Mathematical proof?

Hello! What methods are there to determine whether a natural number is a square number? I don't mean taking square roots or anything like that. For example, for a number with a special pattern (it doesn't have to be; it can also be a completely normal number), is there a mathematical theorem to show that…

From what number does every number always have 3 divisors where no fractions arise?

Is there a number from which every number has 3 divisors and no number only 2 divisors, for example the number 7 has only 2 divisors, the number 8 has 4 divisors but the number 11 again only 2 divisors? Is there then a number from which all numbers have 4 divisors, etc.