Find all positive integer $m$ such $$2^{m}+1\mid5^m-1\,.$$
It seem there no solution, I think it might be necessary to use quadratic reciprocity knowledge to solve this problem.
Let $M=2^m+1$, so we have $$5^{m}\equiv 1\pmod{M}. $$ If $m$ is odd, then we have $$\left(5^{\frac{m+1}{2}}\right)^2=5\pmod {M}\Longrightarrow \left(\dfrac{5}{M}\right)=1,$$ so $$\left(\dfrac{M}{5}\right)(-1)^{\frac{(5-1)(M-1)}{4}}=\left(\dfrac{M}{5}\right)(-1)^M=-\left(\dfrac{M}{5}\right).$$