I have the following question.
Calculate the Legendre symbol $\bigl(\frac{77}{5^{200}+1}\bigr)$.
I know the following: $5^6\equiv1\pmod7$, $5^{10}\equiv1\pmod{11}$. Thus I approached this problem as follows:
\begin{eqnarray} \left(\frac{77}{5^{200}+1}\right) & = & \left(\frac{7}{5^{200}+1}\right)\left(\frac{11}{5^{200}+1}\right)\\\\ & = & \left(\frac{5^{200}+1}{7}\right)\left(\frac{5^{200}+1}{11}\right)\\\\ & = & \left(\frac{5^{(6)33+2}+1}{7}\right)\left(\frac{5^{(20)(10)}+1}{11}\right)\\\\ & = & \left(\frac{25+1}{7}\right)\left(\frac{1+1}{11}\right)\\\\ & = & \left(\frac{26}{7}\right)\left(\frac{2}{11}\right)\\\\ & = & \left(\frac{5}{7}\right)(-1)=(-1)\left(\frac{7}{5}\right)=(-1)\left(\frac{2}{5}\right)\\\\ & = & (-1)(-1)\left(\frac{1}{2}\right)=1 \end{eqnarray}
I am skeptical about my answer because of the beginning part where I flipped the Legendre symbol. Any suggestion?
Thank you for your time and thanks in advance for your feedback.