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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ... In mathematics, a perfect cube or cube number, is an integer that can be written as the cube (arithmetic) of some other integer. ...

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: In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

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: Euler redirects here. ... Christian Goldbach (March 18, 1690 - November 20, 1764), was a Prussian mathematician, who was born in Königsberg, Prussia, as son of a pastor. ... Converge denotes Converge PL a programming language developed by Laurence Tratt Converge, a metalcore band from Massachusetts For the mathematical meaning of this term see Convergence. ...

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 factors of n are then one of the values must be equal to n if n is indeed a perfect power. As a branch of the theory of computation in computer science, computational complexity theory describes the scalability of algorithms, and the inherent difficulty in providing scalable algorithms for specific computational problems. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

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. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ... The term composite can refer to several different things: A dental composite is an type of tooth filling material made of a plastic matrix containing high-strength quartz filler particles. ... In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ...

See also

• Mihăilescu's theorem

Mihăilescus theorem is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and finally proven in 2002 by Preda Mihăilescu. ...

References

• Daniel J. Bernstein (1998). "Detecting perfect powers in essentially linear time". Mathematics of Computation 67 (223): 1253–1283.

External links

• On a series of Goldbach and Euler

• A072103 at the OIES
• Perfect power at Mathworld