Can someone help me to prove there are infinitely many solutions to the Diophantine equation: $$x^2 − 3y^2 = 1$$ using the method of ascent.
The Method of Ascent:
We can do this by showing how, given one solution $(u, v)$, we can compute another solution $(w, z)$ that is larger is some suitable sense. Then my proof will involve finding a pair of formulas, something like: $w = x + y$ and $z = x − y$. However I tried these formulas and they don't work. So I asked my teacher and she said that there is a pair of second degree formulas which do work; one of them has a cross term and one of them involves the number 3.
I asked this question earlier, but it was put on hold because it looked as a duplicate. However, I am asking to prove this using another method, the method of ascent which is quite different and has not yet been proven in such a way. Can someone please help me on this. I have been wanting an answer for a few days now. I know that the other page has the answer, but I don't care about the answer, I want to know how to do it.