The Ramanujan-Nagell equation is $$ x^2+7=2^n, $$ where it has been proven (using non-elementary methods) that the complete solution is $n \in \{3, 4, 5, 7, 15\}$.
I've found an elementary way to reduce the theorem to the following conjecture:
Conjecture: If $r$ and $s$ are coprime integers, and $k$ is a positive integer, such that $$ (2^{k+1}-1)(r^2-2rs-s^2)=r^2+5s^2, $$ then $k \in \{1,2,6\}$.
Are there results about factors of $2^q-1$ that might help me prove this easily? It may be a coincidence (e.g., Strong Law of Small Numbers), but the $1,2,6$ made me think of Zagier's famous problem.
EDIT: In case it helps, I can also prove the condition $(r^2-2rs-s^2) \mid 14$.