In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = 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 0 and 1 are also considered perfect powers (0k = 0 for any k > 0, 1k = 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 A072103 in the OEIS):
The sum of the reciprocals of the perfect powers (including duplicates such as 34 and 92, both of which equal 81) 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 . So if the divisors of are then one of the values 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 for a composite where p is prime, then this can simply be rewritten as . Because of this result, the minimal value of k must necessarily be prime.
If the full factorization of n is known, say where the are distinct primes, then n is a perfect power if and only if where gcd denotes the greatest common divisor. As an example, consider n = 296·360·724. Since gcd(96, 60, 24) = 12, n is a perfect 12th power (and a perfect 6th power, 4th power, cube and square, since 6, 4, 3 and 2 divide 12).
In 2002 Romanian mathematician Preda Mihăilescu proved that the only pair of consecutive perfect powers is 23 = 8 and 32 = 9, thus proving Catalan's conjecture.
Pillai's conjecture states that for any given positive integer k there are only a finite number of pairs of perfect powers whose difference is k. This is an unsolved problem.