Copyright	(c) 2016 Andrew Lelechenko
License	MIT
Maintainer	Andrew Lelechenko <andrew.lelechenko@gmail.com>
Stability	Provisional
Portability	Non-portable (GHC extensions)
Safe Haskell	None
Language	Haskell2010

Math.NumberTheory.UniqueFactorisation

Description

An abstract type class for unique factorisation domains.

Synopsis

type family Prime (f :: *) :: *
class UniqueFactorisation a where

Documentation

type family Prime (f :: *) :: * Source #

Type of primes of a given unique factorisation domain.

abs (unPrime n) == unPrime n must hold for all n of type Prime t

Instances

type Prime Int Source #
Instance details Defined in Math.NumberTheory.Primes.Types type Prime Int
type Prime Integer Source #
Instance details Defined in Math.NumberTheory.Primes.Types type Prime Integer
type Prime Natural Source #
Instance details Defined in Math.NumberTheory.Primes.Types type Prime Natural
type Prime Word Source #
Instance details Defined in Math.NumberTheory.Primes.Types type Prime Word
type Prime GaussianInteger Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation type Prime GaussianInteger
type Prime EisensteinInteger Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation type Prime EisensteinInteger
type Prime (Prefactored a) Source #
Instance details Defined in Math.NumberTheory.Prefactored type Prime (Prefactored a) = Prime a

class UniqueFactorisation a where Source #

The following invariant must hold for n /= 0:

abs n == abs (product (map (\(p, k) -> unPrime p ^ k) (factorise n)))

The result of factorise should not contain zero powers and should not change after multiplication of the argument by domain's unit.

Minimal complete definition

unPrime, factorise

Methods

unPrime :: Prime a -> a Source #

factorise :: a -> [(Prime a, Word)] Source #

isPrime :: a -> Maybe (Prime a) Source #

isPrime :: (Eq a, Num a) => a -> Maybe (Prime a) Source #

Instances

UniqueFactorisation Int Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation Methods unPrime :: Prime Int -> Int Source # factorise :: Int -> [(Prime Int, Word)] Source # isPrime :: Int -> Maybe (Prime Int) Source #
UniqueFactorisation Integer Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation Methods unPrime :: Prime Integer -> Integer Source # factorise :: Integer -> [(Prime Integer, Word)] Source # isPrime :: Integer -> Maybe (Prime Integer) Source #
UniqueFactorisation Natural Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation Methods unPrime :: Prime Natural -> Natural Source # factorise :: Natural -> [(Prime Natural, Word)] Source # isPrime :: Natural -> Maybe (Prime Natural) Source #
UniqueFactorisation Word Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation Methods unPrime :: Prime Word -> Word Source # factorise :: Word -> [(Prime Word, Word)] Source # isPrime :: Word -> Maybe (Prime Word) Source #
UniqueFactorisation GaussianInteger Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation Methods unPrime :: Prime GaussianInteger -> GaussianInteger Source # factorise :: GaussianInteger -> [(Prime GaussianInteger, Word)] Source # isPrime :: GaussianInteger -> Maybe (Prime GaussianInteger) Source #
UniqueFactorisation EisensteinInteger Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation Methods unPrime :: Prime EisensteinInteger -> EisensteinInteger Source # factorise :: EisensteinInteger -> [(Prime EisensteinInteger, Word)] Source # isPrime :: EisensteinInteger -> Maybe (Prime EisensteinInteger) Source #
(Eq a, Num a, UniqueFactorisation a) => UniqueFactorisation (Prefactored a) Source #
Instance details Defined in Math.NumberTheory.Prefactored Methods unPrime :: Prime (Prefactored a) -> Prefactored a Source # factorise :: Prefactored a -> [(Prime (Prefactored a), Word)] Source # isPrime :: Prefactored a -> Maybe (Prime (Prefactored a)) Source #