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For a nonnegative integer $n$, its integer square root is the unique nonnegative integer $r$ satisfying $r^2\le n<(r+1{)}^2$, equivalently $r=\lfloor \sqrt{n}\rfloor$. Testing $r^2=n$ decides whether $n$ is a perfect square using exact integer arithmetic.

It performs a trial division test for a prime number. An integer below $2$ is rejected; otherwise the function searches for a divisor from $2$ through $\lfloor \sqrt{n}\rfloor$. A composite number has a prime factor no larger than its square root, so finding no such divisor proves primality. For example, mystery2(97) returns True, while mystery2(91) returns False, since $91=7\cdot 13$.

This describes the intended mathematical test. For extremely large integers, the floating-point computation n**0.5 can round or overflow; an implementation intended for arbitrary-size integers should use an exact integer square root.

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.