Factory.Math.PrimeFactorisation

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.

Type-classes

class Algorithmic algorithm where

Defines the methods expected of a factorisation-algorithm.

Methods

primeFactors

Arguments

:: (NFData base, Integral base)

=> algorithm

-> base

The operand

-> Factors base Int

Instances

Algorithmic Algorithm

Functions

maxBoundPrimeFactor :: Integral i => i -> i

• 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]

• A constant, zero-indexed, conceptually infinite, list, of the smoothness of all positive integers.
• http://en.wikipedia.org/wiki/Smooth_number.
• http://mathworld.wolfram.com/SmoothNumber.html.

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

• A constant, zero-indexed, conceptually infinite, list of the power-smoothness of all positive integers.
• http://en.wikipedia.org/wiki/Smooth_number#Powersmooth_numbers.

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

• Filters smoothness, to derive the constant list of Hamming-numbers.
• http://en.wikipedia.org/wiki/Regular_number.

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

• 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

• The number of coprimes less than or equal to the specified positive integer.
• http://en.wikipedia.org/wiki/Euler%27s_totient_function.
• http://mathworld.wolfram.com/TotientFunction.html.
• AKA EulerPhi.

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

• A constant, zero-indexed, conceptually infinite, list of the small omega numbers (i.e. the number of distinct prime factors); cf. big omega.
• http://oeis.org/wiki/Omega%28n%29,_number_of_distinct_primes_dividing_n.
• http://mathworld.wolfram.com/DistinctPrimeFactors.html
• http://planetmath.org/encyclopedia/NumberOfDistinctPrimeFactorsFunction.html.

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

• A constant, conceptually infinite, list of the square-free numbers, i.e. those which aren't divisible by any perfect square.
• http://en.wikipedia.org/wiki/Square-free_integer.