How to calculate Jacobi Symbol $\left(\dfrac{27}{101}\right)$? 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,

I think it's better to make sure that the number in the lower case is a prime, since there are examples, if I remember rightly, that the Jacobi symbol is 1 but the corresponding quadratic congruence is not solvable; in addition, as already mentioned, you cannot say that $\left(\dfrac{2^b}{p}\right)\ = (-1)^{(p^2 -1)/8}$; it is a mistake without second thought, and I think it can be well avoided if you know the quadratic reciprocity law well, thanks.
