I was asked to prove the Pell's equation
$$x^2-7y^2=1$$
has infinitely many solution. Here is what I did
By using Brahmagupta method we can generate infinitely many integer solutions. Is that supposed to be a proof? If not could you show me how can I prove this formally?
Equation:
$$x^2-7y^2=1$$
Using the first solution: $(x_1 ; y_1) - (8;3)$
You can find the rest of them.
$$x_{n + 1} = 8 x_n + 21 y_n$$
$$y_{n + 1} = 3 x_n + 8 y_n$$