### 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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