A module to factor numbers in Python. The module uses several techniques to factor an integer, including
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Lookup tables for numbers below 10^6
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Trial division of primes below 1200
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Miller-Rabin primality test (deterministic up to the published limit)
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Brent's variant of the Pollard Rho algorithm, by default running for up to 1 second
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An interface to a DLL of a fast C++ implementation, for 64 bit numbers. The C++ implementation can be found at https://github.com/armchaircaver/Factors
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Python interface to YAFU, running YAFU as a subprocess, to factorise very large numbers (YAFU can be obtained from https://sourceforge.net/projects/yafu/)
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Using Martin G Kelly's implementation of the Elliptic Curve Method (ECM) if the Brent algorithm fails to find a factor within its time limit. In practice, ECM is only useful if GMP arithmetic is available (via module gmpy2). pyecm can be found at https://github.com/martingkelly/pyecm
The main functions of the module are:
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factorise(n, maxBrentTime=1.0) return a list of the prime factors of n. maxBrentTime specifies the number of seconds that the Pollard Rho algorithm should run for, before resorting to ECM or YAFU
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millerRabin(n) returns True if the Miller Rabin algorithm determines that n is probably prime, False if it identifies n as a composite. The algorithm is deterministic up to 2^64
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brent(N,maxBrentTime=1.0) implements the Pollard Rho algorithm with Brent's vaiation, and returns a prime factor of N, or raises and exception if it times out without finding a factor
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brent_gmp(n,maxBrentTime=1.0) GMP implemtation of the Pollard Rho algorithm. Requires module gmpy2, available at https://pypi.org/project/gmpy2/
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carmichael(n) returns the Carmichael number of n
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totient(n) returns the Euler totient number of n
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all_factors(n) returns all the factors, aka divisors of n, e.g. all_factors(12) returns [1,2,3,4,6,12]
A test suite is also provided in "factors test cases.py"