Exercise 1.5.4

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Since $9^2-5\times 4^2=1$, the equation $x^2-5y^2=1$ has a solution

which is distinct of the trivial solution $(x_0,y_0)=(1,0)$.

If $(x,y)$ and $(x^{\prime },y^{\prime })$ are solutions, then

thus

so $(x{x}^{\prime }+5y{y}^{\prime },y{x}^{\prime }+x{y}^{\prime })$ is a solution of our equation. Write this solution

We define recursively $(x_n,y_n)$ by

By induction, $(x_n,y_n)$ is a solution of $x^2-5y^2=1$, with

This gives the solutions

To prove that these solutions are distinct, we show that the sequence ${(x_n)}_{n\in \mathbb{N}}$ is strictly increasing.

First, for all $n\in \mathbb{N},n\ne 0$, $x_n>0$ and $y_n>0$ by trivial induction. Thus, if $n>0$,

and $x_0=1<x_1=9$. Therefore ${(x_n)}_{n\in \mathbb{N}}$ is strictly increasing. This shows that the equation $x^2-5y^2=1$ has infinitely many solutions.