 Square factor

A square factor of an integer $n$ is a positive perfect square $m^2$ dividing $n$. Every integer has the trivial square factor one; usually the useful question is whether a nontrivial one, with $m>1$, occurs.

A positive integer has a nontrivial square factor if and only if some prime number $p$ satisfies $p^2\mid n$. One direction takes $m=p$; for the other, choose any prime factor of $m$. Equivalently, some prime exponent in $n$ is at least two. For example, twelve has square factor four and eighteen has square factor nine, whereas thirty has none other than one. A positive integer without a nontrivial square factor is a square-free integer.