combinat-0.2.4.1: Generation of various combinatorial objects.Source codeContentsIndex
Math.Combinat.Numbers.Primes
Contents
List of prime numbers
Prime factorization
Integer logarithm
Integer square root
Modulo m arithmetic
Prime testing
Description
Prime numbers and related number theoretical stuff.
Synopsis
primes :: [Integer]
primesSimple :: [Integer]
primesTMWE :: [Integer]
groupIntegerFactors :: [Integer] -> [(Integer, Int)]
integerFactorsTrialDivision :: Integer -> [Integer]
integerLog2 :: Integer -> Integer
ceilingLog2 :: Integer -> Integer
isSquare :: Integer -> Bool
integerSquareRoot :: Integer -> Integer
ceilingSquareRoot :: Integer -> Integer
integerSquareRoot' :: Integer -> (Integer, Integer)
integerSquareRootNewton' :: Integer -> (Integer, Integer)
powerMod :: Integer -> Integer -> Integer -> Integer
millerRabinPrimalityTest :: Integer -> Integer -> Bool
List of prime numbers
primes :: [Integer]Source
Infinite list of primes, using the TMWE algorithm.
primesSimple :: [Integer]Source
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]Source
List of primes, using tree merge with wheel. Code by Will Ness.
Prime factorization
groupIntegerFactors :: [Integer] -> [(Integer, Int)]Source
Groups integer factors. Example: from [2,2,2,3,3,5] we produce [(2,3),(3,2),(5,1)]
integerFactorsTrialDivision :: Integer -> [Integer]Source
The naive trial division algorithm.
Integer logarithm
integerLog2 :: Integer -> IntegerSource
Largest integer k such that 2^k is smaller or equal to n
ceilingLog2 :: Integer -> IntegerSource
Smallest integer k such that 2^k is larger or equal to n
Integer square root
isSquare :: Integer -> BoolSource
integerSquareRoot :: Integer -> IntegerSource
Integer square root (largest integer whose square is smaller or equal to the input) using Newton's method, with a faster (for large numbers) inital guess based on bit shifts.
ceilingSquareRoot :: Integer -> IntegerSource
Smallest integer whose square is larger or equal to the input
integerSquareRoot' :: Integer -> (Integer, Integer)Source

We also return the excess residue; that is

 (a,r) = integerSquareRoot' n

means that

 a*a + r = n
 a*a <= n < (a+1)*(a+1)
integerSquareRootNewton' :: Integer -> (Integer, Integer)Source
Newton's method without an initial guess. For very small numbers (<10^10) it is somewhat faster than the above version.
Modulo m arithmetic
powerMod :: Integer -> Integer -> Integer -> IntegerSource

Efficient powers modulo m.

 powerMod a k m == (a^k) `mod` m
Prime testing
millerRabinPrimalityTest :: Integer -> Integer -> BoolSource

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.

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