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:

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

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:

$\backslash sum\_\{p\}\backslash frac\{1\}\{p-1\}=\; \{\backslash frac\{1\}\{3\}\; +\; \backslash frac\{1\}\{7\}\; +\; \backslash frac\{1\}\{8\}+\; \backslash frac\{1\}\{15\}\; +\; \backslash frac\{1\}\{24\}\; +\; \backslash frac\{1\}\{26\}+\; \backslash frac\{1\}\{31\}\}+\; \backslash cdots\; =\; 1$

This is sometimes known as the Goldbach-Euler theorem.

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 $k\; \backslash leq\; \backslash log\_2\; n$. So if the factors of $n$ are $n\_1\backslash cdot\; n\_2\backslash cdot\; \backslash dots\; n\_j$ then one of the values $n\_1^2,\; n\_2^2,\; \backslash dots,\; n\_j^2,\; n\_1^3,\; n\_2^3,\; \backslash dots$ 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 $k\; =\; ap$ where a is composite and p is prime, then this can simply be rewritten as $n\; =\; m^k\; =\; m^\{ap\}\; =\; (m^a)^p$. Because of this result, the minimal value of k must necessarily be prime.

See also

• Mihăilescu's theorem

References

External links

• On a series of Goldbach and Euler

• A072103 at the OIES
• Perfect power at Mathworld

Mathematics

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