Question

Why is a prime number not a square number?

Answer 1

A square number is of the form `axxa`.

Explanation:

This means it has factors other than `1` and itself.

OK, I admit:
`1=1xx1` is square, but `1` is (by definition) excluded from the list of primes. This is because a prime has two different dividors (1 and itself) and 1 has only one.

Answer 2

A prime number has `2` factors and squares have an odd number of factors.

Explanation:

All square numbers have an odd number of factors.

`1` has the factor `1`
`4` has the factors `1,2,4`
`9` has the factors `1,3,9`
`16` has the factors `1,2,4,8,16`

`x^2` has the factors `1, x, x^2`

A prime number by definition has exactly `2` factors - `1` and itself.

Therefore no prime number is a square and no square number is prime.