galois.prime_factors¶

galois.prime_factors(x)[source]¶

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

The integer \(x\) can be factored into \(x = p_1^{k_1} p_2^{k_2} \dots p_{n-1}^{k_{n-1}}\).

Parameters

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

Returns

numpy.ndarray – Sorted array of prime factors \(p = [p_1, p_2, \dots, p_{n-1}]\) with \(p_1 < p_2 < \dots < p_{n-1}\).
numpy.ndarray – Array of corresponding prime powers \(k = [k_1, k_2, \dots, k_{n-1}]\).

Examples

In [1]: p, k = galois.prime_factors(120)

In [2]: p, k
Out[2]: (array([2, 3, 5]), array([3, 1, 1]))

# The product of the prime powers is the factored integer
In [3]: np.multiply.reduce(p ** k)
Out[3]: 120

# Prime factorization of 1 less than a large prime
In [4]: p, k = galois.prime_factors(1000000000000000035000061 - 1)

In [5]: p, k
Out[5]: 
(array([2, 3, 5, 17, 19, 51599587203302375387], dtype=object),
 array([2, 1, 1, 1, 1, 1]))

In [6]: np.multiply.reduce(p ** k)
Out[6]: 1000000000000000035000060