No. To be prime it has to be divisible only by itself and one. That means that no numbers multiply to be it. The only possible exception is one. In the math world it may have been decided whether or not one is a Prime number, but that is debatable and I'm not aware of the answer. If 1 was considered a prime number, then it would be the only prime square.
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4, which is the square of the prime number 2, is the smallest composite number.
The square root of 169 is 13 which is a rational and also a Prime number
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.
4
The square root of 121 is 11 which is a rational number and also a prime number