Factors of Perfect Powers

A perfect power is a number \(n = m^k\), where \(m \ge 1\) and \(k \ge 2\).

If the prime factorization of \(m\) is \(p_1^{a_1}\:p_2^{a_2}\cdots\:p_j^{a_j}\) then

\(n=(p_1^{a_1}\:p_2^{a_2}\cdots\:p_j^{a_j})^k\)
\(\quad=p_1^{a_1k}\:p_2^{a_2k}\cdots\:p_j^{a_jk}\)

All of the exponents of the prime factors of \(n\) are multiples of \(k\).

We can work this in reverse to test if \(n\) is a perfect power.

Given the prime factorization of \(n\) is \(p_1^{a_1}\:p_2^{a_2}\cdots\:p_j^{a_j}\), \(n\) is a perfect power if and only if \(\mathrm{GCD}(a_1,a_2,\cdots,a_j)\ge 2\).

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