prime number

[Sidharth]

If both p and $p^2+8$ are prime then prove that $p^3+4$ will also be a prime.

[asif e elahi]

If $p=2$,Then $p^{2}+8=12$ is not a prime.If $p$ is odd,then $p^{2}+8 \equiv 1+8\equiv 9\equiv 0(mod 3)$. So $3$ divides $p^{2}+8$.So there is no prime $p$ such that,$p^{2}+8$ is a prime.

[Sidharth]

let,p=3.Now find $p^2+8$ and $p^3+4$

[asif e elahi]

oh,$3$ is the only solution.