arithmoi-0.12.0.0: Math/NumberTheory/Quadratic/EisensteinIntegers.hs
-- |
-- Module: Math.NumberTheory.EisensteinIntegers
-- Copyright: (c) 2018 Alexandre Rodrigues Baldé
-- Licence: MIT
-- Maintainer: Alexandre Rodrigues Baldé <alexandrer_b@outlook.com>
--
-- This module exports functions for manipulating Eisenstein integers, including
-- computing their prime factorisations.
--
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TypeFamilies #-}
module Math.NumberTheory.Quadratic.EisensteinIntegers
( EisensteinInteger(..)
, ω
, conjugate
, norm
, associates
, ids
-- * Primality functions
, findPrime
, primes
) where
import Prelude hiding (quot, quotRem, gcd)
import Control.DeepSeq
import Data.Coerce
import Data.Euclidean
import Data.List (mapAccumL, partition)
import Data.Maybe
import Data.Ord (comparing)
import qualified Data.Semiring as S
import GHC.Generics (Generic)
import Math.NumberTheory.Moduli.Sqrt
import Math.NumberTheory.Primes.Types
import qualified Math.NumberTheory.Primes as U
import Math.NumberTheory.Utils (mergeBy)
import Math.NumberTheory.Utils.FromIntegral
infix 6 :+
-- | An Eisenstein integer is @a + bω@, where @a@ and @b@ are both integers.
data EisensteinInteger = !Integer :+ !Integer
deriving (Eq, Ord, Generic)
instance NFData EisensteinInteger
-- | 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 - d) + a * d)
abs = fst . absSignum
negate (a :+ b) = (-a) :+ (-b)
fromInteger n = n :+ 0
signum = snd . absSignum
instance S.Semiring EisensteinInteger where
plus = (+)
times = (*)
zero = 0 :+ 0
one = 1 :+ 0
fromNatural n = naturalToInteger 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)
absSignum 0 = (0, 0)
absSignum z@(a :+ b)
-- first sextant: 0 ≤ Arg(z) < π/3
| a > b && b >= 0 = (z, 1)
-- second sextant: π/3 ≤ Arg(z) < 2π/3
| b >= a && a > 0 = (b :+ (b - a), 1 :+ 1)
-- third sextant: 2π/3 ≤ Arg(z) < π
| b > 0 && 0 >= a = ((b - a) :+ (-a), 0 :+ 1)
-- fourth sextant: -π ≤ Arg(z) < -2π/3
| a < b && b <= 0 = (-z, -1)
-- fifth sextant: -2π/3 ≤ Arg(η) < -π/3
| b <= a && a < 0 = ((-b) :+ (a - b), (-1) :+ (-1))
-- sixth sextant: -π/3 ≤ Arg(η) < 0
| otherwise = ((a - b) :+ a, 0 :+ (-1))
-- | List of all Eisenstein units, counterclockwise across all sextants,
-- starting with @1@.
ids :: [EisensteinInteger]
ids = take 6 (iterate ((1 + ω) *) 1)
-- | Produce a list of an @EisensteinInteger@'s associates.
associates :: EisensteinInteger -> [EisensteinInteger]
associates e = map (e *) ids
instance GcdDomain EisensteinInteger
instance Euclidean EisensteinInteger where
degree = fromInteger . norm
quotRem x (d :+ 0) = quotRemInt x d
quotRem x y = (q, x - q * y)
where
(q, _) = quotRemInt (x * conjugate y) (norm y)
quotRemInt :: EisensteinInteger -> Integer -> (EisensteinInteger, EisensteinInteger)
quotRemInt z 1 = ( z, 0)
quotRemInt z (-1) = (-z, 0)
quotRemInt (a :+ b) c = (qa :+ qb, (ra - bumpA) :+ (rb - bumpB))
where
halfC = abs c `quot` 2
bumpA = signum a * halfC
bumpB = signum b * halfC
(qa, ra) = (a + bumpA) `quotRem` c
(qb, rb) = (b + bumpB) `quotRem` c
-- | Conjugate a Eisenstein integer.
conjugate :: EisensteinInteger -> EisensteinInteger
conjugate (a :+ b) = (a - b) :+ (-b)
-- | The square of the magnitude of a Eisenstein integer.
norm :: EisensteinInteger -> Integer
norm (a :+ b) = a*a - a * b + b*b
-- | Checks if a given @EisensteinInteger@ is prime. @EisensteinInteger@s
-- whose norm is a prime congruent to @0@ or @1@ modulo 3 are prime.
