Math.Combinat.Numbers.Primes

Prime numbers and related number theoretical stuff.

Elementary number theory

divides :: Integer -> Integer -> Bool

d divides n

divisors :: Integer -> [Integer]

Divisors of n (note: the result is not ordered!)

squareFreeDivisors :: Integer -> [(Integer, Sign)]

List of square-free divisors together with their Mobius mu value (note: the result is not ordered!)

squareFreeDivisors_ :: Integer -> [Integer]

List of square-free divisors (note: the result is not ordered!)

divisorSum :: Integer -> Integer

Sum ofthe of the divisors

divisorSum' :: Int -> Integer -> Integer

Sum of k-th powers of the divisors

moebiusMu :: (Integral a, Num b) => a -> b

The Moebius mu function

eulerTotient :: Integer -> Integer

Euler's totient function

liouvilleLambda :: (Integral a, Num b) => a -> b

The Liouville lambda function

List of prime numbers

primes :: [Integer]

Infinite list of primes, using the TMWE algorithm.

primesSimple :: [Integer]

A relatively simple but still quite fast implementation of list of primes. By Will Ness http://www.haskell.org/pipermail/haskell-cafe/2009-November/068441.html

primesTMWE :: [Integer]

List of primes, using tree merge with wheel. Code by Will Ness.

Prime factorization

factorize :: Integer -> [(Integer, Int)]

factorizeNaive :: Integer -> [(Integer, Int)]

productOfFactors :: [(Integer, Int)] -> Integer

integerFactorsTrialDivision :: Integer -> [Integer]

The naive trial division algorithm.

groupIntegerFactors :: [Integer] -> [(Integer, Int)]

Groups integer factors. Example: from [2,2,2,3,3,5] we produce [(2,3),(3,2),(5,1)]

Modulo m arithmetic

powerMod :: Integer -> Integer -> Integer -> Integer

Efficient powers modulo m.

powerMod a k m == (a^k) `mod` m

Prime testing

millerRabinPrimalityTest :: Integer -> Integer -> Bool

Miller-Rabin Primality Test (taken from Haskell wiki). We test the primality of the first argument n by using the second argument a as a candidate witness. If it returs False, then n is composite. If it returns True, then n is either prime or composite.

A random choice between 2 and (n-2) is a good choice for a.

isProbablyPrime :: Integer -> Bool

For very small numbers, we use trial division, for larger numbers, we apply the Miller-Rabin primality test log4(n) times, with candidate witnesses derived deterministically from n using a pseudo-random sequence (which should be based on a cryptographic hash function, but isn't, yet).

Thus the candidate witnesses should behave essentially like random, but the resulting function is still a deterministic, pure function.

TODO: implement the hash sequence, at the moment we use Random instead...

isVeryProbablyPrime :: Integer -> Bool

A more exhaustive version of isProbablyPrime, this one tests candidate witnesses both the first log4(n) prime numbers and then log4(n) pseudo-random numbers