Math.NumberTheory.Quadratic.EisensteinIntegers

This module exports functions for manipulating Eisenstein integers, including computing their prime factorisations.

Documentation

data EisensteinInteger

An Eisenstein integer is a + bω, where a and b are both integers.

Constructors

!Integer :+ !Integer infix 6

Instances

Eq EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

(==) :: EisensteinInteger -> EisensteinInteger -> Bool

(/=) :: EisensteinInteger -> EisensteinInteger -> Bool

Num EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

(+) :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

(-) :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

(*) :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

negate :: EisensteinInteger -> EisensteinInteger

abs :: EisensteinInteger -> EisensteinInteger

signum :: EisensteinInteger -> EisensteinInteger

fromInteger :: Integer -> EisensteinInteger

Ord EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

compare :: EisensteinInteger -> EisensteinInteger -> Ordering

(<) :: EisensteinInteger -> EisensteinInteger -> Bool

(<=) :: EisensteinInteger -> EisensteinInteger -> Bool

(>) :: EisensteinInteger -> EisensteinInteger -> Bool

(>=) :: EisensteinInteger -> EisensteinInteger -> Bool

max :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

min :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

Show EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

showsPrec :: Int -> EisensteinInteger -> ShowS

show :: EisensteinInteger -> String

showList :: [EisensteinInteger] -> ShowS

Generic EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Associated Types

type Rep EisensteinInteger :: Type -> Type

Methods

from :: EisensteinInteger -> Rep EisensteinInteger x

to :: Rep EisensteinInteger x -> EisensteinInteger

NFData EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

rnf :: EisensteinInteger -> ()

GcdDomain EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

divide :: EisensteinInteger -> EisensteinInteger -> Maybe EisensteinInteger

gcd :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

lcm :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

coprime :: EisensteinInteger -> EisensteinInteger -> Bool

Euclidean EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

quotRem :: EisensteinInteger -> EisensteinInteger -> (EisensteinInteger, EisensteinInteger)

quot :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

rem :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

degree :: EisensteinInteger -> Natural

Semiring EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

plus :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

zero :: EisensteinInteger

times :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger

one :: EisensteinInteger

fromNatural :: Natural -> EisensteinInteger

Ring EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

negate :: EisensteinInteger -> EisensteinInteger

UniqueFactorisation EisensteinInteger

See the source code and Haddock comments for the factorise and isPrime functions in this module (they are not exported) for implementation details.

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

Methods

factorise :: EisensteinInteger -> [(Prime EisensteinInteger, Word)]

isPrime :: EisensteinInteger -> Maybe (Prime EisensteinInteger)

type Rep EisensteinInteger

Defined in Math.NumberTheory.Quadratic.EisensteinIntegers

type Rep EisensteinInteger = D1 ('MetaData "EisensteinInteger" "Math.NumberTheory.Quadratic.EisensteinIntegers" "arithmoi-0.12.0.2-1LBW47IaiR6Hp9NDnAlXcl" 'False) (C1 ('MetaCons ":+" ('InfixI 'NotAssociative 6) 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Integer) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Integer)))

ω :: 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.

conjugate :: EisensteinInteger -> EisensteinInteger

Conjugate a Eisenstein integer.

norm :: EisensteinInteger -> Integer

The square of the magnitude of a Eisenstein integer.

associates :: EisensteinInteger -> [EisensteinInteger]

Produce a list of an EisensteinInteger's associates.

ids :: [EisensteinInteger]

List of all Eisenstein units, counterclockwise across all sextants, starting with 1.

Primality functions

findPrime :: Prime Integer -> Prime EisensteinInteger

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

primes :: [Prime EisensteinInteger]

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]