-- See <http://thekeep.eiu.edu/theses/2467 Bandara, Sarada, "An Exposition of the Eisenstein Integers" (2016)>,
-- page 12.
isPrime :: EisensteinInteger -> Bool
isPrime e | e == 0 = False
-- 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 = isJust $ U.isPrime a'
| nE `mod` 3 == 1 = isJust $ U.isPrime nE
| otherwise = False
where nE = norm e
a' :+ b' = abs e
-- | Remove @1 - ω@ factors from an @EisensteinInteger@, and calculate that
-- prime's multiplicity in the number's factorisation.
divideByThree :: EisensteinInteger -> (Word, EisensteinInteger)
divideByThree = go 0
where
go :: Word -> EisensteinInteger -> (Word, EisensteinInteger)
go !n z@(a :+ b) | r1 == 0 && r2 == 0 = go (n + 1) (q1 :+ q2)
| otherwise = (n, abs z)
where
-- @(a + a - b) :+ (a + b)@ is @z * (2 :+ 1)@, and @z * (2 :+ 1)/3@
-- is the same as @z / (1 :+ (-1))@.
(q1, r1) = divMod (a + a - b) 3
(q2, r2) = divMod (a + b) 3
-- | Find an Eisenstein integer whose norm is the given prime number
-- in the form @3k + 1@.
--
-- >>> import Math.NumberTheory.Primes (nextPrime)
-- >>> findPrime (nextPrime 7)
-- Prime 3+2*ω
findPrime :: Prime Integer -> U.Prime EisensteinInteger
findPrime p = case (r, sqrtsModPrime (9 * q * q - 1) p) of
(1, z : _) -> Prime $ abs $ gcd (unPrime p :+ 0) ((z - 3 * q) :+ 1)
_ -> error "findPrime: argument must be prime p = 6k + 1"
where
(q, r) = unPrime p `quotRem` 6
-- | An infinite list of Eisenstein primes. Uses primes in @Z@ to exhaustively
-- generate all Eisenstein primes in order of ascending norm.
--
-- * Every prime is in the first sextant, so the list contains no associates.
-- * Eisenstein primes from the whole complex plane can be generated by
-- applying 'associates' to each prime in this list.
--
-- >>> take 10 primes
-- [Prime 2+ω,Prime 2,Prime 3+2*ω,Prime 3+ω,Prime 4+3*ω,Prime 4+ω,Prime 5+3*ω,Prime 5+2*ω,Prime 5,Prime 6+5*ω]
primes :: [Prime EisensteinInteger]
primes = coerce $ (2 :+ 1) : mergeBy (comparing norm) l r
where
leftPrimes, rightPrimes :: [Prime Integer]
(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)]]
-- | [Implementation notes for factorise function]
--
-- Compute the prime factorisation of a Eisenstein integer.
--
-- 1. This function works by factorising the norm of an Eisenstein integer
-- and then, for each prime factor, finding the Eisenstein prime whose norm
-- is said prime factor with @findPrime@.
-- 2. This is only possible because the norm function of the Euclidean Domain of
-- Eisenstein integers is multiplicative: @norm (e1 * e2) == norm e1 * norm e2@
-- for any two @EisensteinInteger@s @e1, e2@.
-- 3. In the previously mentioned work <http://thekeep.eiu.edu/theses/2467 Bandara, Sarada, "An Exposition of the Eisenstein Integers" (2016)>,
-- in Theorem 8.4 in Chapter 8, a way is given to express any Eisenstein
-- integer @μ@ as @(-1)^a * ω^b * (1 - ω)^c * product [π_i^a_i | i <- [1..N]]@
-- where @a, b, c, a_i@ are nonnegative integers, @N > 1@ is an integer and
-- @π_i@ are Eisenstein primes.
--
-- Aplying @norm@ to both sides of the equation from Theorem 8.4:
--
-- 1. @norm μ = norm ( (-1)^a * ω^b * (1 - ω)^c * product [ π_i^a_i | i <- [1..N]] ) ==@
-- 2. @norm μ = norm ((-1)^a) * norm (ω^b) * norm ((1 - ω)^c) * norm (product [ π_i^a_i | i <- [1..N]]) ==@
-- 3. @norm μ = (norm (-1))^a * (norm ω)^b * (norm (1 - ω))^c * product [ norm (π_i^a_i) | i <- [1..N]] ==@
-- 4. @norm μ = (norm (-1))^a * (norm ω)^b * (norm (1 - ω))^c * product [ (norm π_i)^a_i) | i <- [1..N]] ==@
-- 5. @norm μ = 1^a * 1^b * 3^c * product [ (norm π_i)^a_i) | i <- [1..N]] ==@
-- 6. @norm μ = 3^c * product [ (norm π_i)^a_i) | i <- [1..N]] ==@
--
-- where @a, b, c, a_i@ are nonnegative integers, and @N > 1@ is an integer.
