Deduce that a prime $p$ is a quadratic residue mod $17$ where $kp^2 \equiv t^4 \mod{17}$

Question

Wondering if it's possible to deduce that a prime $p$ is a quadratic residue mod $17$ based on the information that

$kp^2 \equiv t^4 \mod{17}$, where $k, r \in \mathbb{Z}, r \neq 0$? Or if this isn't necessarily true (in which case I've gone wrong somewhere).

Answer 1

Take $t=1, k=p^{-2}:$ this shows that every prime ($\not=17$) satisfies the condition, while not every prime is a residue w.r.t. $17.$
Hope this helps.