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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?

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Equation:

$$x^2-7y^2=1$$

Using the first solution: $(x_1 ; y_1) - (8;3)$

You can find the rest of them.

$$x_2=8x_1+21y_1$$

$$y_2=3x_1+8y_1$$

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