Module

Data.ModularArithmetic

Package: purescript-modular-arithmetic
Repository: hdgarrood/purescript-modular-arithmetic

#Z Source

newtype Z m

Integers modulo some positive integer m.

The type argument should be a positive integer of the kind defined by purescript-typelevel. This way, the modulus that you're working with is specified in the type. Note that even though the modulus is captured at the type level, you can still use modulus values which are not known at compile time, with the reifyIntP function.

This type forms a commutative ring for any positive integer m, and additionally a field when m is prime. Unlike Int and Number, though, all of these instances are fully law-abiding.

The runtime representation is identical to that of Int, except that values are guaranteed to be between 0 and m-1.

Instances

Eq (Z m)
Ord (Z m)
Show (Z m)
(Pos m) => Bounded (Z m)
(Pos m) => Enum (Z m)
(Pos m) => BoundedEnum (Z m)
(Pos m) => Semiring (Z m)
(Pos m) => Ring (Z m)
(Pos m) => CommutativeRing (Z m)
(Prime m) => DivisionRing (Z m)
(Prime m) => EuclideanRing (Z m)

#mkZ Source

mkZ :: forall m. Pos m => Int -> Z m

Smart constructor for Z values.

#runZ Source

runZ :: forall m. Z m -> Int

Get at the underlying Int.

#modulus Source

modulus :: forall m. Pos m => Z m -> Int

Convenience function for accessing m at the value level.

#inverse Source

inverse :: forall m. Pos m => Z m -> Maybe (Z m)

Compute a multiplicative inverse of some nonzero number in Z_m. Note that an inverse exists if and only if the input and m are coprime. If this is not the case, this function returns Nothing.

#enumerate Source

enumerate :: forall m. Pos m => NonEmpty Array (Z m)

List all members of Z_m.

#genZ Source

genZ :: forall m gen. MonadGen gen => Pos m => gen (Z m)

A random generator for elements of Z_m; selects any value of Z_m with each value being equally likely to be selected.

Re-exports from Data.ModularArithmetic.Primality

#Prime Source

class (Pos m) <= Prime m

This class specifies that a type-level integer is prime; that is, it has exactly 2 divisors: itself, and 1.

All primes up to 100 have instances. For larger primes, you will need to use the reifyPrime function.

Instances

#reifyPrime Source

reifyPrime :: forall r. Int -> (forall p. Prime p => p -> r) -> Maybe r

Reify a prime number at the type level. If the first argument provided is not prime, this function returns Nothing.

#primeFactors Source

primeFactors :: Int -> List Int

Find prime factors by trial division; first attempting to divide by 2 and then by every odd number after that. This is the most basic prime factorisation algorithm possible but it is more than enough for this case (specifically, when the input is guaranteed to be no more than 2^31).

Prime factors are returned in increasing order. For example:

> primeFactors 12
(2 : 2 : 3 : Nil)

Passing in any number less than 2 will return an empty list.

> primeFactors (-12)
Nil

For all positive integers, the following properties are satisfied:

> product (primeFactors n) == n
> all isPrime (primeFactors n)

#isPrime Source

isPrime :: Int -> Boolean

Check if a number is prime. Note that 1 is not a prime number.

Modules: Data.ModularArithmetic; Data.ModularArithmetic.Primality