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I've been trying to a find a solution to this for a while now, but I can't get anywhere. I need to find nontrivial rational values for x and y such that $$ \sqrt{x^{8}y^{8}+14x^{8}y^{4}+x^{8}+8x^{7}y^{7}+8x^{7}y^{5}-8x^{7}y^{3}-8x^{7}y-16x^{6}y^{6}+32x^{6}y^{4}-16x^{6}y^{2}+8x^{5}y^{7}+8x^{5}y^{5}-8x^{5}y^{3}-8x^{5}y+14x^{4}y^{8}+32x^{4}y^{6}+132x^{4}y^{4}+32x^{4}y^{2}+14x^{4}-8x^{3}y^{7}-8x^{3}y^{5}+8x^{3}y^{3}+8x^{3}y-16x^{2}y^{6}+32x^{2}y^{4}-16x^{2}y^{2}-8xy^{7}-8xy^{5}+8xy^{3}+8xy+y^{8}+14y^{4}+1} $$ is rational. I'm defining "nontrivial" here as not equal to $$ \left(\left(x^{2}+1\right)\left(y^{2}+1\right)\right)^{2} $$ This polynomial is the numerator to $$ \sqrt{1-\left(\frac{\left(\frac{2x+x^{2}-1}{x^{2}+1}\right)^{2}-\left(\frac{2y+y^{2}-1}{y^{2}+1}\right)^{2}}{2}\right)^{2}} $$ and I need to know if it is ever rational for rational x and y. It is for a personal project of mine, but this equation has been a major roadblock for a while now.

Chixen
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    Did you try to use numerical experiments? If yes, which values of $x$ and $y$ are not good? – Botnakov N. Nov 14 '21 at 08:25
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    @BotnakovN. I was not aware of those. I can't seem to find anything about them that I understand as I'm only a beginner to this kind of stuff. What are they and how would I apply some? – Chixen Nov 14 '21 at 18:13
  • There is a lot of information on the Internet about what numerical experiments are and how they are used – Botnakov N. Nov 14 '21 at 19:41

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