Bashing Fibonacci

Algebra Level 4

\[ \begin{eqnarray} a_{1}+a_{2}+a_{3}+a_{4}+a_{5}&=&1\\ a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}&=&1\\ a_{1}^{3}+a_{2}^{3}+a_{3}^{3}+a_{4}^{3}+a_{5}^{3}&=&2\\ a_{1}^{4}+a_{2}^{4}+a_{3}^{4}+a_{4}^{4}+a_{5}^{4}&=&3\\ a_{1}^{5}+a_{2}^{5}+a_{3}^{5}+a_{4}^{5}+a_{5}^{5}&=&5\\ \end{eqnarray} \]

The value of \[a_{1}^{6}+a_{2}^{6}+a_{3}^{6}+a_{4}^{6}+a_{5}^{6}=\dfrac{m}{n}\]

where \(m , n\) are coprime positive integers.

Find the value of \(m-n\).

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