Copyright	(c) 2016 Chris Fredrickson Google Inc.
License	MIT
Maintainer	Chris Fredrickson <chris.p.fredrickson@gmail.com>
Stability	Provisional
Portability	Non-portable (GHC extensions)
Safe Haskell	None
Language	Haskell2010

Math.NumberTheory.Quadratic.GaussianIntegers

Description

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

Synopsis

data GaussianInteger = (:+) {
- real :: !Integer
- imag :: !Integer
}
ι :: GaussianInteger
conjugate :: GaussianInteger -> GaussianInteger
norm :: GaussianInteger -> Integer
(.^) :: Integral a => GaussianInteger -> a -> GaussianInteger
isPrime :: GaussianInteger -> Bool
primes :: [GaussianInteger]
gcdG :: GaussianInteger -> GaussianInteger -> GaussianInteger
gcdG' :: GaussianInteger -> GaussianInteger -> GaussianInteger
findPrime :: Integer -> GaussianInteger
findPrime' :: Integer -> GaussianInteger
factorise :: GaussianInteger -> [(GaussianInteger, Int)]

Documentation

data GaussianInteger Source #

A Gaussian integer is a+bi, where a and b are both integers.

Constructors

(:+) infix 6
Fields real :: !Integer imag :: !Integer

Instances

Eq GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers Methods (==) :: GaussianInteger -> GaussianInteger -> Bool # (/=) :: GaussianInteger -> GaussianInteger -> Bool #
Num GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers Methods (+) :: GaussianInteger -> GaussianInteger -> GaussianInteger # (-) :: GaussianInteger -> GaussianInteger -> GaussianInteger # (*) :: GaussianInteger -> GaussianInteger -> GaussianInteger # negate :: GaussianInteger -> GaussianInteger # abs :: GaussianInteger -> GaussianInteger # signum :: GaussianInteger -> GaussianInteger # fromInteger :: Integer -> GaussianInteger #
Ord GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers Methods compare :: GaussianInteger -> GaussianInteger -> Ordering # (<) :: GaussianInteger -> GaussianInteger -> Bool # (<=) :: GaussianInteger -> GaussianInteger -> Bool # (>) :: GaussianInteger -> GaussianInteger -> Bool # (>=) :: GaussianInteger -> GaussianInteger -> Bool # max :: GaussianInteger -> GaussianInteger -> GaussianInteger # min :: GaussianInteger -> GaussianInteger -> GaussianInteger #
Show GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers Methods showsPrec :: Int -> GaussianInteger -> ShowS # show :: GaussianInteger -> String # showList :: [GaussianInteger] -> ShowS #
Generic GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers Associated Types type Rep GaussianInteger :: * -> * # Methods from :: GaussianInteger -> Rep GaussianInteger x # to :: Rep GaussianInteger x -> GaussianInteger #
NFData GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers Methods rnf :: GaussianInteger -> () #
Euclidean GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers Methods quotRem :: GaussianInteger -> GaussianInteger -> (GaussianInteger, GaussianInteger) Source # divMod :: GaussianInteger -> GaussianInteger -> (GaussianInteger, GaussianInteger) Source # quot :: GaussianInteger -> GaussianInteger -> GaussianInteger Source # rem :: GaussianInteger -> GaussianInteger -> GaussianInteger Source # div :: GaussianInteger -> GaussianInteger -> GaussianInteger Source # mod :: GaussianInteger -> GaussianInteger -> GaussianInteger Source # gcd :: GaussianInteger -> GaussianInteger -> GaussianInteger Source # lcm :: GaussianInteger -> GaussianInteger -> GaussianInteger Source # coprime :: GaussianInteger -> GaussianInteger -> Bool Source # extendedGCD :: GaussianInteger -> GaussianInteger -> (GaussianInteger, GaussianInteger, GaussianInteger) 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 #
type Rep GaussianInteger Source #
Instance details Defined in Math.NumberTheory.Quadratic.GaussianIntegers type Rep GaussianInteger = D1 (MetaData "GaussianInteger" "Math.NumberTheory.Quadratic.GaussianIntegers" "arithmoi-0.8.0.0-6Rtnbx2jJER74A6C7rjrHd" False) (C1 (MetaCons ":+" PrefixI True) (S1 (MetaSel (Just "real") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Integer) :*: S1 (MetaSel (Just "imag") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Integer)))
type Prime GaussianInteger Source #
Instance details Defined in Math.NumberTheory.UniqueFactorisation type Prime GaussianInteger

ι :: GaussianInteger Source #

The imaginary unit, where

ι .^ 2 == -1

conjugate :: GaussianInteger -> GaussianInteger Source #

Conjugate a Gaussian integer.

norm :: GaussianInteger -> Integer Source #

The square of the magnitude of a Gaussian integer.

(.^) :: Integral a => GaussianInteger -> a -> GaussianInteger infixr 8 Source #

Deprecated: Use (^) instead.

Raise a Gaussian integer to a given power.

isPrime :: GaussianInteger -> Bool Source #

Compute whether a given Gaussian integer is prime.

primes :: [GaussianInteger] Source #

An infinite list of the Gaussian primes. Uses primes in Z to exhaustively generate all Gaussian primes (up to associates), in order of ascending magnitude.

gcdG :: GaussianInteger -> GaussianInteger -> GaussianInteger Source #

Deprecated: Use gcd instead.

Compute the GCD of two Gaussian integers. Result is always in the first quadrant.

gcdG' :: GaussianInteger -> GaussianInteger -> GaussianInteger Source #

Deprecated: Use gcd instead.

findPrime :: Integer -> GaussianInteger Source #

Find a Gaussian integer whose norm is the given prime number of form 4k + 1 using Hermite-Serret algorithm.

findPrime' :: Integer -> GaussianInteger Source #

Deprecated: Use findPrime instead.

factorise :: GaussianInteger -> [(GaussianInteger, Int)] Source #

Compute the prime factorisation of a Gaussian integer. This is unique up to units (+- 1, +- i). Unit factors are not included in the result.