-- |
-- 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 PostfixOperators #-}
{-# 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)
import Data.List.Infinite (Infinite(..), (...))
import qualified Data.List.Infinite as Inf
import Data.List.NonEmpty (NonEmpty(..))
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 (EisensteinInteger -> EisensteinInteger -> Bool
(EisensteinInteger -> EisensteinInteger -> Bool)
-> (EisensteinInteger -> EisensteinInteger -> Bool)
-> Eq EisensteinInteger
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
$c== :: EisensteinInteger -> EisensteinInteger -> Bool
== :: EisensteinInteger -> EisensteinInteger -> Bool
$c/= :: EisensteinInteger -> EisensteinInteger -> Bool
/= :: EisensteinInteger -> EisensteinInteger -> Bool
Eq, Eq EisensteinInteger
Eq EisensteinInteger =>
(EisensteinInteger -> EisensteinInteger -> Ordering)
-> (EisensteinInteger -> EisensteinInteger -> Bool)
-> (EisensteinInteger -> EisensteinInteger -> Bool)
-> (EisensteinInteger -> EisensteinInteger -> Bool)
-> (EisensteinInteger -> EisensteinInteger -> Bool)
-> (EisensteinInteger -> EisensteinInteger -> EisensteinInteger)
-> (EisensteinInteger -> EisensteinInteger -> EisensteinInteger)
-> Ord EisensteinInteger
EisensteinInteger -> EisensteinInteger -> Bool
EisensteinInteger -> EisensteinInteger -> Ordering
EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a.
Eq a =>
(a -> a -> Ordering)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> a)
-> (a -> a -> a)
-> Ord a
$ccompare :: EisensteinInteger -> EisensteinInteger -> Ordering
compare :: EisensteinInteger -> EisensteinInteger -> Ordering
$c< :: EisensteinInteger -> EisensteinInteger -> Bool
< :: EisensteinInteger -> EisensteinInteger -> Bool
$c<= :: EisensteinInteger -> EisensteinInteger -> Bool
<= :: EisensteinInteger -> EisensteinInteger -> Bool
$c> :: EisensteinInteger -> EisensteinInteger -> Bool
> :: EisensteinInteger -> EisensteinInteger -> Bool
$c>= :: EisensteinInteger -> EisensteinInteger -> Bool
>= :: EisensteinInteger -> EisensteinInteger -> Bool
$cmax :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
max :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
$cmin :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
min :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
Ord, (forall x. EisensteinInteger -> Rep EisensteinInteger x)
-> (forall x. Rep EisensteinInteger x -> EisensteinInteger)
-> Generic EisensteinInteger
forall x. Rep EisensteinInteger x -> EisensteinInteger
forall x. EisensteinInteger -> Rep EisensteinInteger x
forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
$cfrom :: forall x. EisensteinInteger -> Rep EisensteinInteger x
from :: forall x. EisensteinInteger -> Rep EisensteinInteger x
$cto :: forall x. Rep EisensteinInteger x -> EisensteinInteger
to :: forall x. Rep EisensteinInteger x -> EisensteinInteger
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 :: EisensteinInteger
ω = Integer
0 Integer -> Integer -> EisensteinInteger
:+ Integer
1

instance Show EisensteinInteger where
    show :: EisensteinInteger -> String
show (Integer
a :+ Integer
b)
        | Integer
b Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0     = Integer -> String
forall a. Show a => a -> String
show Integer
a
        | Integer
a Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0     = String
s String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
b'
        | Bool
otherwise  = Integer -> String
forall a. Show a => a -> String
show Integer
a String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
op String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
b'
        where
            b' :: String
b' = if Integer -> Integer
forall a. Num a => a -> a
abs Integer
b Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
1 then String
"ω" else Integer -> String
forall a. Show a => a -> String
show (Integer -> Integer
forall a. Num a => a -> a
abs Integer
b) String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"*ω"
            op :: String
op = if Integer
b Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
0 then String
"+" else String
"-"
            s :: String
s  = if Integer
b Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
0 then String
"" else String
"-"

