Doing some personal research I just stumbled upon this problem:
Given an integer $m\in \mathbb{Z}$ that is coprime to 10, I am interested in whether or not there exists an integer $n\in\mathbb{Z}$ such that $10^{n} \equiv 91^{\phi(m)-1}$ mod m.
We can further ask for if and how we can construct such a number n and one might also investigate the function you get by varying the number 91. But for now I am mainly interested in this very specific case.
I hope I can gather some inspiration from you guys. Thanks in advance :)