galois.prime_factors

class galois.prime_factors(n)

Computes the prime factors of the positive integer \(n\).

The integer \(n\) can be factored into \(n = p_1^{e_1} p_2^{e_2} \dots p_{k-1}^{e_{k-1}}\).

Steps:

  1. Test if \(n\) is prime. If so, return [n], [1].

  2. Use trial division with a list of primes up to \(10^6\). If no residual factors, return the discovered prime factors.

  3. Use Pollard’s Rho algorithm to find a non-trivial factor of the residual. Continue until all are found.

Parameters

n (int) – The positive integer to be factored.

Returns

  • list – Sorted list of \(k\) prime factors \(p = [p_1, p_2, \dots, p_{k-1}]\) with \(p_1 < p_2 < \dots < p_{k-1}\).

  • list – List of corresponding prime powers \(e = [e_1, e_2, \dots, e_{k-1}]\).

Examples

In [610]: p, e = galois.prime_factors(120)

In [611]: p, e
Out[611]: ([2, 3, 5], [3, 1, 1])

# The product of the prime powers is the factored integer
In [612]: np.multiply.reduce(np.array(p) ** np.array(e))
Out[612]: 120

Prime factorization of 1 less than a large prime.

In [613]: prime =1000000000000000035000061

In [614]: galois.is_prime(prime)
Out[614]: True

In [615]: p, e = galois.prime_factors(prime - 1)

In [616]: p, e
Out[616]: ([2, 3, 5, 17, 19, 112850813, 457237177399], [2, 1, 1, 1, 1, 1, 1])

In [617]: np.multiply.reduce(np.array(p) ** np.array(e))
Out[617]: 2003764205241896700