instance Num EisensteinInteger where
    + :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
(+) (Integer
a :+ Integer
b) (Integer
c :+ Integer
d) = (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
c) Integer -> Integer -> EisensteinInteger
:+ (Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
d)
    * :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
(*) (Integer
a :+ Integer
b) (Integer
c :+ Integer
d) = (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
c Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
d) Integer -> Integer -> EisensteinInteger
:+ (Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* (Integer
c Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
d) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
d)
    abs :: EisensteinInteger -> EisensteinInteger
abs = (EisensteinInteger, EisensteinInteger) -> EisensteinInteger
forall a b. (a, b) -> a
fst ((EisensteinInteger, EisensteinInteger) -> EisensteinInteger)
-> (EisensteinInteger -> (EisensteinInteger, EisensteinInteger))
-> EisensteinInteger
-> EisensteinInteger
forall b c a. (b -> c) -> (a -> b) -> a -> c
. EisensteinInteger -> (EisensteinInteger, EisensteinInteger)
absSignum
    negate :: EisensteinInteger -> EisensteinInteger
negate (Integer
a :+ Integer
b) = (-Integer
a) Integer -> Integer -> EisensteinInteger
:+ (-Integer
b)
    fromInteger :: Integer -> EisensteinInteger
fromInteger Integer
n = Integer
n Integer -> Integer -> EisensteinInteger
:+ Integer
0
    signum :: EisensteinInteger -> EisensteinInteger
signum = (EisensteinInteger, EisensteinInteger) -> EisensteinInteger
forall a b. (a, b) -> b
snd ((EisensteinInteger, EisensteinInteger) -> EisensteinInteger)
-> (EisensteinInteger -> (EisensteinInteger, EisensteinInteger))
-> EisensteinInteger
-> EisensteinInteger
forall b c a. (b -> c) -> (a -> b) -> a -> c
. EisensteinInteger -> (EisensteinInteger, EisensteinInteger)
absSignum

instance S.Semiring EisensteinInteger where
    plus :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
plus          = EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
(+)
    times :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
times         = EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
(*)
    zero :: EisensteinInteger
zero          = Integer
0 Integer -> Integer -> EisensteinInteger
:+ Integer
0
    one :: EisensteinInteger
one           = Integer
1 Integer -> Integer -> EisensteinInteger
:+ Integer
0
    fromNatural :: Natural -> EisensteinInteger
fromNatural Natural
n = Natural -> Integer
naturalToInteger Natural
n Integer -> Integer -> EisensteinInteger
:+ Integer
0

instance S.Ring EisensteinInteger where
    negate :: EisensteinInteger -> EisensteinInteger
negate = EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a
negate

-- | Returns an @EisensteinInteger@'s sign, and its associate in the first
-- sextant.
absSignum :: EisensteinInteger -> (EisensteinInteger, EisensteinInteger)
absSignum :: EisensteinInteger -> (EisensteinInteger, EisensteinInteger)
absSignum EisensteinInteger
0 = (EisensteinInteger
0, EisensteinInteger
0)
absSignum z :: EisensteinInteger
z@(Integer
a :+ Integer
b)
  -- first sextant: 0 ≤ Arg(z) < π/3
  | Integer
a Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
b Bool -> Bool -> Bool
&& Integer
b Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
>= Integer
0 = (EisensteinInteger
z, EisensteinInteger
1)
  -- second sextant: π/3 ≤ Arg(z) < 2π/3
  | Integer
b Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
>= Integer
a Bool -> Bool -> Bool
&& Integer
a Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
0 = (Integer
b Integer -> Integer -> EisensteinInteger
:+ (Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
a), Integer
1 Integer -> Integer -> EisensteinInteger
:+ Integer
1)
  -- third sextant: 2π/3 ≤ Arg(z) < π
  | Integer
b Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
0 Bool -> Bool -> Bool
&& Integer
0 Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
>= Integer
a = ((Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
a) Integer -> Integer -> EisensteinInteger
:+ (-Integer
a), Integer
0 Integer -> Integer -> EisensteinInteger
:+ Integer
1)
  -- fourth sextant: -π ≤ Arg(z) < -2π/3
  | Integer
a Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< Integer
b Bool -> Bool -> Bool
&& Integer
b Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
<= Integer
0 = (-EisensteinInteger
z, -EisensteinInteger
1)
  -- fifth sextant: -2π/3 ≤ Arg(η) < -π/3
  | Integer
b Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
<= Integer
a Bool -> Bool -> Bool
&& Integer
a Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< Integer
0 = ((-Integer
b) Integer -> Integer -> EisensteinInteger
:+ (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
b), (-Integer
1) Integer -> Integer -> EisensteinInteger
:+ (-Integer
1))
  -- sixth sextant: -π/3 ≤ Arg(η) < 0
  | Bool
otherwise       = ((Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
b) Integer -> Integer -> EisensteinInteger
:+ Integer
a, Integer
0 Integer -> Integer -> EisensteinInteger
:+ (-Integer
1))

