Question

Aratati ca nu exista numere naturale n si m pentru care n³- 3 la m =2022​

Answer 1

$\displaystyle\it\\un~cub~perfect~poate~fi~de~forma~\mathcal{M}_9,~\mathcal{M}_9+1~sau~\mathcal{M}_9+8.\\----------------\\n^3-3^m=2022 \Leftrightarrow n^3=2022+3^m.\\daca~m\geq 2,~atunci~2022+3^m=\mathcal{M}_{9}+6,~ceea~ce~ne~indica~ca~\\un~cub~perfect~poate~fi~de~forma~\mathcal{M}_9+6,~ceea~ce~evident~este~fals.\\deci,~m\in\left\{0,1\right\}.\\daca~m=0 \implies n^3=2023,~nu~convine.\\daca~m=1 \implies n^3=2025,~nu~convine.$