Don't Guess 1

Find all the positive integral solutions to the equation \(y^2 = x^2 + x + 2 \).

If the solutions of \((x,y) \) are \( (x_1, y_1) , (x_2, y_2) , \ldots, (x_n, y_n) \), find \( \displaystyle \sum_{i=1}^n x_i y_i \).

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