-- | List of all Eisenstein units, counterclockwise across all sextants,
-- starting with @1@.
ids :: [EisensteinInteger]
ids :: [EisensteinInteger]
ids = Int -> [EisensteinInteger] -> [EisensteinInteger]
forall a. Int -> [a] -> [a]
take Int
6 ((EisensteinInteger -> EisensteinInteger)
-> EisensteinInteger -> [EisensteinInteger]
forall a. (a -> a) -> a -> [a]
iterate ((EisensteinInteger
1 EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
+ EisensteinInteger
ω) *) EisensteinInteger
1)

-- | Produce a list of an @EisensteinInteger@'s associates.
associates :: EisensteinInteger -> [EisensteinInteger]
associates :: EisensteinInteger -> [EisensteinInteger]
associates EisensteinInteger
e = (EisensteinInteger -> EisensteinInteger)
-> [EisensteinInteger] -> [EisensteinInteger]
forall a b. (a -> b) -> [a] -> [b]
map (EisensteinInteger
e *) [EisensteinInteger]
ids

instance GcdDomain EisensteinInteger

instance Euclidean EisensteinInteger where
  degree :: EisensteinInteger -> Natural
degree = Integer -> Natural
forall a. Num a => Integer -> a
fromInteger (Integer -> Natural)
-> (EisensteinInteger -> Integer) -> EisensteinInteger -> Natural
forall b c a. (b -> c) -> (a -> b) -> a -> c
. EisensteinInteger -> Integer
norm
  quotRem :: EisensteinInteger
-> EisensteinInteger -> (EisensteinInteger, EisensteinInteger)
quotRem EisensteinInteger
x (Integer
d :+ Integer
0) = EisensteinInteger
-> Integer -> (EisensteinInteger, EisensteinInteger)
quotRemInt EisensteinInteger
x Integer
d
  quotRem EisensteinInteger
x EisensteinInteger
y = (EisensteinInteger
q, EisensteinInteger
x EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
- EisensteinInteger
q EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
* EisensteinInteger
y)
    where
      (EisensteinInteger
q, EisensteinInteger
_) = EisensteinInteger
-> Integer -> (EisensteinInteger, EisensteinInteger)
quotRemInt (EisensteinInteger
x EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
* EisensteinInteger -> EisensteinInteger
conjugate EisensteinInteger
y) (EisensteinInteger -> Integer
norm EisensteinInteger
y)

quotRemInt :: EisensteinInteger -> Integer -> (EisensteinInteger, EisensteinInteger)
quotRemInt :: EisensteinInteger
-> Integer -> (EisensteinInteger, EisensteinInteger)
quotRemInt EisensteinInteger
z   Integer
1  = ( EisensteinInteger
z, EisensteinInteger
0)
quotRemInt EisensteinInteger
z (-1) = (-EisensteinInteger
z, EisensteinInteger
0)
quotRemInt (Integer
a :+ Integer
b) Integer
c = (Integer
qa Integer -> Integer -> EisensteinInteger
:+ Integer
qb, (Integer
ra Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
bumpA) Integer -> Integer -> EisensteinInteger
:+ (Integer
rb Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
bumpB))
  where
    halfC :: Integer
halfC    = Integer -> Integer
forall a. Num a => a -> a
abs Integer
c Integer -> Integer -> Integer
forall a. Euclidean a => a -> a -> a
`quot` Integer
2
    bumpA :: Integer
bumpA    = Integer -> Integer
forall a. Num a => a -> a
signum Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
halfC
    bumpB :: Integer
bumpB    = Integer -> Integer
forall a. Num a => a -> a
signum Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
halfC
    (Integer
qa, Integer
ra) = (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
bumpA) Integer -> Integer -> (Integer, Integer)
forall a. Euclidean a => a -> a -> (a, a)
`quotRem` Integer
c
    (Integer
qb, Integer
rb) = (Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
bumpB) Integer -> Integer -> (Integer, Integer)
forall a. Euclidean a => a -> a -> (a, a)
`quotRem` Integer
c

-- | Conjugate a Eisenstein integer.
conjugate :: EisensteinInteger -> EisensteinInteger
conjugate :: EisensteinInteger -> EisensteinInteger
conjugate (Integer
a :+ Integer
b) = (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
b) Integer -> Integer -> EisensteinInteger
:+ (-Integer
b)

