A positive integer of the form p^k where p is a prime number and k is a positive integer (k ≥ 1).

Principle

Construct integers by repeated multiplication of a single prime factor; a prime power has exactly one distinct prime in its factorization and exponent k indicating multiplicity.

Demonstration

27 = 3^3 is a prime power with p=3 and k=3; its only prime divisor is 3 and its divisors are 1,3,9,27.

Misapplication

Treating any integer with a single prime factor up to multiplicity errors (for example mistaking p^0=1 as a prime power in contexts where k must be positive) or confusing prime powers with primes themselves when k>1.

Consequence

Prime powers occupy a clear place in multiplicative structure: their multiplicative order modulo other numbers and behavior under arithmetic functions are simpler to analyze because only one prime contributes.

Reversal

The inverse class consists of integers that are not pure powers of a single prime, i.e., those with at least two distinct prime factors; these have more complex multiplicative structure.

Boundary

Includes 1 only if k=0 is allowed by convention, but standard definition requires k ≥ 1 so 1 is excluded; applies strictly to positive integers and single-prime factorization, excluding composite numbers with multiple distinct primes.

Semantic Tension

Close to the concept of 'prime' but different in that primes correspond to k=1; tensions arise when statements valid for primes fail for higher prime powers (e.g., unique factorization properties of units versus nonunits).

Synthesis

A prime power is an integer formed by taking a single prime number to a positive integer exponent, yielding numbers whose factorization involves exactly one distinct prime.