Math.NumberTheory.Canon

A Canon is exponentation-based representation for arbitrarily massive numbers, including prime towers and hyper-expressions.

Documentation

data Canon

Canon: GADT for either Bare (Integer) or some variation of a Can(onical) form (see CanonValueType).

Instances

Enum Canon

Defined in Math.NumberTheory.Canon

Methods

succ :: Canon -> Canon

pred :: Canon -> Canon

toEnum :: Int -> Canon

fromEnum :: Canon -> Int

enumFrom :: Canon -> [Canon]

enumFromThen :: Canon -> Canon -> [Canon]

enumFromTo :: Canon -> Canon -> [Canon]

enumFromThenTo :: Canon -> Canon -> Canon -> [Canon]

Eq Canon

Instances for Canon

Defined in Math.NumberTheory.Canon

Methods

(==) :: Canon -> Canon -> Bool

(/=) :: Canon -> Canon -> Bool

Fractional Canon

Defined in Math.NumberTheory.Canon

Methods

(/) :: Canon -> Canon -> Canon

recip :: Canon -> Canon

fromRational :: Rational -> Canon

Integral Canon

Defined in Math.NumberTheory.Canon

Methods

quot :: Canon -> Canon -> Canon

rem :: Canon -> Canon -> Canon

div :: Canon -> Canon -> Canon

mod :: Canon -> Canon -> Canon

quotRem :: Canon -> Canon -> (Canon, Canon)

divMod :: Canon -> Canon -> (Canon, Canon)

toInteger :: Canon -> Integer

Num Canon

Defined in Math.NumberTheory.Canon

Methods

(+) :: Canon -> Canon -> Canon

(-) :: Canon -> Canon -> Canon

(*) :: Canon -> Canon -> Canon

negate :: Canon -> Canon

abs :: Canon -> Canon

signum :: Canon -> Canon

fromInteger :: Integer -> Canon

Ord Canon

Defined in Math.NumberTheory.Canon

Methods

compare :: Canon -> Canon -> Ordering

(<) :: Canon -> Canon -> Bool

(<=) :: Canon -> Canon -> Bool

(>) :: Canon -> Canon -> Bool

(>=) :: Canon -> Canon -> Bool

max :: Canon -> Canon -> Canon

min :: Canon -> Canon -> Canon

Real Canon

Defined in Math.NumberTheory.Canon

Methods

toRational :: Canon -> Rational

Show Canon

Defined in Math.NumberTheory.Canon

Methods

showsPrec :: Int -> Canon -> ShowS

show :: Canon -> String

showList :: [Canon] -> ShowS

CanonConv Canon

Instance of CanonConv class

Defined in Math.NumberTheory.Canon

Methods

toSC :: Canon -> SimpleCanon

toRC :: Canon -> RationalSimpleCanon

makeCanon :: Integer -> Canon

Create a Canon from an Integer. This may involve expensive factorization.

makeCanon' :: Integer -> (Canon, Bool)

Create a Canon from an Integer. Also return True if the number is fully factored

data BareStatus

BareStatus: A "Bare Simp" number means a prime number, +/-1 or 0. The code must set the flag properly A "Bare NSim" number is an Integer that has not been checked (to see if it can be factored).

Constructors

Simp

NSim

Instances

Eq BareStatus

Defined in Math.NumberTheory.Canon

Methods

(==) :: BareStatus -> BareStatus -> Bool

(/=) :: BareStatus -> BareStatus -> Bool

Ord BareStatus

Defined in Math.NumberTheory.Canon

Methods

compare :: BareStatus -> BareStatus -> Ordering

(<) :: BareStatus -> BareStatus -> Bool

(<=) :: BareStatus -> BareStatus -> Bool

(>) :: BareStatus -> BareStatus -> Bool

(>=) :: BareStatus -> BareStatus -> Bool

max :: BareStatus -> BareStatus -> BareStatus

min :: BareStatus -> BareStatus -> BareStatus

Show BareStatus

Defined in Math.NumberTheory.Canon

Methods

showsPrec :: Int -> BareStatus -> ShowS

show :: BareStatus -> String

showList :: [BareStatus] -> ShowS

data CanonValueType

CanonValueType: 3 possibilities for this GADT (integral, non-integral rational, irrational). Imaginary/complex numbers are not supported