-- | The square of the magnitude of a Eisenstein integer.
norm :: EisensteinInteger -> Integer
norm :: EisensteinInteger -> Integer
norm (Integer
a :+ Integer
b) = Integer
aInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
b Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
bInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
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 :: EisensteinInteger -> Bool
isPrime EisensteinInteger
e | EisensteinInteger
e EisensteinInteger -> EisensteinInteger -> Bool
forall a. Eq a => a -> a -> Bool
== EisensteinInteger
0                     = Bool
False
          -- Special case, @1 - ω@ is the only Eisenstein prime with norm @3@,
          --  and @abs (1 - ω) = 2 + ω@.
          | Integer
a' Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
2 Bool -> Bool -> Bool
&& Integer
b' Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
1         = Bool
True
          | Integer
b' Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0 Bool -> Bool -> Bool
&& Integer
a' Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
`mod` Integer
3 Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
2 = Maybe (Prime Integer) -> Bool
forall a. Maybe a -> Bool
isJust (Maybe (Prime Integer) -> Bool) -> Maybe (Prime Integer) -> Bool
forall a b. (a -> b) -> a -> b
$ Integer -> Maybe (Prime Integer)
forall a. UniqueFactorisation a => a -> Maybe (Prime a)
U.isPrime Integer
a'
          | Integer
nE Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
`mod` Integer
3 Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
1            = Maybe (Prime Integer) -> Bool
forall a. Maybe a -> Bool
isJust (Maybe (Prime Integer) -> Bool) -> Maybe (Prime Integer) -> Bool
forall a b. (a -> b) -> a -> b
$ Integer -> Maybe (Prime Integer)
forall a. UniqueFactorisation a => a -> Maybe (Prime a)
U.isPrime Integer
nE
          | Bool
otherwise = Bool
False
  where nE :: Integer
nE       = EisensteinInteger -> Integer
norm EisensteinInteger
e
        Integer
a' :+ Integer
b' = EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a
abs EisensteinInteger
e

-- | Remove @1 - ω@ factors from an @EisensteinInteger@, and calculate that
-- prime's multiplicity in the number's factorisation.
divideByThree :: EisensteinInteger -> (Word, EisensteinInteger)
divideByThree :: EisensteinInteger -> (Word, EisensteinInteger)
divideByThree = Word -> EisensteinInteger -> (Word, EisensteinInteger)
go Word
0
  where
    go :: Word -> EisensteinInteger -> (Word, EisensteinInteger)
    go :: Word -> EisensteinInteger -> (Word, EisensteinInteger)
go !Word
n z :: EisensteinInteger
z@(Integer
a :+ Integer
b) | Integer
r1 Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0 Bool -> Bool -> Bool
&& Integer
r2 Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0 = Word -> EisensteinInteger -> (Word, EisensteinInteger)
go (Word
n Word -> Word -> Word
forall a. Num a => a -> a -> a
+ Word
1) (Integer
q1 Integer -> Integer -> EisensteinInteger
:+ Integer
q2)
                     | Bool
otherwise          = (Word
n, EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a
abs EisensteinInteger
z)
      where
        -- @(a + a - b) :+ (a + b)@ is @z * (2 :+ 1)@, and @z * (2 :+ 1)/3@
        -- is the same as @z / (1 :+ (-1))@.
        (Integer
q1, Integer
r1) = Integer -> Integer -> (Integer, Integer)
forall a. Integral a => a -> a -> (a, a)
divMod (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
b) Integer
3
        (Integer
q2, Integer
r2) = Integer -> Integer -> (Integer, Integer)
forall a. Integral a => a -> a -> (a, a)
divMod (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
b) Integer
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 :: Prime Integer -> Prime EisensteinInteger
findPrime Prime Integer
p = case (Integer
r, Integer -> Prime Integer -> [Integer]
sqrtsModPrime (Integer
9 Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
q Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
q Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
1) Prime Integer
p) of
  (Integer
1, Integer
z : [Integer]
_) -> EisensteinInteger -> Prime EisensteinInteger
forall a. a -> Prime a
Prime (EisensteinInteger -> Prime EisensteinInteger)
-> EisensteinInteger -> Prime EisensteinInteger
forall a b. (a -> b) -> a -> b
$ EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a
abs (EisensteinInteger -> EisensteinInteger)
-> EisensteinInteger -> EisensteinInteger
forall a b. (a -> b) -> a -> b
$ EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. GcdDomain a => a -> a -> a
gcd (Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p Integer -> Integer -> EisensteinInteger
:+ Integer
0) ((Integer
z Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
3 Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
q) Integer -> Integer -> EisensteinInteger
:+ Integer
1)
  (Integer, [Integer])
_ -> String -> Prime EisensteinInteger
forall a. HasCallStack => String -> a
error String
"findPrime: argument must be prime p = 6k + 1"
  where
    (Integer
q, Integer
r) = Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p Integer -> Integer -> (Integer, Integer)
forall a. Euclidean a => a -> a -> (a, a)
`quotRem` Integer
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 :: Infinite (Prime EisensteinInteger)
primes :: Infinite (Prime EisensteinInteger)
primes = Infinite EisensteinInteger -> Infinite (Prime EisensteinInteger)
forall a b. Coercible a b => a -> b
coerce (Infinite EisensteinInteger -> Infinite (Prime EisensteinInteger))
-> Infinite EisensteinInteger -> Infinite (Prime EisensteinInteger)
forall a b. (a -> b) -> a -> b
$ (Integer
2 Integer -> Integer -> EisensteinInteger
:+ Integer
1) EisensteinInteger
-> Infinite EisensteinInteger -> Infinite EisensteinInteger
forall a. a -> Infinite a -> Infinite a
:< (EisensteinInteger -> EisensteinInteger -> Ordering)
-> Infinite EisensteinInteger
-> Infinite EisensteinInteger
-> Infinite EisensteinInteger
forall a.
(a -> a -> Ordering) -> Infinite a -> Infinite a -> Infinite a
mergeBy ((EisensteinInteger -> Integer)
-> EisensteinInteger -> EisensteinInteger -> Ordering
forall a b. Ord a => (b -> a) -> b -> b -> Ordering
comparing EisensteinInteger -> Integer
norm) Infinite EisensteinInteger
l Infinite EisensteinInteger
r
  where
    leftPrimes, rightPrimes :: Infinite (Prime Integer)
    (Infinite (Prime Integer)
leftPrimes, Infinite (Prime Integer)
rightPrimes) = (Prime Integer -> Bool)
-> Infinite (Prime Integer)
-> (Infinite (Prime Integer), Infinite (Prime Integer))
forall a. (a -> Bool) -> Infinite a -> (Infinite a, Infinite a)
Inf.partition (\Prime Integer
p -> Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
`mod` Integer
3 Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
2) (Integer -> Prime Integer
forall a.
(Bits a, Integral a, UniqueFactorisation a) =>
a -> Prime a
U.nextPrime Integer
2...)

