arithmoi-0.11.0.0: 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 :: Type -> Type # Methods Source # Instance details Methodsrnf :: GaussianInteger -> () # Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details type Rep GaussianInteger = D1 (MetaData "GaussianInteger" "Math.NumberTheory.Quadratic.GaussianIntegers" "arithmoi-0.11.0.0-GC1gQiNKoSR4YxT4GQqfcr" False) (C1 (MetaCons ":+" PrefixI True) (S1 (MetaSel (Just "real") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Integer) :*: S1 (MetaSel (Just "imag") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Integer)))

The imaginary unit, where

ι .^ 2 == -1

Conjugate a Gaussian integer.

The square of the magnitude of a Gaussian integer.

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]


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

>>> findPrime (nextPrime 5)
Prime 2+ι