Pell-Fermat equation I was reading Pell-Fermat equation from Arthur Engel's book Problem Solving Strategies. The equation is $x^2-dy^2=1$, where $ d$ is not a square. It says about finding all the solutions of this equation. Can anyone explain it?I mean  I am not getting the complete solution about what they are trying to do? I don't get it how we are considering $x_0+y_0\sqrt{d}$  a solution for the above equation if $(x_0,y_0)$ is  a solution.Also the portion where it says,"we could  have integrated its square factors",I am not sure what they are trying to mean .... Well, the method demonstrated as follows:

We may assume that d is also square free. If it were not square free we could have integrated its square factors. We associate each number $x+y\sqrt{d}$ with every integer $(x,y)$.
We have the factorization $x^2-dy^2=(x+y\sqrt{d})(x-y\sqrt{d})$. It follows from  here the product and quotient of two solutions of the equation is again a solution of this equation. If $x,y>0$ ,then it follows from $x^2-dy^2=1$ that $(x+y\sqrt{d})$ and $(x-y\sqrt{d})$ are positive. In addition the first one is >1 and the second<1. We consider the smallest solutions $x_{0}+y_{0}\sqrt{d}$. Then we will show that all the solutions given by $(x_0+y_0\sqrt{d})^n$, $n\in\mathbb{Z}$. We will prove this by the ingenious method of descent.  Suppose there is another solution $u+v\sqrt{d}$ which is not a power of $x_0+y_0\sqrt{d}$. Then it must lie between two succeeding powers of $x_0+y_0\sqrt{d}$ that is for some $n$,
$(x_0+y_0\sqrt{d})<u+v\sqrt{d}<(x_0+y_0\sqrt{d})^{n+1}$.
Multiplying with the solution $x_0-y_0\sqrt{d}$, we get
$$1<(u-v\sqrt{d}(x_0+y_0\sqrt{d})^n<(x_0+y_0\sqrt{d})$$
The middle term of the inequality chain is a solution and because it is larger than 1, it is a positive solution. This is a contradiction because we have found a positive solution which is smaller than the smallest positive solution. Thus every solution is a power of the smallest positive solution.

 A: We are simply associating the number $x + y\sqrt d$ to every solution $(x,y)$, then by "abuse of notation" we consider $x + y\sqrt d$ to be the solution itself, instead of $(x,y)$.
No information is lost, and when they say "consider the solution $x_0 + y_0 \sqrt d$" that is abuse of notation for "consider the solution $(x_0, y_0)$".
But the advantage of using a real number $x + y \sqrt d$ is that you can now order these solutions, which is the essential part of the quoted text, whose result is that every solution (as a real number) is $\pm$ of a power of the smallest positive solution (as a real number).

When they say "integrate the square factors", they mean "integrate the square factors into the equation", i.e. if you want to solve $x^2 - 12 y^2 = 1$, where $12 = 2^2 \times 3$, you can rewrite this as $x^2 - 3 (2y)^2 = 1$, where the square factor $2$ is now part of the variable $2y$, and you can make the variable change $z=2y$ to make the equation becomes $x^2 - 3z^2 = 1$, which will include more solutions (since $z$ doesn't need to be even).
