\[\begin{array}{l} {\text{Пусть }}k > 7,m > 3 \hfill \\ {2^k} = {5^m} + 3 \Leftrightarrow {2^7} \cdot \left( {{2^{k - 7}} - 1} \right) = {5^3} \cdot \left( {{5^{m - 3}} - 1} \right) \hfill \\ {\operatorname{ord} _{125}}2 = 100 \Rightarrow 41|{2^{100}} - 1|{2^{k - 7}} - 1 \hfill \\ {\operatorname{ord} _{41}}5 = 20 \Rightarrow 71|{5^{20}} - 1|{5^{m - 3}} - 1 \hfill \\ {\operatorname{ord} _{71}}2 = 35 \Rightarrow 127|{2^{35}} - 1|{2^{k - 7}} - 1 \hfill \\ {\operatorname{ord} _{127}}5 = 42 \Rightarrow 449|{5^{42}} - 1|{5^{m - 3}} - 1 \hfill \\ {\operatorname{ord} _{449}}2 = 224 \Rightarrow 257|{2^{224}} - 1|{2^{k - 7}} - 1 \hfill \\ {\operatorname{ord} _{257}}5 = 256 \Rightarrow {2^{10}}|{5^{256}} - 1|{5^{m - 3}} - 1.{\text{ Противоречие}}{\text{.}} \hfill \\ {\text{Ответ: }}\left( {3;1} \right),{\text{ }}\left( {7;3} \right). \hfill \\ \end{array}\]