Prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
first im trying to proof if p is prime number there is positive integer n such that $p+1=n^2$ implies $p=3$
suppose there is positive n integer such that $p+1=n^2$ and prove that $p=3$. $p=n^2-1=(n+1)(n-1)$ since $p$ is prime $n-1=1$,$ n = 2 $ $p+1=4$ $p=3$ hence it is proved that $p=3$
now if $p=3$ implies there is positive integer n such that $p+1=n^2 $
$p=3$ and p is prime
$p+1=n^2$,
$4=n^2$,
$n2$ or $n=-2$
hence it is proved that there is positive integer $n$ such that $p+1=n^2 $
is my proof correct? and also for prime number can i write $n+1=1$ instead $n-1=1$?