arithmoi-0.8.0.0: Efficient basic number-theoretic functions.

Description

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

Synopsis

# Documentation

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

Constructors

 (:+) infix 6 Fieldsreal :: !Integer imag :: !Integer
Instances
 Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details MethodsshowList :: [GaussianInteger] -> ShowS # Source # Instance details Associated Typestype Rep GaussianInteger :: * -> * # Methods Source # Instance details Methodsrnf :: GaussianInteger -> () # Source # Instance details Methods Source # Instance detailsDefined in Math.NumberTheory.UniqueFactorisation Methods Source # Instance details 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))) Source # Instance detailsDefined in Math.NumberTheory.UniqueFactorisation

The imaginary unit, where

ι .^ 2 == -1

Conjugate a Gaussian integer.

The square of the magnitude of a Gaussian integer.

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

Raise a Gaussian integer to a given power.

Compute whether a given Gaussian integer is prime.

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.

Deprecated: Use gcd instead.

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

Deprecated: Use gcd instead.

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

Deprecated: Use findPrime instead.

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