The inverse for totient function.

The return value is parameterized by a Semiring , which allows various applications by providing different (multiplicative) embeddings. E. g., list all preimages (see a helper asSetOfPreimages ):

>>> import qualified Data.Set as S
>>> import Data.Semigroup
>>> S.mapMonotonic getProduct (inverseTotient (S.singleton . Product) 120)
fromList [143,155,175,183,225,231,244,248,286,308,310,350,366,372,396,450,462]

Count preimages:

>>> inverseTotient (const 1) 120
17

Sum preimages:

>>> inverseTotient id 120
4904

Find minimal and maximal preimages:

>>> unMinWord (inverseTotient MinWord 120)
143
>>> unMaxWord (inverseTotient MaxWord 120)
462

The inverse for jordan function.

Generalizes the inverseTotient function, which is inverseJordan 1.

The return value is parameterized by a Semiring , which allows various applications by providing different (multiplicative) embeddings. E. g., list all preimages (see a helper asSetOfPreimages ):

>>> import qualified Data.Set as S
>>> import Data.Semigroup
>>> S.mapMonotonic getProduct (inverseJordan 2 (S.singleton . Product) 192)
fromList [15,16]

Similarly to inverseTotient , it is possible to count and sum preimages, or get the maximum/minimum preimage.

Note: it is the user's responsibility to use an appropriate type for inverseSigmaK . Even low values of k with Int or Word will return invalid results due to over/underflow, or throw exceptions (i.e. division by zero).

>>> asSetOfPreimages (inverseJordan 15) (jordan 15 19) :: S.Set Int
fromList []

>>> asSetOfPreimages (inverseJordan 15) (jordan 15 19) :: S.Set Integer
fromList [19]

The inverse for sigma 1 function.

The return value is parameterized by a Semiring , which allows various applications by providing different (multiplicative) embeddings. E. g., list all preimages (see a helper asSetOfPreimages ):

>>> import qualified Data.Set as S
>>> import Data.Semigroup
>>> :set -XFlexibleContexts
>>> S.mapMonotonic getProduct (inverseSigma (S.singleton . Product) 120)
fromList [54,56,87,95]

Count preimages:

>>> inverseSigma (const 1) 120
4

Sum preimages:

>>> inverseSigma id 120
292

Find minimal and maximal preimages:

>>> unMinWord (inverseSigma MinWord 120)
54
>>> unMaxWord (inverseSigma MaxWord 120)
95

The inverse for sigma function.

Generalizes the inverseSigma function, which is inverseSigmaK 1.

The return value is parameterized by a Semiring , which allows various applications by providing different (multiplicative) embeddings. E. g., list all preimages (see a helper asSetOfPreimages ):

>>> import qualified Data.Set as S
>>> import Data.Semigroup
>>> :set -XFlexibleContexts
>>> S.mapMonotonic getProduct (inverseSigmaK 2 (S.singleton . Product) 850)
fromList [24,26]

Similarly to inverseSigma , it is possible to count and sum preimages, or get the maximum/minimum preimage.

Note: it is the user's responsibility to use an appropriate type for inverseSigmaK . Even low values of k with Int or Word will return invalid results due to over/underflow, or throw exceptions (i.e. division by zero).

>>> asSetOfPreimages (inverseSigmaK 17) (sigma 17 13) :: S.Set Int
fromList *** Exception: divide by zero

Wrapper to use in conjunction with inverseTotient and inverseSigma . Extracts the minimal preimage of function.

Constructors

Instances

Wrapper to use in conjunction with inverseTotient and inverseSigma . Extracts the maximal preimage of function.

Constructors

Instances

Wrapper to use in conjunction with inverseTotient and inverseSigma . Extracts the minimal preimage of function.

Constructors

Instances

Wrapper to use in conjunction with inverseTotient and inverseSigma . Extracts the maximal preimage of function.

Constructors

Instances

Helper to extract a set of preimages for inverseTotient or inverseSigma .