I've heard that if a number $$$n$$$ has quadratic surplus under a odd prime $$$p$$$ so it has around 50% accuracy to be a perfect square number.

Is it correct? How to prove?

If so,perhaps there exists a method to judge whether a big number $$$n$$$ is a perfect square number or not is that random amount of odd primes which are not divisors of $$$n$$$ and find Quadratic surplus of $$$n$$$ mod $$$p$$$. Failure appears means $$$n$$$ isn't a perfect square number.