Are you smarter than an 8th grader? #3

Number Theory Level pending

This problem is part of the question set Mathematics in Anime. It appears in the anime Puella Magi Madoka Magica. Students attending the Math class are only \(14\) years old, yet they are expected to solve the following problem on the spot!

The integer solutions of the equation \(a^{3}+a^{2}-ab+b-1=0\) can be expressed in \((a_1, b_1), (a_2, b_2), \ldots (a_n, b_n)\).

Find \( \sum_{i=1}^n (a_i+b_i)\)

Image and translation credits: Puella Magi Wiki


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