Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - a x - b = 0$ with $\operatorname { Im } ( \alpha ) < \operatorname { Im } ( \beta )$. Let $P _ { n } = \alpha ^ { n } - \beta ^ { n }$. If $\mathrm { P } _ { 3 } = - 5 \sqrt { 7 } i , \mathrm { P } _ { 4 } = - 3 \sqrt { 7 } i , \mathrm { P } _ { 5 } = 11 \sqrt { 7 } i$ and $\mathrm { P } _ { 6 } = 45 \sqrt { 7 } i$, then $\left| \alpha ^ { 4 } + \beta ^ { 4 } \right|$ is equal to