proof or disprove :If p is a prime number of 5 or more, there is no integer n satisfying $p + 1 = n^2$
first attempt: suppose $p>5$ and $p$ is prime and prove that there is no n integer that satisfy $p+1=n^2$. let $ p=7$, $n= \sqrt {p+1}$ , $n= \sqrt 8$ and is not integer
i think this proof is wrong? because i only proved for p=7
second attempt: let $p=n^2-1 $,$ p=(n+1 )(n-1)$, since $p$ is prime and bigger than 5 by assumption , $n-1=1$, $n=2$, $p=3.2$ , this contradict the fact that p is prime and bigger than 5, hence there is no integer n that satisfy $p + 1 = n^2$ is this looks fine?
now im trying to proof by contrapositive, there is n integer satisfy $ p+1=n^2$ and trying to proof p is prime and p<5
$n=\sqrt {p+1}$ , suppose $p=3$ ,$ n= \sqrt 4$, $n=2$ so there is n integer satisfy $p+1=n^2$, hence If p is a prime number of 5 or more, there is no integer n satisfying $p + 1 = n^2$
please give review thanks! if im not sure that the proposition is correct, how can i proof by contradiction ?