I have asked a similar question here before, and received a nice answer. I think that the next question here is equivalent, but can't seem to be able to prove it. Here goes:
Given an odd $n$, I want to find an $0\leq m\leq n-1$ s.t $m\in\left(\mathbb{Z}/n\mathbb{Z}\right)^{*}$ (i.e, $m$ is invertible modulu $n$), and also $\left(1-m^{-1}\right)\in\left(\mathbb{Z}/n\mathbb{Z}\right)^{*}$ and $\left(\frac{m}{n}\right)=1$ when $\left(\frac{m}{n}\right)$ is the Jacobi symbol.
I can prove that if we disregard the Jacobi symbol requirement, this is equivalent to finding to succeeding invertible numbers modulu $n$. But when we throw the Jacobi symbol in to the equation, I'm not sure if this is the same as my earlier question. If it is, I would greatly appreciate a proof. If not, a new answer :).
Thanks alot!