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Example:

What is the Legendre Symbol $ (\frac{3^{24671}}{105953}) $?

Since ($\frac{3}{105953}$) $= -1$ and the exponent p = prime = $24671$ is odd, would this mean the answer would be -1?

Please advise.

Thanks!

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    $\begingroup$ Your reasoning is correct! $\endgroup$ – Greg Martin Dec 6 '19 at 8:44
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By quadratic reciprocity we have $\left(\frac{3}{105953}\right)=\left(\frac{105953}{3}\right)=\left(\frac{2}{3}\right)=-1$, since $105953\equiv 1\bmod 4$, and therefore $$ \left(\frac{3^p}{105953}\right)=\left(\frac{3}{105953}\right)^p=\left(\frac{2}{3}\right)^p=(-1)^p=-1. $$

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