilpotents: " ++ show n)+ (all (\xs -> 0 `elem` xs) iteratedLists)+ | n <- sampleMixed+ , let ns = map fromIntegral $ nilpotents n+ , let iteratedLists = map (\x -> take (fromIntegral n) $ iterate (\l -> canon (l * x) n) x) ns+ ]+ , TestList $ limit [ TestCase $ assertBool+ ("test idempotents: " ++ show n)+ (all (\i -> canon (i * i) n == i) is)+ | n <- sampleMixed+ , let is = idempotents n+ ]+ , TestCase $ assertEqual "test roots" [3, 5, 6, 7, 10, 11, 12, 14] (roots (17 :: Integer))+ , TestCase $ assertEqual "test almostRoots" [2, 7, 8, 13] (almostRoots (15 :: Integer))+ , TestCase $ assertEqual "test orders" [1, 4, 2, 4, 4, 2, 4, 2] (orders (15 :: Integer))+ , TestCase $ assertEqual "test expressAsRoots" [(-2, 1), (7, 3), (-8, 3), (13, 1)] (expressAsRoots 13 (15 :: Integer))+ , TestCase $ assertEqual "test powerCong" [2] (powerCong 11 3 (5 :: Integer))+ ]++arithmeticFnsTests :: Test+arithmeticFnsTests = TestList+ [ TestList $ limit [ TestCase $ assertEqual "totient counts number of coprimes <=n" c c'+ | n <- sampleMixed+ , let c = totient n+ , let c' = genericLength $ filter (areCoprime n) [1 .. n]+ ]+ , TestCase $ assertEqual "legendre 3 5" (-1 :: Integer) (legendre 3 5)+ , TestCase $ assertEqual "kronecker 6 5" (1 :: Integer) (kronecker 6 5)+ , TestCase $ assertEqual "tau 60" (12 :: Integer) (tau 60)+ , TestCase $ assertEqual "sigma 1 60" (168 :: Integer) (sigma 1 60)+ , TestCase $ assertEqual "sigma 4 60" (14013636 :: Integer) (sigma 4 60)+ , TestCase $ assertEqual "mobius 9 (non-squarefree)" (0 :: Integer) (mobius 9)+ , TestCase $ assertEqual "mobius 5" (-1 :: Integer) (mobius 5)+ , TestCase $ assertEqual "littleOmega 60" (3 :: Integer) (littleOmega 60)+ , TestCase $ assertEqual "bigOmega 60" (4 :: Integer) (bigOmega 60)+ ]++gaussianIntTests :: Test+gaussianIntTests = TestList+ [ TestList $ limit [ TestCase $ assertEqual "conjugate with 0i" g g'+ | n <- sampleMixed+ , let g = n :+ 0+ , let g' = conjugate g+ ]+ , TestList $ limit [ TestCase $ assertEqual "conjugate mixed ints" (a :+ b) (a' :+ (-b'))+ | g@(a :+ b) <- sampleMixedGaussInts+ , let (a' :+ b') = conjugate g+ ]+ , TestCase $ assertEqual "Gaussian int multiplication" ((2 :: Integer) :+ 42) ((5 :+ 3) .* (4 :+ 6))+ , TestCase $ assertEqual "Gaussian div on even division" ((4 :: Integer) :+ 6) ((2 :+ 42) ./ (5 :+ 3))+ , TestCase $ assertEqual "Gaussian div on uneven division" ((4 :: Integer) :+ 6) ((2 :+ 43) ./ (5 :+ 3))+ , TestCase $ assertEqual "Gaussian div on negative divisor" ((4 :: Integer) :+ 6) (((-2) :+ (-43)) ./ ((-5) :+ (-3)))+ , TestCase $ assertEqual "Gaussian mod on positive case" ((0 :: Integer) :+ 1) ((2 :+ 43) .% (5 :+ 3))+ , TestCase $ assertEqual "Gaussian mod on negative case" ((0 :: Integer) :+ (-1)) (((-2) :+ (-43)) .% (5 :+ 3))+ , TestCase $ assertEqual "magnitude on integer case" (25 :: Integer) (magnitude (5 :+ 0))+ , TestCase $ assertEqual "magnitude on 5 :+ 3" (34 :: Integer) (magnitude (5 :+ 3))+ , TestCase $ assertBool "gIsPrime on prime" (gIsPrime ((2 :: Integer) :+ 5))+ , TestCase $ assertBool "gIsPrime on composite" (not $ gIsPrime ((3 :: Integer) :+ 5))+ , TestList $ limit [ TestCase $ assertBool "gPrimes generates primes" (gIsPrime p)+ | p <- take 100 (gPrimes :: [GaussInt Integer])+ ]+ , TestCase $ assertEqual "gGCD on even multiple" ((2 :: Integer) :+ 4) (gGCD (2 :+ 4) (12 :+ 24))+ , TestCase $ assertEqual "gGCD on uneven multiple" ((1 :: Integer) :+ 1) (gGCD (2 :+ 4) (5 :+ 3))+ , TestCase $ assertBool "gGCD on uneven multiple (division rounding test)"+ (gGCD ((12::Int) :+ 23) (23 :+ 34) `elem` [x :+ y | x <- [(-1)..1], y <- [(-1)..1], abs x + abs y == 1])+ , TestCase $ assertBool "gFindPrime 5" (head (gFindPrime (5::Int)) `elem` [ a :+ b | a <- [2, -2], b <- [1, -1]])+ , TestCase $ assertEqual "gFindPrime 7" [] (gFindPrime (7::Int))+ , TestList $ limit [ TestCase $ assertEqual "gExponentiate on real ints" ((a ^ pow) :+ 0) (gExponentiate g pow)+ | a <- sampleMixed+ , pow <- [1 .. 5] :: [Integer]+ , let g = a :+ 0+ ]+ , TestCase $ assertEqual "gExponentiate on 1st complex int" ((-119 :: Integer) :+ (-120)) (gExponentiate (2 :+ 3) (4 :: Integer))+ , TestCase $ assertEqual "gExponentiate on 2nd complex int" ((122 :: Integer) :+ (-597)) (gExponentiate (2 :+ 3) (5 :: Integer))+ , TestList $ limit [ TestCase $ assertEqual "gFactorize, gMultiply, gExponentiate recover original GaussInt"+ g prod+ | g <- sampleMixedGaussInts+ , let factors = gFactorize g+ , let condensedFactors = map (uncurry gExponentiate) factors+ , let prod = foldl (.*) (1 :+ 0) condensedFactors+ ]+ ]++continuedFractionTests :: Test+continuedFractionTests = TestList+ [ TestCase $ assertBool ("Test conversion to and from continued fraction: (" ++ show m ++ "+ sqrt(" ++ show d ++ "))/" ++ show q)+ (abs (((fromIntegral m + (sqrt :: Double -> Double) (fromIntegral d)) / fromIntegral q) -+ (fromRational . continuedFractionToRational $ continuedFractionFromQuadratic m d q))+ < 0.00000000000001)+ | m <- [0 .. 20]+ , d <- [0 .. 20]+ , q <- [1 .. 20]+ ]
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain