In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo pn (that is, the group of units of the ring Z/pnZ) is cyclic.
[1] The number of elements of a finite field is always a prime power and conversely, every prime power occurs as the number of elements in some finite field (which is unique up to isomorphism).
[3] The totient function (φ) and sigma functions (σ0) and (σ1) of a prime power are calculated by the formulas All prime powers are deficient numbers.
It is not known whether a prime power pn can be a member of an amicable pair.