Math.NumberTheory.Prefactored

Type for numbers, accompanied by their factorisation.

Documentation

data Prefactored a

A container for a number and its pairwise coprime (but not neccessarily prime) factorisation. It is designed to preserve information about factors under multiplication. One can use this representation to speed up prime factorisation and computation of arithmetic functions.

For instance, let p and q be big primes:

> let p, q :: Integer
>     p = 1000000000000000000000000000057
>     q = 2000000000000000000000000000071

It would be difficult to compute prime factorisation of their product as is: factorise would take ages. Things become different if we simply change types of p and q to prefactored ones:

> let p, q :: Prefactored Integer
>     p = 1000000000000000000000000000057
>     q = 2000000000000000000000000000071

Now prime factorisation is done instantly:

> factorise (p * q)
[(PrimeNat 1000000000000000000000000000057, 1), (PrimeNat 2000000000000000000000000000071, 1)]
> factorise (p^2 * q^3)
[(PrimeNat 1000000000000000000000000000057, 2), (PrimeNat 2000000000000000000000000000071, 3)]

Moreover, we can instantly compute totient and its iterations. It works fine, because output of totient is also prefactored.

> prefValue $ totient (p^2 * q^3)
8000000000000000000000000001752000000000000000000000000151322000000000000000000000006445392000000000000000000000135513014000000000000000000001126361040
> prefValue $ totient $ totient (p^2 * q^3)
2133305798262843681544648472180210822742702690942899511234131900112583590230336435053688694839034890779375223070157301188739881477320529552945446912000

Let us look under the hood:

> prefFactors $ totient (p^2 * q^3)
[(2, 4), (41666666666666666666666666669, 1), (3, 3), (111111111111111111111111111115, 1),
(1000000000000000000000000000057, 1), (2000000000000000000000000000071, 2)]
> prefFactors $ totient $ totient (p^2 * q^3)
[(2, 22), (39521, 1), (5, 3), (199937, 1), (3, 8), (6046667, 1), (227098769, 1),
(85331809838489, 1), (361696272343, 1), (22222222222222222222222222223, 1),
(41666666666666666666666666669, 1), (2000000000000000000000000000071, 1)]

Pairwise coprimality of factors is crucial, because it allows us to process them independently, possibly even in parallel or concurrent fashion.

Following invariant is guaranteed to hold:

abs (prefValue x) = abs (product (map (uncurry (^)) (prefFactors x)))

Instances

(Integral a, UniqueFactorisation a) => Num (Prefactored a)

Methods

(+) :: Prefactored a -> Prefactored a -> Prefactored a

(-) :: Prefactored a -> Prefactored a -> Prefactored a

(*) :: Prefactored a -> Prefactored a -> Prefactored a

negate :: Prefactored a -> Prefactored a

abs :: Prefactored a -> Prefactored a

signum :: Prefactored a -> Prefactored a

fromInteger :: Integer -> Prefactored a

Show a => Show (Prefactored a)

Methods

showsPrec :: Int -> Prefactored a -> ShowS

show :: Prefactored a -> String

showList :: [Prefactored a] -> ShowS

UniqueFactorisation a => UniqueFactorisation (Prefactored a)

Methods

unPrime :: Prime (Prefactored a) -> Prefactored a

factorise :: Prefactored a -> [(Prime (Prefactored a), Word)]

isPrime :: Prefactored a -> Maybe (Prime (Prefactored a))

type Prime (Prefactored a)

type Prime (Prefactored a) = Prime a

fromValue :: a -> Prefactored a

Create Prefactored from a given number.

> fromValue 123
Prefactored {prefValue = 123, prefFactors = [(123, 1)]}

fromFactors :: Integral a => [(a, Word)] -> Prefactored a

Create Prefactored from a given list of pairwise coprime (but not neccesarily prime) factors with multiplicities. If you cannot ensure coprimality, use splitIntoCoprimes.

> fromFactors (splitIntoCoprimes [(140, 1), (165, 1)])
Prefactored {prefValue = 23100, prefFactors = [(5, 2), (28, 1), (33, 1)]}
> fromFactors (splitIntoCoprimes [(140, 2), (165, 3)])
Prefactored {prefValue = 88045650000, prefFactors = [(5, 5), (28, 2), (33, 3)]}