arithmoi-0.13.0.0: Efficient basic number-theoretic functions.
Copyright(c) 2016 Chris Fredrickson Google Inc.
LicenseMIT
MaintainerChris Fredrickson <chris.p.fredrickson@gmail.com>
Safe HaskellSafe-Inferred
LanguageHaskell2010

Math.NumberTheory.Quadratic.GaussianIntegers

Description

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

Synopsis

Documentation

data GaussianInteger Source #

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

Constructors

(:+) infix 6 

Fields

Instances

Instances details
UniqueFactorisation GaussianInteger Source # 
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Defined in Math.NumberTheory.Quadratic.GaussianIntegers

Generic GaussianInteger Source # 
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Associated Types

type Rep GaussianInteger :: Type -> Type #

Num GaussianInteger Source # 
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Show GaussianInteger Source # 
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NFData GaussianInteger Source # 
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Methods

rnf :: GaussianInteger -> () #

Eq GaussianInteger Source # 
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Ord GaussianInteger Source # 
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Euclidean GaussianInteger Source # 
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GcdDomain GaussianInteger Source # 
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Ring GaussianInteger Source # 
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Semiring GaussianInteger Source # 
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type Rep GaussianInteger Source # 
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Defined in Math.NumberTheory.Quadratic.GaussianIntegers

type Rep GaussianInteger = D1 ('MetaData "GaussianInteger" "Math.NumberTheory.Quadratic.GaussianIntegers" "arithmoi-0.13.0.0-LV4QEqFnDk8FOh3Yrp60Gv" 'False) (C1 ('MetaCons ":+" 'PrefixI 'True) (S1 ('MetaSel ('Just "real") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Integer) :*: S1 ('MetaSel ('Just "imag") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Integer)))

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.

primes :: Infinite (Prime 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.

>>> take 10 primes
[Prime 1+ι,Prime 2+ι,Prime 1+2*ι,Prime 3,Prime 3+2*ι,Prime 2+3*ι,Prime 4+ι,Prime 1+4*ι,Prime 5+2*ι,Prime 2+5*ι]

findPrime :: Prime Integer -> Prime GaussianInteger Source #

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

>>> import Math.NumberTheory.Primes (nextPrime)
>>> findPrime (nextPrime 5)
Prime 2+ι