Evaluate the Jacobi Symbol $\dfrac{1111}{8193}$
The way I approached this was:
$$\frac{1111}{8193} = \frac{416}{1111}=\frac{279}{416}=\frac{137}{249}= \frac{5}{137}=\frac{2}{5}=-1$$
And following the Jacobi properties $$\left(\frac{2}{n}\right)=(-1)^{\frac{n^2-1}{8}}= -1$$ if $n=3,5.$
Is my computation correct?
It looks correct. As an alternative, $$\left(\frac{1111}{8193}\right)=\left(\frac{1111}{3}\right)\left(\frac{11}{2731}\right)\left(\frac{101}{2731}\right)=\left(\frac{3}{11}\right)\left(\frac{4}{101}\right)=\left(\frac{11}{3}\right)=-1. $$