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A [perfect] square number, by definition, has a factor which is its square root. As a result it CANNOT be a prime!
No, there are no prime numbers that are also square numbers. Prime numbers are only divisible by 1 and themselves, while square numbers have integer square roots. Since the square root of a prime number is not an integer, a prime number cannot be a square number.
All real numbers are irrational. For example, Pi is an irrational number that is a real number. Other irrational numbers can be the square root of an imperfect square.
Square numbers, greater than 1, have more than two factors.
The likely square numbers are 36 and 1. (6^2 - 1^2) However, the square numbers 324 and 289 also work. (18^2 - 17^2)