Bashing unavailable - part 4

Algebra Level 5

{a+b+c=1a2+b2+c2=2a3+b3+c3=3 \begin{cases} a+b+c= 1 \\ a^2+b^2+c^2= 2 \\ a^3+b^3+c^3= 3 \end{cases}

Let a,ba,b and cc be complex numbers satisfying the system of equations above.

Given that the following equations are true as well for positive integers a1,a2,a3,a4,a5,a6a_1, a_2, a_3, a_4, a_5, a_6 and a7a_7, with gcd(a1,a2)=gcd(a3,a4)=1=gcd(a6,a7)=1\gcd(a_1, a_2) = \gcd(a_3, a_4) = 1 = \gcd(a_6, a_7) = 1 .

Find a1+a2+a3+a4+a5+a6+a7a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7.

{a4+b4+c4=a1a2a5+b5+c5=a3a6+b6+c6=a4a5a7+b7+c7=a6a7 \begin{cases} a^4+b^4+c^4 = \dfrac{a_1}{a_2} \\ a^5+b^5+c^5 = a_3 \\ a^6+b^6+c^6 = \dfrac{a_4}{a_5} \\ a^7+b^7+c^7 = \dfrac{a_6}{a_7} \end{cases}

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