\[\boxed{\text{669\ (669).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[Пусть\ x_{1},\ x_{2} -\]
\[последовательные\ \]
\[натуральные\ числа.\]
\[x_{2}^{3} - x_{1}^{3} = 919,\ \ x_{2} = x_{1} + 1\]
\[Составим\ уравнение:\]
\[\left( x_{1} + 1 \right)^{3} - x_{1}^{3} = 919\]
\[x_{1}^{2} + 2x_{1} + 1 + x_{1}^{2} + x_{1} + x_{1}^{2} =\]
\[= 919\]
\[3x_{1}^{2} + 3x_{1} - 918 = 0\ \ \ \ \ \ \ \ |\ :3\]
\[x_{1}^{2} + x_{1} - 306 = 0\]
\[D = 1 + 1224 = 1225 = 35^{2}\]
\[x_{1,2} = \frac{- 1 \pm 35}{2} =\]
\[= 17;\ - 18 < 0\ \ \ \ \varnothing\]
\[x_{1} = 17 \Longrightarrow x_{2} = 18\]
\[Ответ:17\ и\ 18.\ \]