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I want to find a solution to the equation below (or show that it doesn't have any solutions) \begin{equation} 2x^2 - 44 \equiv 368y + 138z\pmod{46} \end{equation} So far I've only learned how to solve equations on the form $x^2 + 1 \equiv 0 \pmod{p}$, and I can't find any examples in my textbook on how to solve equations as the one above. I have no idea how to solve this. Any help is much appreciated.

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$$2x^2 - 44 \equiv 368y + 138z\pmod{46} \iff x^2 - 22 \equiv 184y + 69z\pmod{23}$$

$$\iff x^2 +1 \equiv 23(8y+3z)\pmod{23}\iff x^2 +1 \equiv 0\pmod{23}$$

As $23\equiv3\pmod{4}$, $-1$ is not a quadratic residue $\pmod{23}$, and the equation is insoluble.

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