\[\begin{array}{l} k = 1 \Rightarrow 9 = {m^3},{\text{ }}\emptyset \hfill \\ k = 2 \Rightarrow 35 = {m^3},{\text{ }}\emptyset \hfill \\ k = 3 \Rightarrow 16355 = {m^3},{\text{ }}\emptyset \hfill \\ {\text{Пусть }}k > 3. \hfill \\ {{\text{5}}^{k!}} = {5^{1 \cdot 2 \cdot 3 \cdot 4 \cdot ... \cdot k}} = {\left( {{5^{2 \cdot 3}}} \right)^{4 \cdot 5 \cdot ... \cdot k}} \equiv 1\left( {\bmod 7} \right) \hfill \\ {3^{k!}} = {\left( {{3^{2 \cdot 3}}} \right)^{4 \cdot 5 \cdot ... \cdot k}} \equiv 1\left( {\bmod 7} \right) \hfill \\ \left. \begin{array}{l} {{\text{5}}^{k!}} + {3^{k!}} + 1 \equiv 3\left( {\bmod 7} \right) \hfill \\ {m^3}\cancel{ \equiv }3\left( {\bmod 7} \right) \hfill \\ \end{array} \right\} \Rightarrow {{\text{5}}^{k!}} + {3^{k!}} + 1 \ne {m^3}. \hfill \\ \end{array}\]

\[\emptyset \]