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I need to solve the Diophantine equation: $$\frac{x}{\sqrt{2x+3y-5z+6}}+\frac{y}{\sqrt{6x-y+z-8}}+\frac{z}{\sqrt{-8x-2y+4z+5}}=23$$ I found the solution when the arguments of the square roots are perfect squares, but I am unable to prove that it is the only solution.

I was thinking about proving that if: x,y,z are rationals, p,q,r non perfect squares, then: $x.\sqrt{p}+y.\sqrt{q}+z.\sqrt{r}$ is irrational, but I am stuck.

Shaktyai
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    FYI, I found this duplicate using an Approach0 search. To see why it's basically the same question, note that we have $\frac{x}{\sqrt{p}}+\frac{y}{\sqrt{q}}+\frac{z}{\sqrt{r}}=\sqrt{\frac{x^2}{p}}+\sqrt{\frac{y^2}{q}}+\sqrt{\frac{z^2}{r}}$, where each term on the RHS is the square root of a rational that's not a perfect square and, thus, each term is not rational, which means by the duplicate the sum also isn't rational. – John Omielan Sep 21 '24 at 20:42

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