Show that $p^3+4$ is prime
If $p$ and $p^2 +8$ are both prime number, prove that $p^3 +4$ is also prime.
I was thinking using two cases for even and odd. So $p$ is even thus there is no prime, the statement does not hold. So, $p$ is odd, $p$ has to be of the form of $2k+1$.
let $p=2k+1$ which is prime
$(2k+1)^2 + 8 = 4k^2+4k+1+8 = 2(2k^2+2k+4)+1$ which is prime
so $(2k+1)^3 +4 = 8k^3+12k^2+6k+1 +4= 2(4k^3+6k^2+3k+2)+1$ which is also prime.
is that all that i need to show
No, as commenters have points out, all you've shown is that $p^3+4$ is odd.
Hint: What must $p$ be, modulo $3$?