factory-0.2.1.1: Rational arithmetic in an irrational world.

Factory.Math.PrimeFactorisation

Contents

Description

`AUTHOR`
Dr. Alistair Ward
`DESCRIPTION`
• http://en.wikipedia.org/wiki/Integer_factorization.
• Exports a common interface to permit decomposition of positive integers, into the unique combination of prime-factors known to exist according to the Fundamental Theorem of Arithmetic; http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic.
• Leveraging this abstract capability, it derives the smoothness, power-smoothness, omega-numbers and square-free integers.
• Filters the list of regular-numbers from the list of smoothness.
• CAVEAT: to avoid wasting time, it may be advantageous to check Factory.Math.Primality.isPrime first.

Synopsis

# Type-classes

class Algorithmic algorithm where Source

Defines the methods expected of a factorisation-algorithm.

Methods

Arguments

 :: (NFData base, Integral base) => algorithm -> base The operand -> Factors base Int

Instances

 Algorithmic Algorithm

# Functions

maxBoundPrimeFactor :: Integral i => i -> i Source

• The upper limit for a prime to be considered as a candidate factor of the specified number.
• One might naively think that this limit is `(x div 2)` for an even number, but though a prime-factor greater than the square-root of the number can exist, its smaller cofactor decomposes to a prime which must be less than the square-root.
• NB: rather then using `(primeFactor <= sqrt numerator)` to filter the candidate prime-factors of a given numerator, one can alternatively use `(numerator >= primeFactor ^ 2)` to filter what can potentially be factored by a given prime-factor.
• CAVEAT: suffers from rounding-errors, though no consequence has been witnessed.

smoothness :: (Algorithmic algorithm, NFData base, Integral base) => algorithm -> [base] Source

powerSmoothness :: (Algorithmic algorithm, NFData base, Integral base) => algorithm -> [base] Source

regularNumbers :: (Algorithmic algorithm, NFData base, Integral base) => algorithm -> [base] Source

primePowerTotient :: (Integral base, Integral exponent) => Exponential base exponent -> base Source

• Euler's Totient for a power of a prime-number.
• By Olofsson; `(phi(n^k) = n^(k - 1) * phi(n))` and since `(phi(prime) = prime - 1)`
• CAVEAT: checks neither the primality nor the bounds of the specified value; therefore for internal use only.

eulersTotient :: (Algorithmic algorithm, NFData i, Integral i, Show i) => algorithm -> i -> i Source

omega :: (Algorithmic algorithm, NFData i, Integral i) => algorithm -> [i] Source

squareFree :: (Algorithmic algorithm, NFData i, Integral i) => algorithm -> [i] Source