ModMethod IndeterminateEquation Difficult

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.

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