[tex]un~cub~perfect~poate~fi~de~forma~\mathcal{M}_9,~\mathcal{M}_9+1,\mathcal{M}_9+8.\\3^x~este~\mathcal{M}_9~daca~x\neq 1,x\in\mathbb{N}.\\----------------\\n^3-3^m=2022~\Big|+3^m \implies n^3=2022+3^m,\\observam~ca~membrul~drept~este~de~forma~\mathcal{M}_9+6,~pentru~n\geq 2,\\dar~un~cub~perfect~nu~poate~fi~de~forma~\mathcal{M}_9+6,~deci~n\in\left\{0,1 \right\}.\\daca~n=0 \implies 2023=n^3,~nu~convine.\\daca~n=1 \implies 2025=n^3,~nu~convine.\\[/tex]
[tex]prin~urmare,~nu~exista~numere~naturale~m~si~n~care~sa~verifice~:~\\n^3-3^m=2022.[/tex]
