Pell's equation (simple solutions)

Pell's equation is any Diophantine equation of the form $x^2 – Dy^2 = 1,$ where $D$ is a given positive nonsquare integer, and integer solutions are sought for $x$ and $y.$

Denote the sequence of solutions ${x_i, y_i}.$ It is clear that ${x_0, y_0} = {1,0}.$ During the solution we need: a) to construct a recurrent sequence ${x_{i+1}, y_{i+1}} = f({x_i, y_i})$ or two sequences ${x_{i+1}} = f({x_i}), {y_{ i+1}} = g({y_i});$ b) to prove that the equation has no other integer solutions.

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Pell's equation (simple solutions)

Equation of the form $x^2 – 2y^2 = 1$