Square number ($n^2$)

A square number is an integer of the form $n^2$ for some integer $n$. Consecutive nonnegative squares differ by $(n+1{)}^2-n^2=2n+1$.

The square part of a positive integer $n={\prod }_p{p}^{e_p}$ is its largest square divisor, ${\prod }_p{p}^{2\lfloor e_p/2\rfloor }$. Dividing $n$ by its square part leaves the squarefree part.

For a nonnegative integer $n$, compute its integer square root $r=\lfloor \sqrt{n}\rfloor$ and accept exactly when $r^2=n$. An approximate square root must be corrected with exact integer comparisons before this equality is tested. This procedure distinguishes a squareness test from merely approximating a square root.