## Task 13/79

All natural numbers $x$ and $y$ are to be determined for which the equation $5^{x}-2^{y}=3$ holds.
If $y>1$, then $2^{y} \equiv 0(\bmod 4)$. Since $5^{x} \equiv 1(\bmod 4)$, it would then follow that $5 x-2^{y} \equiv 1-0 \equiv 1 \neq 3(\bmod 4)$. Therefore, there is no solution for $y>1$. For $y=1$, the given equation takes the form $5 x-2=3$, from which $x=1$ immediately follows. Thus, $(x ; y)=(1 ; 1)$ is the only solution pair.