A number Theory Question: Solve 2^x=3^y+509 over positive integers

[littlemathquark]

Homework Statement

Question about number theory

Relevant Equations

Solve $2^x=3^y+509$ over positive integers.

My attempt and solution :
$$2^x=3^y+509\Longrightarrow 2^x-512=3^y+509-512\Longrightarrow 2^x-2^9=3^y-3$$
$$\Longrightarrow 2^9(2^{x-9}-1)=3(3^{y-1}-1)$$
$$\Longrightarrow (x,~y)=\boxed{(9,~1)}$$
İs there any solution?

 

[nuuskur]

Yes there is, you found one.

It would help to explain the last implication to the uninitiated. As for uniqueness:

Suppose $y\geqslant 2$. It is enough to verify that $2^x\in\{6,7\}\pmod{13}$ and $3^y\in \{1,3,9\} \pmod{13}$. None of these combinations satisfy $2^x \equiv 3^y+2\pmod{13}$. So $y=1$ is mandatory.