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November 9th, 2018, 11:38 PM   #1
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Number of rational points

The number of rational points on the circle with center $(\sqrt{2},-\sqrt{2})$ and which passes through $(1,-1)$ is
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November 10th, 2018, 12:47 AM   #2
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At a guess, 1.
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November 10th, 2018, 06:10 PM   #3
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For $x=1$ and $y=-1$,

$\displaystyle (x-\sqrt2)^2+(y+\sqrt2)^2=6-4\sqrt2$

Generally, if $x$ and $y$ are rational,

$\displaystyle x^2-2\sqrt2x+2+y^2+2\sqrt2y+2=6-4\sqrt2$

$\displaystyle x^2+y^2+2\sqrt2(y-x)=2-4\sqrt2$

$\displaystyle \implies x^2+y^2=2$

$\displaystyle \implies x-y=2$

$\displaystyle (y+2)^2+y^2=2$

$\displaystyle 2y^2+4y+2=0\implies y=\frac{-4\pm0}{4}$

$\displaystyle x^2+(x-2)^2=2$

$\displaystyle 2x^2-4x+2=0\implies x=\frac{4\pm0}{4}$

Looks like skipjack's guess is correct.
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Last edited by greg1313; November 10th, 2018 at 06:44 PM.
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