\[\begin{array}{l}
{\text{Решите в натуральных числах уравнение:}} \hfill \\
{{\text{5}}^{k!}} - k \cdot {3^{k!}} + 1 = {m^3}. \hfill \\
\end{array}\]
\[\begin{array}{l}
k = 1 \Rightarrow 3 = {m^3},{\text{ }}\emptyset \hfill \\
k = 2 \Rightarrow 8 = {m^3} \Rightarrow \left( {2;2} \right) \hfill \\
{\text{При }}k > 2 \hfill \\
\left. \begin{array}{l}
{{\text{5}}^{k!}} - k \cdot {3^{k!}} + 1 \equiv 2\left( {\bmod 9} \right) \hfill \\
{m^3}\cancel{ \equiv }2\left( {\bmod 9} \right) \hfill \\
\end{array} \right\} \Rightarrow \emptyset \hfill \\
\end{array}\]
\[\left( {2;2} \right)\]