    rightPrimes' :: Infinite (Prime Integer)
    rightPrimes' :: Infinite (Prime Integer)
rightPrimes' = (Prime Integer -> Bool)
-> Infinite (Prime Integer) -> Infinite (Prime Integer)
forall a. (a -> Bool) -> Infinite a -> Infinite a
Inf.filter (\Prime Integer
prime -> Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
prime Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
`mod` Integer
3 Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
1) (Infinite (Prime Integer) -> Infinite (Prime Integer))
-> Infinite (Prime Integer) -> Infinite (Prime Integer)
forall a b. (a -> b) -> a -> b
$ Infinite (Prime Integer) -> Infinite (Prime Integer)
forall a. Infinite a -> Infinite a
Inf.tail Infinite (Prime Integer)
rightPrimes

    l :: Infinite EisensteinInteger
    l :: Infinite EisensteinInteger
l = (Prime Integer -> EisensteinInteger)
-> Infinite (Prime Integer) -> Infinite EisensteinInteger
forall a b. (a -> b) -> Infinite a -> Infinite b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (\Prime Integer
p -> Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p Integer -> Integer -> EisensteinInteger
:+ Integer
0) Infinite (Prime Integer)
leftPrimes

    r :: Infinite EisensteinInteger
    r :: Infinite EisensteinInteger
r = (Prime Integer -> NonEmpty EisensteinInteger)
-> Infinite (Prime Integer) -> Infinite EisensteinInteger
forall a b. (a -> NonEmpty b) -> Infinite a -> Infinite b
Inf.concatMap
        (\Prime Integer
p -> let Integer
x :+ Integer
y = Prime EisensteinInteger -> EisensteinInteger
forall a. Prime a -> a
unPrime (Prime Integer -> Prime EisensteinInteger
findPrime Prime Integer
p) in (Integer
x Integer -> Integer -> EisensteinInteger
:+ Integer
y) EisensteinInteger
-> [EisensteinInteger] -> NonEmpty EisensteinInteger
forall a. a -> [a] -> NonEmpty a
:| [Integer
x Integer -> Integer -> EisensteinInteger
:+ (Integer
x Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
y)])
        Infinite (Prime Integer)
rightPrimes'

