Perfect power

In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that m^k = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (1^k = 1 for any k).

A sequence of perfect powers can be generated by iterating through the possible values for m and k. The first few ascending perfect powers in numerical order (showing duplicate powers) are (sequence in the OEIS):

The sum of the reciprocals of the perfect powers (including duplicates) is 1:

which can be proved as follows:

The first perfect powers without duplicates are:

The sum of the reciprocals of the perfect powers p without duplicates is:

where μ(k) is the Möbius function and ζ(k) is the Riemann zeta function.

According to Euler, Goldbach showed (in a now lost letter) that the sum of 1/(p−1) over the set of perfect powers p, excluding 1 and excluding duplicates, is 1:

This is sometimes known as the Goldbach-Euler theorem.

Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity. One of the simplest such methods is to consider all possible values for k across each of the divisors of n, up to ${\displaystyle k\leq \log _{2}n}$. So if the divisors of ${\displaystyle n}$ are ${\displaystyle n_{1},n_{2},\dots ,n_{j}}$ then one of the values ${\displaystyle n_{1}^{2},n_{2}^{2},\dots ,n_{j}^{2},n_{1}^{3},n_{2}^{3},\dots }$ must be equal to n if n is indeed a perfect power.

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Wikipedia