- Dr. Alistair Ward
- 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.
- class Algorithmic algorithm where
- maxBoundPrimeFactor :: Integral i => i -> i
- smoothness :: (Algorithmic algorithm, NFData base, Integral base) => algorithm -> [base]
- powerSmoothness :: (Algorithmic algorithm, NFData base, Integral base) => algorithm -> [base]
- regularNumbers :: (Algorithmic algorithm, NFData base, Integral base) => algorithm -> [base]
- primePowerTotient :: (Integral base, Integral exponent) => Exponential base exponent -> base
- eulersTotient :: (Algorithmic algorithm, NFData i, Integral i, Show i) => algorithm -> i -> i
- omega :: (Algorithmic algorithm, NFData i, Integral i) => algorithm -> [i]
- squareFree :: (Algorithmic algorithm, NFData i, Integral i) => algorithm -> [i]
Defines the methods expected of a factorisation-algorithm.
- 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
(xfor 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.
- A constant, zero-indexed, conceptually infinite, list of the power-smoothness of all positive integers.
- 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.
- The number of coprimes less than or equal to the specified positive integer.
- AKA EulerPhi.
- A constant, zero-indexed, conceptually infinite, list of the small omega numbers (i.e. the number of distinct prime factors); cf. big omega.