Solve $x^2 + x + 47 ≡ 0 \pmod {7^3}$.
I don't know how to go about doing this question. I've tried completing the square and other routes but I always seem to end up with horrible answers for $x$ involving surds.
Any help or hints are appreciated.
2026-03-27 08:41:32.1774600892
Solve $x^2 + x + 47 ≡ 0 \pmod {7^3}$
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Let $y=2x+1$, then $x^2+x+47\equiv 0$ is equivalent to $y^2+187\equiv 0$. For $p=7$ this is answered by quadratic reciprocity; for powers we could apply Hensel's Lemma. Actually Qiaochu Yuan has answered the question in general here on MSE.