\[\begin{array}{l} {\text{Пусть }}x > 2,{\text{ }}y > 1. \hfill \\ {4^x} = {7^y} + 9 \Leftrightarrow {2^4} \cdot \left( {{4^{x - 2}} - 1} \right) = 7 \cdot \left( {{7^{y - 1}} - 1} \right) \hfill \\ {\operatorname{ord} _7}4 = 3 \Rightarrow {3^2} \cdot 7 = {4^3} - 1|{4^{x - 2}} - 1 \hfill \\ {\operatorname{ord} _{16}}7 = 2 \Rightarrow {2^4} \cdot 3 = {7^2} - 1|{7^{y - 1}} - 1 \hfill \\ {\operatorname{ord} _9}7 = 3 \Rightarrow 2 \cdot {3^2} \cdot 19 = {7^3} - 1|{7^{y - 1}} - 1 \hfill \\ {\operatorname{ord} _{19}}4 = 9 \Rightarrow {3^3} \cdot 7 \cdot 19 \cdot 73 = {4^9} - 1|{4^{x - 2}} - 1 \hfill \\ {\operatorname{ord} _{73}}7 = 24 \Rightarrow {2^6}|{7^{24}} - 1|{7^{y - 1}} - 1.{\text{ Противоречие}}{\text{.}} \hfill \\ \left( {x;y} \right) = \left( {2;1} \right). \hfill \\ \end{array}\]

\[\left( {x;y} \right) = \left( {2;1} \right)\]