Condition for this expression to be prime.

$p^2\equiv 0,1 \mod{3}$ and $p^2+8 \equiv p^2+2 \equiv 2 \text{ or } 0 \mod{3}$ . For $p^2 + 8$ to be a prime number, it should not be divisible by 3. Therefore, $p^2 \equiv 0 \mod{3} \implies p = 3$ . Then, $p^3 +ap+2 = 29+3a$ and the smallest $a$ that makes this prime is $a = 4$ where $29+3a = 41$ .