Definition:Square-Free Integer

Definition

Let $n \in \Z$ be an integer.

Then $n$ is a square-free integer if and only if $n$ has no divisor which is the square of a prime.

That is, if and only if the prime decomposition $n = {p_1}^{k_1} {p_2}^{k_2} \cdots {p_r}^{k_r}$ is such that:

$\forall i: 1 \le i \le r: k_i = 1$

Examples

Example: $15$

The integer $15$ is square-free:

$15 = 3 \times 5$

Non-Example: $28$

The integer $28$ is not square-free:

$28 = 2^4 \times 7$

Also see

• Definition:Radical of Integer

• Results about square-free integers can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): square-free
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): square-free