Prime Power — Definition, Formula & Examples

A prime power is a positive integer that can be written as a prime number raised to a positive integer exponent. For example, 8 is a prime power because

8 = 2^3

A positive integer $n$ is a prime power if there exist a prime $p$ and a positive integer $k \geq 1$ such that $n = p^k$ . Under this definition, primes themselves (where $k = 1$ ) are prime powers.

Key Formula

n = p^k

Where:

$n$ = The prime power (a positive integer)
$p$ = A prime number
$k$ = A positive integer exponent (k ≥ 1)

How It Works

To determine whether a number is a prime power, try to express it as

p^k

for some prime

p

. Start by finding the prime factorization of the number. If the factorization contains exactly one distinct prime, the number is a prime power. If two or more distinct primes appear, it is not a prime power.

Worked Example

Problem: Determine whether 81 and 72 are prime powers.

Check 81: Find the prime factorization of 81.

81 = 3^4

Conclusion for 81: Only one distinct prime (3) appears, so 81 is a prime power with p = 3 and k = 4.

Check 72: Find the prime factorization of 72.

72 = 2^3 \times 3^2

Conclusion for 72: Two distinct primes (2 and 3) appear, so 72 is not a prime power.

Answer: 81 is a prime power (

3^4

); 72 is not a prime power.

Why It Matters

Prime powers appear throughout number theory and algebra. Finite fields, which are central to cryptography and error-correcting codes, exist only when their order is a prime power. In modular arithmetic, many theorems (such as the Chinese Remainder Theorem) decompose problems into cases involving prime power moduli.

Common Mistakes

Mistake: Thinking that 1 is a prime power (since

p^0 = 1

Correction: The standard definition requires

k \geq 1

, so the exponent must be at least 1. Since 1 has no prime factor, it is generally not considered a prime power.