\[\begin{array}{l}
{\text{Решить уравнение в натуральных числах:}} \hfill \\
{2^a} = {3^b} + 5 \hfill \\
\end{array}\]
\[\begin{array}{l}
{\text{Пусть }}k > 5,{\text{ }}m > 3. \hfill \\
{2^k} = {3^m} + 5 \Leftrightarrow {2^5}\left( {{2^{k - 5}} - 1} \right) = {3^3}\left( {{3^{m - 3}} - 1} \right) \hfill \\
{\operatorname{ord} _{27}}2 = 18 \Rightarrow {3^3} \cdot 7 \cdot 19 \cdot 73 = {2^{18}} - 1|{2^{k - 5}} - 1 \hfill \\
{\operatorname{ord} _{19}}3 = 18 \Rightarrow {2^3} \cdot 7 \cdot 13 \cdot 19 \cdot 37 \cdot 757 = {3^{18}} - 1|{3^{m - 3}} - 1 \hfill \\
{\operatorname{ord} _{757}}2 = 756 \Rightarrow {3^4}|{2^{756}} - 1|{2^{k - 5}} - 1.{\text{ Противоречие}}{\text{.}} \hfill \\
\end{array}\]
\[\left( {3;1} \right),{\text{ }}\left( {5;3} \right)\]
