Bashing Fibonacci

Algebra Level 4

a1+a2+a3+a4+a5=1a12+a22+a32+a42+a52=1a13+a23+a33+a43+a53=2a14+a24+a34+a44+a54=3a15+a25+a35+a45+a55=5 \begin{aligned} 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{aligned}

The value of a16+a26+a36+a46+a56=mna_{1}^{6}+a_{2}^{6}+a_{3}^{6}+a_{4}^{6}+a_{5}^{6}=\dfrac{m}{n}

where m,nm , n are coprime positive integers.

Find the value of mnm-n.

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