The only prime of the form n^2-4 is 5?

[Math100]

Homework Statement

Prove the assertion below:
The only prime of the form n^2-4 is 5.

Relevant Equations

None.

Proof: Suppose p is a prime such that p=n^2-4.
Then we have p=n^2-4=(n+2)(n-2).
Note that prime number is a number that has only two factors,
1 and the number itself.
Since n+2>1 for $\forall n \in \mathbb{N}$,
it follows that n-2=1, and so n=1+2=3.
Thus p=n^2-4=3^2-4
=9-4
=5.
Therefore, the only prime of the form n^2-4 is 5.

Above is my proof for this assertion. Can anyone please verify/review it and see if it's correct?

 

[fresh_42]

Same here. Correct.

 

[Math100]

Same here. Correct.

Thank you!