Does $x^2=83\pmod{101}$ have solutions? without calculating them. I'm not sure how to tackle this without solving, I tried using chinese remainder and quadratic reciprocity.
2026-03-30 00:18:27.1774829907
Does $x^2=83\pmod{101}$ have solutions? without calculating them
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Can you use the supplements to quadratic reciprocity? If yes, the problem is easily solved by noting that $83 \equiv -18 \equiv -1 \cdot 2 \cdot 3^2 \mod 101$. $-1$ is a square modulo $101$ and $2$ isn't, so $83$ is not a square modulo $101$.