1. ## ascent problem

Suppose n is a fixed integer not equal to 0 and d is a positive integer not a perfect square, and suppose

$x_0^2 - dy_0^2 = n$,

where $x_0 \geq 0,\ y_0 \geq 0$. Let $x_1 = ax_0 + bdy_0,\ y_1 = bx_0 + ay_0$, where a>0, b>0 provides a solution to the Fermat-Pell equation, $a^2 - db^2 = 1$. Show that

$x_1^2 - dy_1^2 = n,\ \ \ \ \ x_1 > x_0 \geq 0,\ \ \ \ \ y_1 > y_0 \geq 0$.

Conclude that the equation $x^2 - dy^2 = n$ either has no solutions or infinitely many solutions.

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I do know that the problem is similar to http://www.mathhelpforum.com/math-he...od-ascent.html, but I did not understand something in that thread. How did TPH go from $x^2 - 3y^2 = 1$ to
Originally Posted by ThePerfectHacker
If,
x=a
y=b
Is a solution,
Then,
(a^2+3b^2)^2-3(2ab)^2=(a^2-3b^2)^2=1
Is there a method to that or is it just something you recognize as the right formula to use?

2. Doth this help?

3. Originally Posted by ThePerfectHacker
Doth this help?
honestly, no...the theorem he posted is word for word from the text I am using in my class and I did not really understand it any more than he did. But whereas your answer helped him to understand it, I am still apparently missing something in my comprehension of it...or of how to apply it to my posted problem. So I would still appreciate help on this

4. okay, so in finding another example and then rereading what was here and in the book, I see that those are pretty much just formulas. I was looking for more than that, like you had gotten those from the problem itself somehow but now I see it is just a fomula to be used.

Then back to the problem in my original post. I keep wanting to show the formula simply by substituing in the given formula values and am unsure of whether or not it is that simple or I need to show it in another way. Ideas? Hints? Blatant spoilers?

5. ## Is there anyone out there who can truly answer this question?

Proggy,
I am in the same boat. This class was super easy until this section and then the author just stopped writing. The questions are nothing like the examples. Did you survive the course?

If anyone out there has a solution to this proof please post.