$[$ Arithmetic of residues (miscellaneous) $]$ $[$ Equations in integers $]$

Solve the equation $2^{x}-1=5^{y}$ in integers.
Obviously, $x>0$. For $x>1$, the left side gives a remainder of 3 when divided by 4, while the right side always gives a remainder of 1.

## Answer

$x=1, y=0$.