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Jotadiolyne Dicci
Apr 29, 2020
  ·  Edited: Apr 29, 2020

Linear transformation method for a diophantine equation of 2nd degree with two unknows

in Math Problems

Morning, everyone,


I would like to know how a second-degree diophantine equation with two unknows like this, ax^2 + bxy + cy^2 + dx + ey + f = 0, where a,b,c,d,e and f are integers

can be transformed using linear transformation into this, mu^2 + nv^2 + g = 0, where m,n and g are integers


For example, 9x^2 + 6xy - 13y^2 - 6x - 16y + 20 = 0

There is : 2u^2 - 7v^2 + 45 = 0, where u = 3x + y - 1 and v = 2y + 1


Someone told me that mu^2 + nv^2 + g = 0 could be found using the linear transformation method. How do we get to mu^2 + nv^2 + g = 0 using this ?

Or is there another simpler method?

2 comments
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Faizan Ahmed
Apr 29, 2020

this term contain term xy which indicates that this equation is transformed into some angle with the axes. First we have to remove the xy term by rotating the curve by that angle through which it has been transformed.

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Jotadiolyne Dicci
Apr 29, 2020

So someone just helped me and indeed there are some very quick ways,


indeed
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2 comments