Solve the following equation: $\displaystyle 4^x +2 \equiv 0 \mod 29$.
the equation has no solution: first note that: $\displaystyle 4^x=(2^x)^2.$ now we have $\displaystyle \frac{29 - 1}{2}=14$ and $\displaystyle \frac{29^2 - 1}{8}=105.$ therefore: $\displaystyle \left(\frac{-2}{29} \right)=(-1)^{14}(-1)^{105}=-1. \ \Box$