new: Solved problems 31 to 41

Problems 31-41/problem_31.hs

@@ -0,0 +1,5 @@
+isqrt :: (Integral n) => n -> n
+isqrt = floor . sqrt . fromIntegral
+
+isPrime :: (Integral n) => n -> Bool
+isPrime n = (n >= 2) && all ((/= 0) . (n `mod`)) [2 .. isqrt n]

Problems 31-41/problem_32.hs

@@ -0,0 +1,7 @@
+myGCD :: (Integral n) => n -> n -> n
+myGCD m n =
+  if remainder == 0
+    then abs n
+    else myGCD n remainder
+  where
+    remainder = m `mod` n

Problems 31-41/problem_33.hs

@@ -0,0 +1,10 @@
+myGCD :: (Integral n) => n -> n -> n
+myGCD m n =
+  if remainder == 0
+    then abs n
+    else myGCD n remainder
+  where
+    remainder = m `mod` n
+
+coprime :: (Integral n) => n -> n -> Bool
+coprime m n = myGCD m n == 1

Problems 31-41/problem_34.hs

@@ -0,0 +1,13 @@
+myGCD :: (Integral n) => n -> n -> n
+myGCD m n =
+  if remainder == 0
+    then abs n
+    else myGCD n remainder
+  where
+    remainder = m `mod` n
+
+coprime :: (Integral n) => n -> n -> Bool
+coprime m n = myGCD m n == 1
+
+totient :: (Integral n) => n -> n
+totient m = fromIntegral $ length $ filter (coprime m) [1 .. m - 1]

Problems 31-41/problem_35.hs

@@ -0,0 +1,10 @@
+import Data.List (find)
+
+isqrt :: (Integral n) => n -> n
+isqrt = floor . sqrt . fromIntegral
+
+primeFactors :: (Integral n) => n -> [n]
+primeFactors n =
+  case find ((== 0) . (n `mod`)) [2 .. isqrt n] of
+    Nothing -> [n]
+    Just m -> m : primeFactors (n `div` m)

Problems 31-41/problem_36.hs

@@ -0,0 +1,13 @@
+import Data.List (find, group)
+
+isqrt :: (Integral n) => n -> n
+isqrt = floor . sqrt . fromIntegral
+
+primeFactors :: (Integral n) => n -> [n]
+primeFactors n =
+  case find ((== 0) . (n `mod`)) [2 .. isqrt n] of
+    Nothing -> [n]
+    Just m -> m : primeFactors (n `div` m)
+
+primeFactorsMult :: (Integral n) => n -> [(n, n)]
+primeFactorsMult = map (\g -> (head g, fromIntegral $ length g)) . group . primeFactors

Problems 31-41/problem_37.hs

@@ -0,0 +1,16 @@
+import Data.List (find, group)
+
+isqrt :: (Integral n) => n -> n
+isqrt = floor . sqrt . fromIntegral
+
+primeFactors :: (Integral n) => n -> [n]
+primeFactors n =
+  case find ((== 0) . (n `mod`)) [2 .. isqrt n] of
+    Nothing -> [n]
+    Just m -> m : primeFactors (n `div` m)
+
+primeFactorsMult :: (Integral n) => n -> [(n, n)]
+primeFactorsMult = map (\g -> (head g, fromIntegral $ length g)) . group . primeFactors
+
+phi :: (Integral n) => n -> n
+phi = product . map (\(p, r) -> (p - 1) * p ^ (r - 1)) . primeFactorsMult

Problems 31-41/problem_39.hs

@@ -0,0 +1,8 @@
+isqrt :: (Integral n) => n -> n
+isqrt = floor . sqrt . fromIntegral
+
+isPrime :: (Integral n) => n -> Bool
+isPrime n = (n >= 2) && all ((/= 0) . (n `mod`)) [2 .. isqrt n]
+
+primesR :: (Integral n) => n -> n -> [n]
+primesR m n = filter isPrime [m .. n]

Problems 31-41/problem_40.hs

@@ -0,0 +1,16 @@
+import Data.List (find)
+
+isqrt :: (Integral n) => n -> n
+isqrt = floor . sqrt . fromIntegral
+
+isPrime :: (Integral n) => n -> Bool
+isPrime n = (n >= 2) && all ((/= 0) . (n `mod`)) [2 .. isqrt n]
+
+goldbach :: (Integral n) => n -> (n, n)
+goldbach n =
+  if odd n
+    then error "Odd number was given"
+    else (p, n - p)
+  where
+    Just p = find (\p -> (n - p) `elem` primes) primes
+    primes = filter isPrime [2 .. n - 1]

Problems 31-41/problem_41.hs

@@ -0,0 +1,22 @@
+import Data.List (find)
+
+isqrt :: (Integral n) => n -> n
+isqrt = floor . sqrt . fromIntegral
+
+isPrime :: (Integral n) => n -> Bool
+isPrime n = (n >= 2) && all ((/= 0) . (n `mod`)) [2 .. isqrt n]
+
+goldbach :: (Integral n) => n -> (n, n)
+goldbach n =
+  if odd n
+    then error "Odd number was given"
+    else (p, n - p)
+  where
+    Just p = find (\p -> (n - p) `elem` primes) primes
+    primes = filter isPrime [2 .. n - 1]
+
+goldbachList' :: (Integral n) => n -> n -> n -> [(n, n)]
+goldbachList' m n d = filter (\(p, q) -> p > d && q > d) $ map goldbach $ filter even [m .. n]
+
+goldbachList :: (Integral n) => n -> n -> [(n, n)]
+goldbachList m n = goldbachList' m n 1