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
- unPrime :: Prime a -> a
- factorise :: a -> [(Prime a, Word)]
- isPrime :: a -> Maybe (Prime a)

Documentation

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

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

class UniqueFactorisation a where #

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 #

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

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

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

Instances

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