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QUESTION

Find a solution of $x^2+1\equiv0$ mod 17.



ANSWER

The alert will spot $\pm4$ as roots immediately! if you didn't
notice, then as $17\equiv1$ mod 4, we can appeal to the method of
th.4.6 to deduce that the roots are
$\pm\left(\frac{(p-1)}{2}\right)!$ where $p=17$. Thus the roots
are $\pm(8!)$ mod 17.

$8!=8.7.6.5.4.3.2=56.30.24\equiv5.(-4).7\equiv5.(-28)\equiv5.6\equiv
30\equiv-4$ mod 17, showing that the roots are $\pm4$ mod 17, as
spotted!




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