Problem - 4789
Solve $17^x-15^y=2$ in positive integers.
On obvious solution is $(x, y)=(1,1)$. We are going to show this is the only solution.
If $y\ge 2$, $15^y$ must be a multiple of $9$. Taking MOD $9$ on sides yieds $$(-1)^x -0 \equiv 2\mod{9}$$
which is impossible to hold.