-- | [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 :: EisensteinInteger -> [(Prime EisensteinInteger, Word)]
factorise EisensteinInteger
g = [[(Prime EisensteinInteger, Word)]]
-> [(Prime EisensteinInteger, Word)]
forall (t :: * -> *) a. Foldable t => t [a] -> [a]
concat ([[(Prime EisensteinInteger, Word)]]
 -> [(Prime EisensteinInteger, Word)])
-> [[(Prime EisensteinInteger, Word)]]
-> [(Prime EisensteinInteger, Word)]
forall a b. (a -> b) -> a -> b
$
              (EisensteinInteger, [[(Prime EisensteinInteger, Word)]])
-> [[(Prime EisensteinInteger, Word)]]
forall a b. (a, b) -> b
snd ((EisensteinInteger, [[(Prime EisensteinInteger, Word)]])
 -> [[(Prime EisensteinInteger, Word)]])
-> (EisensteinInteger, [[(Prime EisensteinInteger, Word)]])
-> [[(Prime EisensteinInteger, Word)]]
forall a b. (a -> b) -> a -> b
$
              (EisensteinInteger
 -> (Prime Integer, Word)
 -> (EisensteinInteger, [(Prime EisensteinInteger, Word)]))
-> EisensteinInteger
-> [(Prime Integer, Word)]
-> (EisensteinInteger, [[(Prime EisensteinInteger, Word)]])
forall (t :: * -> *) s a b.
Traversable t =>
(s -> a -> (s, b)) -> s -> t a -> (s, t b)
mapAccumL EisensteinInteger
-> (Prime Integer, Word)
-> (EisensteinInteger, [(Prime EisensteinInteger, Word)])
go (EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a
abs EisensteinInteger
g) (Integer -> [(Prime Integer, Word)]
forall a. UniqueFactorisation a => a -> [(Prime a, Word)]
U.factorise (Integer -> [(Prime Integer, Word)])
-> Integer -> [(Prime Integer, Word)]
forall a b. (a -> b) -> a -> b
$ EisensteinInteger -> Integer
norm EisensteinInteger
g)
  where
    go :: EisensteinInteger -> (Prime Integer, Word) -> (EisensteinInteger, [(Prime EisensteinInteger, Word)])
    go :: EisensteinInteger
-> (Prime Integer, Word)
-> (EisensteinInteger, [(Prime EisensteinInteger, Word)])
go EisensteinInteger
z (Prime Integer
3, Word
e)
      | Word
e Word -> Word -> Bool
forall a. Eq a => a -> a -> Bool
== Word
n    = (EisensteinInteger
q, [(EisensteinInteger -> Prime EisensteinInteger
forall a. a -> Prime a
Prime (Integer
2 Integer -> Integer -> EisensteinInteger
:+ Integer
1), Word
e)])
      | Bool
otherwise = String -> (EisensteinInteger, [(Prime EisensteinInteger, Word)])
forall a. HasCallStack => String -> a
error (String -> (EisensteinInteger, [(Prime EisensteinInteger, Word)]))
-> String -> (EisensteinInteger, [(Prime EisensteinInteger, Word)])
forall a b. (a -> b) -> a -> b
$ String
"3 is a prime factor of the norm of z = " String -> ShowS
forall a. [a] -> [a] -> [a]
++ EisensteinInteger -> String
forall a. Show a => a -> String
show EisensteinInteger
z
                          String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
" with multiplicity " String -> ShowS
forall a. [a] -> [a] -> [a]
++ Word -> String
forall a. Show a => a -> String
show Word
e
                          String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
" but (1 - ω) only divides z " String -> ShowS
forall a. [a] -> [a] -> [a]
++ Word -> String
forall a. Show a => a -> String
show Word
n String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"times."
      where
        -- Remove all @1 :+ (-1)@ (which is associated to @2 :+ 1@) factors
        -- from the argument.
        (Word
n, EisensteinInteger
q) = EisensteinInteger -> (Word, EisensteinInteger)
divideByThree EisensteinInteger
z
    go EisensteinInteger
z (Prime Integer
p, Word
e)
      | Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
`mod` Integer
3 Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
2
      = let e' :: Word
e' = Word
e Word -> Word -> Word
forall a. Euclidean a => a -> a -> a
`quot` Word
2 in (EisensteinInteger
z EisensteinInteger -> Integer -> EisensteinInteger
`quotI` (Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p Integer -> Word -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^ Word
e'), [(EisensteinInteger -> Prime EisensteinInteger
forall a. a -> Prime a
Prime (Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p Integer -> Integer -> EisensteinInteger
:+ Integer
0), Word
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@.
      | Bool
otherwise   = (EisensteinInteger
z', ((Prime EisensteinInteger, Word) -> Bool)
-> [(Prime EisensteinInteger, Word)]
-> [(Prime EisensteinInteger, Word)]
forall a. (a -> Bool) -> [a] -> [a]
filter ((Word -> Word -> Bool
forall a. Ord a => a -> a -> Bool
> Word
0) (Word -> Bool)
-> ((Prime EisensteinInteger, Word) -> Word)
-> (Prime EisensteinInteger, Word)
-> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Prime EisensteinInteger, Word) -> Word
forall a b. (a, b) -> b
snd) [(Prime EisensteinInteger
gp, Word
k), (Prime EisensteinInteger
gp', Word
k')])
      where
        gp :: Prime EisensteinInteger
gp = Prime Integer -> Prime EisensteinInteger
findPrime Prime Integer
p
        Integer
x :+ Integer
y = Prime EisensteinInteger -> EisensteinInteger
forall a. Prime a -> a
unPrime Prime EisensteinInteger
gp
        -- @gp'@ is @gp@'s conjugate.
        gp' :: Prime EisensteinInteger
gp' = EisensteinInteger -> Prime EisensteinInteger
forall a. a -> Prime a
Prime (Integer
x Integer -> Integer -> EisensteinInteger
:+ (Integer
x Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
y))
        (Word
k, Word
k', EisensteinInteger
z') = Prime EisensteinInteger
-> Prime EisensteinInteger
-> Integer
-> Word
-> EisensteinInteger
-> (Word, Word, EisensteinInteger)
divideByPrime Prime EisensteinInteger
gp Prime EisensteinInteger
gp' (Prime Integer -> Integer
forall a. Prime a -> a
unPrime Prime Integer
p) Word
e EisensteinInteger
z

