Perfect power

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In mathematics, a perfect power is a number that can be expressed as a power of any positive, whole number. 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 would be called a perfect square or perfect cube, respectively.

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 duplicates) are:

$2^2 = 4,\ 2^3 = 8,\ 3^2 = 9,\ 2^4 = 4^2 = 16,\ 5^2 = 25,\ 3^3 = 27,\$ $2^5 = 32,\ 6^2 = 36,\ 7^2 = 49,\ 2^6 = 4^3 = 8^2 = 64, \dots$

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 omitting repetitions, converges to 1:

$\sum_{p}\frac{1}{p-1}= {\frac{1}{3} + \frac{1}{7} + \frac{1}{8}+ \frac{1}{15} + \frac{1}{24} + \frac{1}{26}+ \frac{1}{31}}+ \cdots = 1$

This is sometimes known as the Goldbach-Euler theorem.

1 Detecting perfect powers
2 See also
3 References
4 External links

[edit] Detecting perfect powers

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 $k \leq \log_2 n$ . So if the factors of $n$ are $n_1\cdot n_2\cdot \dots n_j$ then one of the values $n_1^2, n_2^2, \dots, n_j^2, n_1^3, n_2^3, \dots$ must be equal to $n$ if $n$ is indeed a perfect power.

This method can immediately be simplified by instead considering only prime values of k. This is because if $n = m k$ and $k = a p$ where a is composite and p is prime, then this can simply be rewritten as $n = m k = m a p = (m a) p$ . Because of this result, the minimal value of k must necessarily be prime.