How to calculate Jacobi Symbol $\left(\dfrac{27}{101}\right)$?
The book solution $$\left(\dfrac{27}{101}\right) = \left(\dfrac{3}{101}\right)^3 = \left(\dfrac{101}{3}\right)^3 = (-1)^3 = -1$$
My solution $$\left(\dfrac{27}{101}\right) = \left(\dfrac{101}{27}\right) = \left(\dfrac{20}{27}\right) = \left(\dfrac{2^2}{27}\right) \cdot \left(\dfrac{5}{27}\right)$$ $$= (-1) \cdot \left(\dfrac{27}{5}\right) = (-1) \cdot \left(\dfrac{2}{5}\right) = (-1) \cdot (-1) = 1.$$
Whenever I encounter $\left(\dfrac{2^b}{p}\right)$, I use the formula $$(-1)^{\frac{p^2 - 1}{8}}$$ I guess mine was wrong, but I couldn't figure out where? Any idea?
Thank you,