Solve the equation $$x^2 \equiv 1729 \mod{7957}. \ \ \ \ (1)$$ What I've tried:
$7957 = 109 \cdot 73$, so $(i) \iff \begin{cases}x^2 \equiv 94 \mod{109} \\ x^2 \equiv 50 \mod 73\end{cases}$. But I don't know how to continue
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There are either $0$ or $4$ solutions mod $7957$, depending on whether $x^2 \equiv 94 \mod{109}$ and $x^2 \equiv 50 \mod 73$ have solutions or not. If one or both of those equations have no solutions, then neither does the original equation. If both of those equations have $2$ solutions each, then the original equation has $4$ solutions mod $7957$. – Geoffrey Trang May 12 '24 at 22:48
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$50\equiv 196 = 14^2 \pmod{73}$ – Sil May 12 '24 at 22:52
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Compute the square roots mod $p$ and mod $q$ using the algorithms in the 2nd link, and lift them to mod $p$ using CRT as in the first dupe link – Bill Dubuque May 12 '24 at 23:13