Factory.Data.PrimeFactors

AUTHOR

Dr. Alistair Ward

DESCRIPTION

• Describes a list of prime factors.
• The product of this list of prime-factors represents the composite integer from which they were originally extracted.

Types

Type-synonyms

type Factors base exponent = [Exponential base exponent]

• Each element of this list represents one prime-factor, expressed as an exponential with a prime base, of the original integer.
• Whilst it only makes sense for both the base and exponent to be integral, these constrains are applied at the function-level as required.

Functions

insert' :: (Ord base, Num exponent) => Exponential base exponent -> Factors base exponent -> Factors base exponent

• Insert a Exponential, into a list representing a product of prime factors, multiplying with any incumbent of like base.
• The list should be sorted by increasing base.
• Preserves the sort-order.
• CAVEAT: this is tolerably efficient for sporadic insertion; to insert a list, use >*<.

product'

Arguments

:: (Num base, Integral exponent)

=> BisectionRatio

-> MinLength

-> Factors base exponent

The list on which to operate.

-> base

The result.

Multiply a list of prime factors.

reduce :: (Ord base, Num exponent, Ord exponent) => Factors base exponent -> Factors base exponent

• Sorts a list representing a product of prime factors by increasing base.
• Multiplies Exponentials of similar base.

Operators

(>*<) :: (Ord base, Num exponent, Ord exponent) => Factors base exponent -> Factors base exponent -> Factors base exponent infixl 7

• Multiplies two lists each representing a product of prime factors, and sorted by increasing base.
• Preserves the sort-order.

(>/<) infixl 7

Arguments

:: (Integral base, Integral exponent)

=> Factors base exponent

The list of prime factors in the numerator.

-> Factors base exponent

The list of prime factors in the denominator.

-> (Factors base exponent, Factors base exponent)

The ratio of numerator and denominator, after like prime factors are cancelled.

• Divides two lists, each representing a product of prime factors, and sorted by increasing base.
• Preserves the sort-order.

(>^) :: Num exponent => Factors base exponent -> exponent -> Factors base exponent infixr 8

• Raise the product of a list prime factors to the specified power.
• CAVEAT: this merely involves raising each element to the specified power; cf. raising a polynomial to a power.