        quotI :: EisensteinInteger -> Integer -> EisensteinInteger
quotI (Integer
a :+ Integer
b) Integer
n = Integer
a Integer -> Integer -> Integer
forall a. Euclidean a => a -> a -> a
`quot` Integer
n Integer -> Integer -> EisensteinInteger
:+ Integer
b Integer -> Integer -> Integer
forall a. Euclidean a => a -> a -> a
`quot` Integer
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 :: Prime EisensteinInteger
-> Prime EisensteinInteger
-> Integer
-> Word
-> EisensteinInteger
-> (Word, Word, EisensteinInteger)
divideByPrime Prime EisensteinInteger
p Prime EisensteinInteger
p' Integer
np Word
k = Word
-> Word -> EisensteinInteger -> (Word, Word, EisensteinInteger)
go Word
k Word
0
    where
        go :: Word -> Word -> EisensteinInteger -> (Word, Word, EisensteinInteger)
        go :: Word
-> Word -> EisensteinInteger -> (Word, Word, EisensteinInteger)
go Word
0 Word
d EisensteinInteger
z = (Word
d, Word
d, EisensteinInteger
z)
        go Word
c Word
d EisensteinInteger
z | Word
c Word -> Word -> Bool
forall a. Ord a => a -> a -> Bool
>= Word
2, Just EisensteinInteger
z' <- EisensteinInteger
z EisensteinInteger -> Integer -> Maybe EisensteinInteger
`quotEvenI` Integer
np = Word
-> Word -> EisensteinInteger -> (Word, Word, EisensteinInteger)
go (Word
c Word -> Word -> Word
forall a. Num a => a -> a -> a
- Word
2) (Word
d Word -> Word -> Word
forall a. Num a => a -> a -> a
+ Word
1) EisensteinInteger
z'
        go Word
c Word
d EisensteinInteger
z = (Word
d Word -> Word -> Word
forall a. Num a => a -> a -> a
+ Word
d1, Word
d Word -> Word -> Word
forall a. Num a => a -> a -> a
+ Word
d2, EisensteinInteger
z'')
            where
                (Word
d1, EisensteinInteger
z') = Word -> Word -> EisensteinInteger -> (Word, EisensteinInteger)
go1 Word
c Word
0 EisensteinInteger
z
                d2 :: Word
d2 = Word
c Word -> Word -> Word
forall a. Num a => a -> a -> a
- Word
d1
                z'' :: EisensteinInteger
z'' = (EisensteinInteger -> EisensteinInteger)
-> EisensteinInteger -> [EisensteinInteger]
forall a. (a -> a) -> a -> [a]
iterate (\EisensteinInteger
g -> EisensteinInteger -> Maybe EisensteinInteger -> EisensteinInteger
forall a. a -> Maybe a -> a
fromMaybe EisensteinInteger
err (Maybe EisensteinInteger -> EisensteinInteger)
-> Maybe EisensteinInteger -> EisensteinInteger
forall a b. (a -> b) -> a -> b
$ (EisensteinInteger
g EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
* Prime EisensteinInteger -> EisensteinInteger
forall a. Prime a -> a
unPrime Prime EisensteinInteger
p) EisensteinInteger -> Integer -> Maybe EisensteinInteger
`quotEvenI` Integer
np) EisensteinInteger
z' [EisensteinInteger] -> Int -> EisensteinInteger
forall a. HasCallStack => [a] -> Int -> a
!! Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 (Word -> Int
wordToInt Word
d2)

