I have been told that a simple linear transformation (or a change of variables) can transform the quadratic $$x^2+45xy-216y^2$$ into the Pell equation $$p^2−321q^2=1$$ However I have been unable to achieve this. Can anyone help me find a simple linear transformation (or a change of variables) to arrive at the Pell equation above? I want to eliminate the xy term in the quadratic.
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Complete the square & put some $'s around your formulae. – Donald Splutterwit Feb 14 '20 at 04:41
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is the quadratic supposed to be an equation? – J. W. Tanner Feb 14 '20 at 04:41
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The quadratic is an expression and it is part of a larger cubic equation. – Derak Feb 14 '20 at 04:44
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How do I complete the square ? – Derak Feb 14 '20 at 04:51
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1$x^2+2cxy+dy^2= (x+cy)^2+dy^2-c^2y^2$ – J. W. Tanner Feb 14 '20 at 04:54
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1I noticed that when you edited you changed the signs on $45$ and $216$ – J. W. Tanner Feb 14 '20 at 04:55
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Yes I corrected the sign. I made a mistake initially. Does a sign change make a lot of difference ? – Derak Feb 14 '20 at 04:57
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@J.W.Tanner When I complete the square , according to the formula you have given, I end up with the Pell Equation $$ p^2 - 722.25 q^2 = 1 $$ – Derak Feb 14 '20 at 05:12
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2Note that $722.25=321(1.5)^2$ – J. W. Tanner Feb 14 '20 at 05:16
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1@J.W.Tanner Thank you very much. So the change of variables I wanted was 1.5y = q and x+ 22.5 y = p . – Derak Feb 14 '20 at 05:21
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Let $p=x+\dfrac{45}2y$ and $q=\dfrac32y$.
Then $p^2-321q^2=x^2+45xy+\dfrac{2025}4y^2-321\dfrac94y^2=x^2+45xy-216y^2.$
J. W. Tanner
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Im very late for this party, but in case you've ever wondered what the context of this apparently random Pell equation was, the OP came across my 2013 post on "Ramanujan's sum of cubes identity" and is trying to specialize the identity, $$(3(3p^2 - 104p q + 909q^2))^3 + (-2(4p^2 - 135p q + 1119q^2))^3 = (6(p^2 - 37p q + 348q^2))^3 + (p^2 - 321q^2)^3$$ to the form $a^3+b^3 = c^3+1$ by solving the Pell equation $p^2 - 321q^2 = 1$. More details in the link. Just in case you wondered. :) – Tito Piezas III Dec 26 '23 at 13:38