When trying to see if a number of the form $n^8-n^4+1$ can be divisible by the square of a prime, I found that it can indeed. The first few values for $n$ are

412, 786, 1417, 1818, 2430, 2640, 2809, 2822, 2899 ...

and the first few such primes $p$ (in increasing order) are

73, 97, 193, 241, 313, 337, 409, 433, ...

Interestingly enough, the latter is precisely the beginning of this sequence which lists the primes of the form $x^2+24y^2$. I am quite sure that this cannot be a pure coincidence and that some deep number theory must be involved. The number $24$ is not accidental either, as $n^8-n^4+1=\Phi_{24}(n)$, with $\Phi_k(x)$ being the $k$th cyclotomic polynomial. Maybe there is some relation to the field $\mathbb{Q}(\zeta_{24})$...

So, the question is if the following is true:

**Conjecture**. A prime $p$ has the form $x^2+24y^2$ if and only if $p^2$ divides $n^8-n^4+1$ for some $n$.