Constructors

IntC

NirC

IrrC

Instances

Eq CanonValueType

Defined in Math.NumberTheory.Canon

Methods

(==) :: CanonValueType -> CanonValueType -> Bool

(/=) :: CanonValueType -> CanonValueType -> Bool

Ord CanonValueType

Defined in Math.NumberTheory.Canon

Methods

compare :: CanonValueType -> CanonValueType -> Ordering

(<) :: CanonValueType -> CanonValueType -> Bool

(<=) :: CanonValueType -> CanonValueType -> Bool

(>) :: CanonValueType -> CanonValueType -> Bool

(>=) :: CanonValueType -> CanonValueType -> Bool

max :: CanonValueType -> CanonValueType -> CanonValueType

min :: CanonValueType -> CanonValueType -> CanonValueType

Show CanonValueType

Defined in Math.NumberTheory.Canon

Methods

showsPrec :: Int -> CanonValueType -> ShowS

show :: CanonValueType -> String

showList :: [CanonValueType] -> ShowS

cShowFull :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowFullAsCode :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowAsCode :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowAsCodeUnf :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowUnf :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cMult :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Multiply Function: Generally speaking, this will be much cheaper than addition/subtraction which requires factoring. You are usually just merging lists of prime, exponent pairs and adding exponents where common primes are found. This notion is the crux of the library.

Note: This can be used instead of the * operator if you want to maintain a CycloMap for performance reasons.

cDiv :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Div function : Multiply by the reciprocal.

cAdd :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Addition and subtraction is generally much more expensive because it requires refactorization. There is logic to look for algebraic forms which can greatly reduce simplify factorization. Note: This can be used instead of the +/- operators if you want to maintain a CycloMap for performance reasons.

cSubtract :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Addition and subtraction is generally much more expensive because it requires refactorization. There is logic to look for algebraic forms which can greatly reduce simplify factorization. Note: This can be used instead of the +/- operators if you want to maintain a CycloMap for performance reasons.

cExp :: Canon -> Canon -> Bool -> CycloMap -> (Canon, CycloMap)

Exponentiation: This does allow for negative exponentiation if the Bool flag is True. Note: This can be used instead of the exponentiation operator if you want to maintain a CycloMap for performance reasons.

cReciprocal :: Canon -> Canon

Compute reciprocal (by negating exponents or equivalent).

(>^) :: Canon -> Canon -> Canon infixr 9

Exponentiation and root operator declarations

(<^) :: Canon -> Canon -> Canon infixr 9

Exponentiation and root operator declarations

cGCD :: Canon -> Canon -> Canon

GCD and LCM functions for Canon

cLCM :: Canon -> Canon -> Canon

GCD and LCM functions for Canon

cMod :: Canon -> Canon -> Canon

Mod function

cOdd :: Canon -> Bool

Check if a Canon is integral and odd/even, respectively. Note: Return False for both if the Canon is not integral. See CanonValueType for possible cases.

cEven :: Canon -> Bool

Check if a Canon is integral and odd/even, respectively. Note: Return False for both if the Canon is not integral. See CanonValueType for possible cases.

cTotient :: Canon -> CycloMap -> (Canon, CycloMap)

Totient functions

cPhi :: Canon -> CycloMap -> (Canon, CycloMap)

