galois.perfect_power¶

galois.perfect_power(n: int) → tuple[int, int]¶

Returns the integer base \(c\) and exponent \(e\) of \(n = c^e\).

If \(n\) is a not perfect power, then \(c = n\) and \(e = 1\).

Parameters

n: int¶

An integer.

Returns

• The potentially composite base \(c\).

• The exponent \(e\).

See also

factors, is_perfect_power, is_prime_power

Examples

Primes are not perfect powers because their exponent is 1.

In [1]: galois.perfect_power(13)
Out[1]: (13, 1)

In [2]: galois.is_perfect_power(13)
Out[2]: False

Products of primes are not perfect powers.

In [3]: galois.perfect_power(5*7)
Out[3]: (35, 1)

In [4]: galois.is_perfect_power(5*7)
Out[4]: False

Products of prime powers where the GCD of the exponents is 1 are not perfect powers.

In [5]: galois.perfect_power(2 * 3 * 5**3)
Out[5]: (750, 1)

In [6]: galois.is_perfect_power(2 * 3 * 5**3)
Out[6]: False

Products of prime powers where the GCD of the exponents is greater than 1 are perfect powers.

In [7]: galois.perfect_power(2**2 * 3**2 * 5**4)
Out[7]: (150, 2)

In [8]: galois.is_perfect_power(2**2 * 3**2 * 5**4)
Out[8]: True

Negative integers can be perfect powers if they can be factored with an odd exponent.

In [9]: galois.perfect_power(-64)
Out[9]: (-4, 3)

In [10]: galois.is_perfect_power(-64)
Out[10]: True

Negative integers that are only factored with an even exponent are not perfect powers.

In [11]: galois.perfect_power(-100)
Out[11]: (-100, 1)

In [12]: galois.is_perfect_power(-100)
Out[12]: False