To The Sixth

Algebra Level 5

Suppose that the following holds for complex numbers \(a\) and \(b:\) \[\begin{align} \left(a - \frac{1}{b}\right)\left(b - \frac{1}{a}\right) & = a^2 + ab + b^2 \\ \frac{1}{a} + \frac{1}{b} + \frac{1}{ab} & = a + b. \end{align}\] Then the largest possible real value of \(a^6 + b^6\) can be expressed in the form \[p + \frac{q}{\sqrt{r}},\] where \(p,q,r\) are positive integers, \(q\) and \(r\) are relatively prime, and \(r\) is square-free.

What is the value of \(p + q + r\)?

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