Primepower

Prime power is a positive integer that can be written as p^k, where p is a prime number and k is a positive integer. If k = 1, the number is simply a prime; otherwise it is a higher prime power. Examples include 2, 3, 5, 7 (primes) and 4 = 2^2, 8 = 2^3, 9 = 3^2, 16 = 2^4.

Characterizations and basic properties. A prime power has exactly one distinct prime factor in its prime factorization. Its divisors are 1, p, p^2, ..., p^k, so there are k + 1 divisors. The Euler totient of a prime power is φ(p^k) = p^k − p^(k−1), and the sum of its divisors is σ(p^k) = (p^(k+1) − 1)/(p − 1).

Occurrence and context. An integer n > 1 is a prime power if and only if its prime factorization contains only a single prime, i.e., n = p^k for some prime p and k ≥ 1. Prime powers arise in various areas of number theory, including p-adic concepts, valuations, and modular arithmetic modulo p^k. In algebraic contexts, rings such as Z/p^kZ frequently feature prime powers in their structure and properties.

Related concepts. A related term is "perfect power," which refers to any number that can be written as a^b with a > 1 and b > 1, not necessarily with a prime base. Prime powers form a subset of perfect powers.