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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).

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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.

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  • $\begingroup$ The inverse here means the inverse modulo $17.$ $\endgroup$ – awllower Feb 24 '14 at 14:19

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