[#118] Add Eisenstein integers module

Math/NumberTheory/EisensteinIntegers.hs

@@ -0,0 +1,68 @@
+-- |
+-- Module:      Math.NumberTheory.EisensteinIntegers
+-- Licence:     MIT
+-- Stability:   Provisional
+-- Portability: Non-portable (GHC extensions)
+--
+-- This module exports functions for manipulating Eisenstein integers, including
+-- computing their prime factorisations.
+--
+
+{-# LANGUAGE DeriveGeneric #-}
+
+module Math.NumberTheory.EisensteinIntegers
+  ( EisensteinInteger(..)
+  ,  ω
+  , divE
+  ) where
+
+import GHC.Generics (Generic)
+
+infix 6 :+
+
+-- |An Eisenstein integer is a + bω, where a and b are both integers.
+data EisensteinInteger = (:+) { real :: !Integer, imag :: !Integer }
+    deriving (Eq, Ord, Generic)
+
+-- | The imaginary unit for Eisenstein integers, where
+--
+-- > ω == (-1/2) + ((sqrt 3)/2)ι == exp(2*pi*ι/3)
+-- and ι is the usual imaginary unit with ι² == -1.
+ω :: EisensteinInteger
+ω = 0 :+ 1
+
+instance Show EisensteinInteger where
+    show (a :+ b)
+        | b == 0     = show a
+        | a == 0     = s ++ b'
+        | otherwise  = show a ++ op ++ b'
+        where
+            b' = if abs b == 1 then "ω" else show (abs b) ++ "*ω"
+            op = if b > 0 then "+" else "-"
+            s  = if b > 0 then "" else "-"
+
+instance Num EisensteinInteger where
+    (+) (a :+ b) (c :+ d) = (a + c) :+ (b + d)
+    (*) (a :+ b) (c :+ d) = (a * c - b * d) :+ (b * c + a * d - b * d)
+    -- An Eisenstein integer @a :+ b@, with @a, b@ integers, can we written as
+    -- @(2*a - b) / 2 + ((b * sqrt 3) * ι) / 2@, but this number is in the
+    -- same quadrant as @(2*a - b) / 2 + (b * ι) / 2@, and this one in the
+    -- same as @(2*a - b) + b * ι@. Divisions or floating points are not
+    -- necessary.
+    abs z@(x :+ y) = abs' (2*x - y) x
+      where
+        abs' a b
+            | a == 0 && b == 0 =   z             -- origin
+            | a >  0 && b >= 0 =   z             -- first quadrant: (0, inf) x [0, inf)ω
+            | a <= 0 && b >  0 =   b  :+ (-a)    -- second quadrant: (-inf, 0] x (0, inf)ω
+            | a <  0 && b <= 0 = (-a) :+ (-b)    -- third quadrant: (-inf, 0) x (-inf, 0]ω
+            | otherwise        = (-b) :+   a     -- fourth quadrant: [0, inf) x (-inf, 0)ω
+    negate (a :+ b) = (-a) :+ (-b)
+    fromInteger n = n :+ 0
+    signum z@(a :+ b)
+        | a == 0 && b == 0 = z               -- hole at origin
+        | otherwise        = z `divE` abs z
+
+-- | Eisenstein integer division, truncating toward negative infinity.
+divE :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
+divE = undefined

arithmoi.cabal

@@ -57,6 +57,7 @@ library
     Math.NumberTheory.ArithmeticFunctions.SieveBlock
     Math.NumberTheory.ArithmeticFunctions.Standard
     Math.NumberTheory.Curves.Montgomery
+    Math.NumberTheory.EisensteinIntegers
     Math.NumberTheory.GaussianIntegers
     Math.NumberTheory.GCD
     Math.NumberTheory.GCD.LowLevel