Totient functions

cNegative :: Canon -> Bool

Functions to check if a canon is negative/positive

cPositive :: Canon -> Bool

Functions to check if a canon is negative/positive

cIntegral :: Canon -> Bool

Functions to check if a Canon is Integral, (Ir)Rational, Simplified, a prime or a prime tower

cRational :: Canon -> Bool

Functions to check if a Canon is Integral, (Ir)Rational, Simplified, a prime or a prime tower

cIrrational :: Canon -> Bool

Functions to check if a Canon is Integral, (Ir)Rational, Simplified, a prime or a prime tower

cPrime :: Canon -> Bool

Functions to check if a Canon is Integral, (Ir)Rational, Simplified, a prime or a prime tower

cSimplified :: Canon -> Bool

Functions to check if a Canon is Integral, (Ir)Rational, Simplified, a prime or a prime tower

cSplit :: Canon -> (Canon, Canon)

Split a Canon into the numerator and denominator.

cNumerator :: Canon -> Canon

cNumerator and cDenominator are for processing "rational" canon reps.

cDenominator :: Canon -> Canon

cNumerator and cDenominator are for processing "rational" canon reps.

cCanonical :: Canon -> Bool

Checks if the Canon is Canonical, a more complex expression.

cBare :: Canon -> Bool

Checks if the Canon just a Bare Integer.

cBareStatus :: Canon -> BareStatus

Returns the status for Bare numbers.

cValueType :: Canon -> CanonValueType

Return the CanonValueType (Integral, etc).

cDelve :: Canon -> [Int] -> Canon

Take a canon and a list of indexes and delve into the canon. This operates on the internal hyper lists

cIsPrimeTower :: Canon -> Bool

Functions to check if a Canon is Integral, (Ir)Rational, Simplified, a prime or a prime tower

cPrimeTowerLevel :: Canon -> Canon

This is used for tetration, etc. It defaults to zero for non-integral reps.

cSuperLog :: Canon -> (SuperPairC, Int)

This is the super or iterated log function. A level and mantissa is returned along with the number's sign.

cSuperLogCmp :: SuperPairC -> SuperPairC -> Ordering

Compare the level and the "mantissa"

cTetration :: Canon -> Canon -> Canon

Tetration Function - Level 4

cPentation :: Canon -> Canon -> Canon

Pentation Function - Level 5

cHexation :: Canon -> Canon -> Canon

Hexation Function - Level 6

cHeptation :: Canon -> Canon -> Canon

Heptation Function - Level 7

cOctation :: Canon -> Canon -> Canon

Octation Function -- Level 8

cNonation :: Canon -> Canon -> Canon

Nonation Function -- Level 9

cTetrationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Tetration List Function

cPentationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Pentation List Function

cHexationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Hexation List Function

cHeptationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Heptation List Function

cOctationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Octation List Function

cNonationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Nonation List Function

(<^>) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(<<^>>) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(<<<^>>>) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(<<<<^>>>>) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(<<<<<^>>>>>) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(|<^>|) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(<^^>) :: Canon -> [Canon] -> Canon infixr 9

Hyperoperation List Operators. On display, the towers will have single caret operators interspersed.

(<<^^>>) :: Canon -> [Canon] -> Canon infixr 9

Hyperoperation List Operators. On display, the towers will have single caret operators interspersed.

(<<<^^>>>) :: Canon -> [Canon] -> Canon infixr 9

Hyperoperation List Operators. On display, the towers will have single caret operators interspersed.

(<<<<^^>>>>) :: Canon -> [Canon] -> Canon infixr 9

Hyperoperation List Operators. On display, the towers will have single caret operators interspersed.

(<<<<<^^>>>>>) :: Canon -> [Canon] -> Canon infixr 9

Hyperoperation List Operators. On display, the towers will have single caret operators interspersed.

(|<^^>|) :: Canon -> [Canon] -> Canon infixr 9

Hyperoperation List Operators. On display, the towers will have single caret operators interspersed.

