Absolutely not. A square number has an integer square root, so by definition it has at least one factor. Prime numbers have no factors
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There are no prime numbers that are square numbers
Every square number (except 1) is composite. Prime numbers only have two factors, one and the numbers themselves. Since square numbers also have at least the square roots as factors, they have to be composite.
No. That isn't possible: A prime number, by definition, has no smaller factors. A square number does have a smaller factor - the number that is squared.
The products of a square of a prime and a prime, like 12.
Prime numbers have two factors. All square numbers (other than 1) have more than that.