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?

 

[littlemathquark]

İf $$x>9$$ and $$y>1$$ then another solution let $$a,b\in\mathbb{Z^+}$$ $$(9+a,1+b)$$. Because of $2^{a+9}=3^{b+1}+509$ and $2^9=3+509$ $$2^a(3+509)=509+3^{b+1}$$ $$(2^a-1)509=3(3^b-2^a)$$ But 3 and 509 numbers prime so $$2^a-1=3$$ and $$3^b-2^a=509$$ . But in that case $$3^b=513$$ so b not be a positive number. Hence only solution is $(9,1)$