Perfect power


In mathematics, a perfect power is a positive integer that can be resolved into equal factors, and whose root can be exactly extracted. i.e., 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 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.

Examples and sums

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 are :
The sum of the reciprocals of the perfect powers 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 is the Möbius function and ζ is the Riemann zeta function.
According to Euler, Goldbach showed that the sum of over the set of perfect powers p, excluding 1 and excluding duplicates, is 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. 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 = 12, n is a perfect 12th power.

Gaps between perfect powers

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.