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. 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.


marked as duplicate by Will Jagy, user147263, apnorton, Lucian, davidlowryduda Feb 28 '15 at 20:50

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    $\begingroup$ How can the same thing 100 times to ask? You especially don't read other topics? $\endgroup$ – individ Feb 28 '15 at 17:44
  • $\begingroup$ @WillJagy It is not a duplicate because I want to prove using the method of Ascent. $\endgroup$ – Rebecca Feb 28 '15 at 19:11

$(2x+3y)^2-3(x+2y)^2=x^2-3 y^2$

  • $\begingroup$ How did you get to that? $\endgroup$ – Rebecca Feb 28 '15 at 18:12
  • $\begingroup$ has this something to with the continued fraction of $\sqrt 3?$ $\endgroup$ – abel Feb 28 '15 at 23:22

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