(~^~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~<^>~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~<<^>>~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~<<<^>>>~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~<<<<^>>>>~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~|^|~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~|<^>|~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~|<<^>>|~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~|<<<^>>>|~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~|<<<<^>>>>|~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~^~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~<^>~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~<<^>>~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~<<<^>>>~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~<<<<^>>>>~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~|^|~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~|<^>|~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~|<<^>>|~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~|<<<^>>>|~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~|<<<<^>>>>|~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~^~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~<^>~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~<<^>>~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~<<<^>>>~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~<<<<^>>>>~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~|^|~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~|<^>|~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~|<<^>>|~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~|<<<^>>>|~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~|<<<<^>>>>|~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~^~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~<^>~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~<<^>>~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~<<<^>>>~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~<<<<^>>>>~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~|^|~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~|<^>|~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~|<<^>>|~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~|<<<^>>>|~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~|<<<<^>>>>|~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

(~~~~~^~~~~~) :: Canon -> Canon -> Canon infixr 9

The thinking around the hyperoperators is that they should look progressively scarier :) | They range from level 4 / tetration (^) to level 50 (~~~~~^~~~~~). Please read .odp file for the naming convention.

cAddOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cMultOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cExpOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cTetrOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cPentOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cHexOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cHeptOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cOctOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cNonOpLevel :: Canon

Levels starting with 1 in the hyperoperation hierarchy

cGetHyperList :: Canon -> [Canon]

Return the list of canons from a hyper expression

cGetHyperOp :: Canon -> Canon

Return the level of hyperoperation from a hyper expression.

maxHyperOpDispLevel :: Integer

Internal value that corresponds with ~~~~~^~~~~~ (level 50 hyperoperation)

maxHyperOpDelveLevel :: Canon

Max hyper operaton level when converting to canonical form (for the sake of combining and reducing terms)

cFactorSum :: Canon -> Bool -> Canon

cFactor : Factor simple terms from a sum. If the flag is true, only factor by the gcd if the gcd is a hyper expression

cConvertToSum :: Canon -> (Canon, Bool)

Convert a hyperexpression to a sum if possible. Useful in comparison. Will expand polynomials to a limited degree.

cMaxExpoToExpand :: Canon

Maximum exponent (of a polynomial) to distribute into a sum of terms.

cFactorHorizon :: Canon -> Canon

cFactorHorizon: Good for polynomial-like expressions like: (1 + 3^4 + 3^5) <^ 3 - 1, where there's a mixture of "canonical" and hE exponents.

cApplyHy :: Canon -> [Canon] -> Bool -> Canon

wrapper to create apply a hyperoperation to a list

cHyperOp :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Generalized Hyperoperation Function (https:/en.wikipedia.orgwiki/Hyperoperation)

cHyperExpr :: Canon -> Bool

Utility functions regarding hyperoperations. Any functions search the entire expression

cHyperExprAny :: Canon -> Bool

Utility functions regarding hyperoperations. Any functions search the entire expression

cMaxHyperOp :: Canon -> Canon

Find the maximum hyperoperation embedded in a Canon

cMinHyperOp :: Canon -> Canon

Find the minimum hyperoperation embedded in a Canon. (If not at all, return zer0

cHyperSum :: Canon -> Bool

Utility functions regarding hyperoperations. Any functions search the entire expression

cHyperProd :: Canon -> Bool

Utility functions regarding hyperoperations. Any functions search the entire expression

cHyperExpo :: Canon -> Bool

Utility functions regarding hyperoperations. Any functions search the entire expression

cHyperSumAny :: Canon -> Bool

Utility functions regarding hyperoperations. Any functions search the entire expression

cHyperize :: Canon -> Canon

Hyperize will take a Canon in quasi-canonized form and try to clean it up in a tidier expression Example: 7 ^ ( 1 + 2 * (49 ^ 7) = 7 * 49 ^ 8. ToDo: Enhancement: Partial hyperizing?

cQuasiCanonize :: Canon -> Canon

This is akin to canonical form except you may have sums in the bases. It converts expression up to a hyperoperational cutoff

cQuasiCanonized :: Canon -> Bool

This checks if the (hyper)expression is in quasi-canonical form

cCleanup :: Canon -> Canon

Break code into a canonized

cGetAddends :: Canon -> [Canon]

If the Canon is a product, return the factors. Otherwise, return the Canon itself.

cGetFactors :: Canon -> [Canon]

