Number of ways of proving that a number given in algebraic form is a perfect square or is not.

Question

I've listed a few ways by which it can be proved. Please correct me if any of these are wrong. Are there other possible ways which I'm missing out?

1. If an even number is a perfect square, it must be of the form $4k$.
2. If an odd number is a perfect square, it must be of the form $8k+1$ .
3. If N is a perfect square it cannot be of the form $3n+2, 5n+2, 5n+3, 7n+3, 7n+5, 7n+6$.
4. If a number is not a square number, it can be proved so if we show that it is divisible by a prime but not by its square.
5. (A special case of 4) If a number is not a square number, it can be proved so if we show that it is divisible by a particular prime with an odd number in the power but not an even number (greater than the odd number referred to).