We need to solve for prime numbers, $p_i$, $i=1,2,\cdots, 8$ such that $$\sum_{i=1}^8p_i^2 + 992 = 4\prod_{i=1}^8p_i$$

If any $p_i$ is odd, then $p_i^2\equiv 1\pmod{8}$. Therefore, if all the $p_i$ are odd, the left side of the above equation is a multiple of $8$, but the right side is not. This means that at least one of $p_i$ must be even.

However, if at least one of $p_i$ is even, the right side is divisible by 8. In this case, the left side cannot be a multiple of $8$ unless all the $p_i$ are even.

There is only one even prime which is $2$. Setting all $p_i$ as $2$ does satisfy the above equation. Therefore, the only solution is $$p_1=p_2=\cdots=p_8=2$$