If the Canon is a sum, return the addends. Otherwise, return the Canon itself.

cCleanupAsNumDenPair :: Canon -> (Canon, Canon)

Split the hyperoperation into a cleaned-up numerator and denominator pair (if denom is 1). This still represents an integral value. e.g. 3 ^ 7 / 3 ^ 4

type CanonElement = (Canon, Canon)

This element is a base, exponent pair. The base is an integer and is generally prime or 0, -1. The exponent is also a Canon (allowing for arbitrary nesting) A Canon conceptually consists of a list of these elements. The first member of the pair will be a Canon raised to the first power. By doing this, we're allow for further generality in the definition of a Canon.

cGetBase :: CanonElement -> Canon

Return the base b from a Canon Element (equivalent to b^e)

cGetExponent :: CanonElement -> Canon

Return the exponent e from a Canon Element (equivalent to b^e)

cGetBases :: Canon -> [Canon]

Return the list of bases from a Canon (conceptually of the form [b^e])>

cGetBasesDeep :: Canon -> [Canon]

Similar to cGetBases except that it will do trial factoring of any hyper sums. So, for obvious reasons, this is not a complete factorization.

cGetExponents :: Canon -> [Canon]

Return the list of exponents from a Canon (conceptually of the form [b^e]).

cGetElements :: Canon -> [CanonElement]

Return the list of CanonElements from a Canon (conceptually of the form [b^e]).

cNumDivisors :: Canon -> Canon

Divisor functions should be called with integral Canons. Restricted to positive divisors. Returns Either String Canon

cTau :: Canon -> Canon

Divisor functions should be called with integral Canons. Restricted to positive divisors. Returns Either String Canon

cDivisors :: Canon -> Either String [Canon]

Efficiently compute all of the divisors based on the canonical representation. | Returns Either an error message or a list of Canons.

cNthDivisor :: Canon -> Canon -> Either String Canon

Compute the nth divisor of a Canon. It operates on the absolute value of the Canon and is zero based. Note: This is deterministic but it's not ordered by the value of the divisor.

cWhichDivisor :: Canon -> Canon -> Either String Canon

Consider this to be the inverse of the cNthDivisor function. This function ignores signs but both parameters must be integral.

cRelativelyPrime :: Canon -> Canon -> Maybe Bool

This will determine if two arbitary expressions are relatively prime or not (if possible). Goes deep.

cGetFirstNDivisors :: Int -> Canon -> Either String [Canon]

Return the first N divisors of a hyper expression (if possible)

cN1 :: Canon

Define some small canons for convenience

c0 :: Canon

Define some small canons for convenience

c1 :: Canon

Define some small canons for convenience

c2 :: Canon

Define some small canons for convenience

c3 :: Canon

Define some small canons for convenience

c4 :: Canon

Define some small canons for convenience

c5 :: Canon

Define some small canons for convenience

c6 :: Canon

Define some small canons for convenience

c7 :: Canon

Define some small canons for convenience

c8 :: Canon

Define some small canons for convenience

c9 :: Canon

Define some small canons for convenience

data CycloMap

CycloMap is a newtype hiding the details of a map of CR_ to pairs of integers and corresponding cyclotomic polynomials.

Instances

Eq CycloMap

Defined in Math.NumberTheory.Canon.AurifCyclo

Methods

(==) :: CycloMap -> CycloMap -> Bool

(/=) :: CycloMap -> CycloMap -> Bool

Show CycloMap

Defined in Math.NumberTheory.Canon.AurifCyclo

Methods

showsPrec :: Int -> CycloMap -> ShowS

show :: CycloMap -> String

showList :: [CycloMap] -> ShowS

getIntegerBasedCycloMap :: CycloMap -> Map Integer CycloPair

Unwrap the CycloMap and convert the internal canon rep keys to Integers, returning a "raw" map

showCyclo :: CR_ -> CycloMap -> [Char]

This will display the cyclotomic polynomials for a CR.

crCycloInitMap :: CycloMap

This is an initial map with the cyclotomic polynomials for 1.