Integer solutions to an exponential equation $2^x\pm2=5^y$

Question

Are there any integer solutions for the equation $$2^x+2=5^y$$ Similarly, are there any solutions to $$2^x-2=5^y$$ I ask the second because I'm not sure if they are answered similarly. Put qualitatively, are there any powers of 2 and 5 that are exactly 2 apart?

Answer 1

The left side is even if $x>0$. If $x$ is $0$, there is no solution. If $x<0$ then $1<2^x+2<5$ and $2^x-2<0$.

Answer 2

Assuming $x$ and $y$ are positive the answer is no. All powers of $5$ end in $5$. Powers of $2$ end in $2,4,6,8$. Since adding or subtracting $2$ to any of those will not result in $5$ the left hand side never ends in $5$ while the right hand side always does and the two are never equal.