--
-- The remainder of the Eisenstein integer factorisation problem is about
-- finding appropriate Eisenstein primes @[e_i | i <- [1..M]]@ such that
-- @map norm [e_i | i <- [1..M]] == map norm [π_i | i <- [1..N]]@
-- where @ 1 < N <= M@ are integers and @==@ is equality on sets
-- (i.e.duplicates do not matter).
--
-- NB: The reason @M >= N@ is because the prime factors of an Eisenstein integer
-- may include a prime factor and its conjugate (both have the same norm),
-- meaning the number may have more Eisenstein prime factors than its norm has
-- integer prime factors.
factorise :: EisensteinInteger -> [(Prime EisensteinInteger, Word)]
factorise g = concat $
snd $
mapAccumL go (abs g) (U.factorise $ norm g)
where
go :: EisensteinInteger -> (Prime Integer, Word) -> (EisensteinInteger, [(Prime EisensteinInteger, Word)])
go z (Prime 3, e)
| e == n = (q, [(Prime (2 :+ 1), e)])
| otherwise = error $ "3 is a prime factor of the norm of z = " ++ show z
++ " with multiplicity " ++ show e
++ " but (1 - ω) only divides z " ++ show n ++ "times."
where
-- Remove all @1 :+ (-1)@ (which is associated to @2 :+ 1@) factors
-- from the argument.
(n, q) = divideByThree z
go z (p, e)
| unPrime p `mod` 3 == 2
= let e' = e `quot` 2 in (z `quotI` (unPrime p ^ e'), [(Prime (unPrime p :+ 0), e')])
-- 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.
-- This @otherwise@ is mandatorily @`mod` 3 == 1@.
| otherwise = (z', filter ((> 0) . snd) [(gp, k), (gp', k')])
where
gp = findPrime p
x :+ y = unPrime gp
-- @gp'@ is @gp@'s conjugate.
gp' = Prime (x :+ (x - y))
(k, k', z') = divideByPrime gp gp' (unPrime p) e z
quotI (a :+ b) n = a `quot` n :+ b `quot` n
-- | Remove @p@ and @conjugate p@ factors from the argument, where
-- @p@ is an Eisenstein prime.
divideByPrime
:: Prime EisensteinInteger -- ^ Eisenstein prime @p@
-> Prime EisensteinInteger -- ^ Conjugate of @p@
-> Integer -- ^ Precomputed norm of @p@, of form @4k + 1@
-> Word -- ^ Expected number of factors (either @p@ or @conjugate p@)
-- in Eisenstein integer @z@
-> EisensteinInteger -- ^ Eisenstein integer @z@
-> ( Word -- Multiplicity of factor @p@ in @z@
, Word -- Multiplicity of factor @conjigate p@ in @z@
, EisensteinInteger -- Remaining Eisenstein integer
)
divideByPrime p p' np k = go k 0
where
go :: Word -> Word -> EisensteinInteger -> (Word, Word, EisensteinInteger)
go 0 d z = (d, d, z)
go c d z | c >= 2, Just z' <- z `quotEvenI` np = go (c - 2) (d + 1) z'
go c d z = (d + d1, d + d2, z'')
where
(d1, z') = go1 c 0 z
d2 = c - d1
z'' = iterate (\g -> fromMaybe err $ (g * unPrime p) `quotEvenI` np) z' !! max 0 (wordToInt d2)
go1 :: Word -> Word -> EisensteinInteger -> (Word, EisensteinInteger)
go1 0 d z = (d, z)
go1 c d z
| Just z' <- (z * unPrime p') `quotEvenI` np
= go1 (c - 1) (d + 1) z'
| otherwise
= (d, z)
err = error $ "divideByPrime: malformed arguments" ++ show (p, np, k)
-- | Divide an Eisenstein integer by an even integer.
quotEvenI :: EisensteinInteger -> Integer -> Maybe EisensteinInteger
quotEvenI (x :+ y) n
| xr == 0 , yr == 0 = Just (xq :+ yq)
| otherwise = Nothing
where
(xq, xr) = x `quotRem` n
(yq, yr) = y `quotRem` n
-------------------------------------------------------------------------------
-- | See the source code and Haddock comments for the @factorise@ and @isPrime@
-- functions in this module (they are not exported) for implementation
-- details.
instance U.UniqueFactorisation EisensteinInteger where
factorise 0 = []
factorise e = coerce $ factorise e
isPrime e = if isPrime e then Just (Prime e) else Nothing