Find integers $x$ and $y$ such that $|5^x - 2^y| = 1$. 
Find integers $x$ and $y$ such that $$\large |5^x - 2^y| = 1$$

Below is a graph of the equation $|5^x - 2^y| = 1$. As can clearly be seen, $(0, 1)$ and $(1, 2)$ are the solutions. I don't know the other answers, maybe there could be none.


$$5^x - 2^y = 1$$
If $y=1$, then there is no integer $x$ such that $5^x=3$.
If $y=2$, then $5^x=5$ implies $x=1$.
If $y\ge 3$, then$$5^x\equiv 1\pmod 8$$
So, we see that $x$ has to be even.
Then, we have
$$1-(-1)^y\equiv 1\pmod 3\implies (-1)^y\equiv 0\pmod 3$$
which is impossible.

$$5^x - 2^y = -1$$
If $y=1$, then there is no integer $x$ such that $5^x=1$.
If $y=2$, then there is no integer $x$ such that $5^x=3$.
If $y\ge 3$, then we get $$5^x\equiv 7\pmod 8$$which is impossible.

Therefore, the only solution for $$|5^x-2^y|=1$$
is $(x,y)=(1,2)$.