        go1 :: Word -> Word -> EisensteinInteger -> (Word, EisensteinInteger)
        go1 :: Word -> Word -> EisensteinInteger -> (Word, EisensteinInteger)
go1 Word
0 Word
d EisensteinInteger
z = (Word
d, EisensteinInteger
z)
        go1 Word
c Word
d EisensteinInteger
z
            | Just EisensteinInteger
z' <- (EisensteinInteger
z EisensteinInteger -> EisensteinInteger -> EisensteinInteger
forall a. Num a => a -> a -> a
* Prime EisensteinInteger -> EisensteinInteger
forall a. Prime a -> a
unPrime Prime EisensteinInteger
p') EisensteinInteger -> Integer -> Maybe EisensteinInteger
`quotEvenI` Integer
np
            = Word -> Word -> EisensteinInteger -> (Word, EisensteinInteger)
go1 (Word
c Word -> Word -> Word
forall a. Num a => a -> a -> a
- Word
1) (Word
d Word -> Word -> Word
forall a. Num a => a -> a -> a
+ Word
1) EisensteinInteger
z'
            | Bool
otherwise
            = (Word
d, EisensteinInteger
z)

        err :: EisensteinInteger
err = String -> EisensteinInteger
forall a. HasCallStack => String -> a
error (String -> EisensteinInteger) -> String -> EisensteinInteger
forall a b. (a -> b) -> a -> b
$ String
"divideByPrime: malformed arguments" String -> ShowS
forall a. [a] -> [a] -> [a]
++ (Prime EisensteinInteger, Integer, Word) -> String
forall a. Show a => a -> String
show (Prime EisensteinInteger
p, Integer
np, Word
k)

-- | Divide an Eisenstein integer by an even integer.
quotEvenI :: EisensteinInteger -> Integer -> Maybe EisensteinInteger
quotEvenI :: EisensteinInteger -> Integer -> Maybe EisensteinInteger
quotEvenI (Integer
x :+ Integer
y) Integer
n
    | Integer
xr Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0 , Integer
yr Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0 = EisensteinInteger -> Maybe EisensteinInteger
forall a. a -> Maybe a
Just (Integer
xq Integer -> Integer -> EisensteinInteger
:+ Integer
yq)
    | Bool
otherwise         = Maybe EisensteinInteger
forall a. Maybe a
Nothing
  where
    (Integer
xq, Integer
xr) = Integer
x Integer -> Integer -> (Integer, Integer)
forall a. Euclidean a => a -> a -> (a, a)
`quotRem` Integer
n
    (Integer
yq, Integer
yr) = Integer
y Integer -> Integer -> (Integer, Integer)
forall a. Euclidean a => a -> a -> (a, a)
`quotRem` Integer
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 :: EisensteinInteger -> [(Prime EisensteinInteger, Word)]
factorise EisensteinInteger
0 = []
  factorise EisensteinInteger
e = [(Prime EisensteinInteger, Word)]
-> [(Prime EisensteinInteger, Word)]
forall a b. Coercible a b => a -> b
coerce ([(Prime EisensteinInteger, Word)]
 -> [(Prime EisensteinInteger, Word)])
-> [(Prime EisensteinInteger, Word)]
-> [(Prime EisensteinInteger, Word)]
forall a b. (a -> b) -> a -> b
$ EisensteinInteger -> [(Prime EisensteinInteger, Word)]
factorise EisensteinInteger
e

  isPrime :: EisensteinInteger -> Maybe (Prime EisensteinInteger)
isPrime EisensteinInteger
e = if EisensteinInteger -> Bool
isPrime EisensteinInteger
e then Prime EisensteinInteger -> Maybe (Prime EisensteinInteger)
forall a. a -> Maybe a
Just (EisensteinInteger -> Prime EisensteinInteger
forall a. a -> Prime a
Prime EisensteinInteger
e) else Maybe (Prime EisensteinInteger)
forall a. Maybe a
Nothing