# 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

## 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

`Prime D2`

`Prime D3`

`Prime D5`

`Prime D7`

`Prime (NumCons D1 D1)`

`Prime (NumCons D1 D3)`

`Prime (NumCons D1 D7)`

`Prime (NumCons D1 D9)`

`Prime (NumCons D2 D3)`

`Prime (NumCons D2 D9)`

`Prime (NumCons D3 D1)`

`Prime (NumCons D3 D7)`

`Prime (NumCons D4 D1)`

`Prime (NumCons D4 D3)`

`Prime (NumCons D4 D7)`

`Prime (NumCons D5 D3)`

`Prime (NumCons D5 D9)`

`Prime (NumCons D6 D1)`

`Prime (NumCons D6 D7)`

`Prime (NumCons D7 D1)`

`Prime (NumCons D7 D3)`

`Prime (NumCons D7 D9)`

`Prime (NumCons D8 D3)`

`Prime (NumCons D8 D9)`

`Prime (NumCons D9 D7)`

### #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)
```