Why is a prime number not a square number?

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Why is a prime number not a square number?

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Answer 1 · 2016-12-27T03:04:48

A square number is of the form $a \times a$ $axxa$ .

Explanation:

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

OK, I admit:
$1 = 1 \times 1$ $1=1xx1$ is square, but $1$ $1$ 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 · 2017-10-07T19:27:10

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

Explanation:

All square numbers have an odd number of factors.

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

$x^{2}$ $x^2$ has the factors $1, x, x^{2}$ $1, x, x^2$

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

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

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Why is a prime number not a square number?

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