Perfect power

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 : 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: (sometimes 0 and 1), 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, ... 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 .