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I am trying to solve the system of diophantine equations below where $x,y$ are coprime

\begin{eqnarray} xr^2+ys^2&=&uva\\ xr+ys&= &uvb\\ x+y&=&uc \end{eqnarray}

I solve 2 out of 3 then replace in the remaining one. However, things get messy at this point. I am unable to close the argument.Any hints?

  • I don't see any tries to solve that. It's understandable, 3 equations with 11 unknowns look sort of challenging, and there's also the nagging "what for?" you didn't try to address. –  Jan 04 '18 at 19:51
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    What is given and what are the variables? – Dietrich Burde Jan 04 '18 at 19:53
  • I am trying to find $x,y$ such that the system remains consistent. –  Jan 04 '18 at 20:01
  • With specific integer values, it's simple. I am trying to find a general method. –  Jan 04 '18 at 20:07

1 Answers1

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The system may not have any solution. For example, if $(r,s,u,v)=(1,1,5,6)$, then there is no solution for $x,y$. In general we can use that $u$ divides $x+y$, $xr+ys$ and $xr^2+ys^2$. Similarly $v$ divides $xr+ys$ and $xr^2+ys^2$.

Dietrich Burde
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