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Let $C$ be a circle with center $(\pi, e)$. Then is it true that there is atmost one rational point (i.e. point with both coordinates rational ) on the circumference of the circle ?

I can prove that there are atmost two rational points on the circumference of the circle; but I don't know whether that bound can be further reduced to One. If there are two distinct rational points then the perpendicular bisector of the line joining those two points has rational coefficients and it passes through $(\pi, e)$, hence $1, e ,\pi$ are linearly dependent over the rationals. I don't know whether this can happen or not ...

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marked as duplicate by Cesareo, KReiser, Lord Shark the Unknown, José Carlos Santos, Jyrki Lahtonen Jan 1 at 10:35

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  • $\begingroup$ this doesn't seem like geometry to me; perhaps remove that tag $\endgroup$ – Wen May 31 '18 at 9:48
  • $\begingroup$ @Wen: some tag like " geometry of numbers" would be appropriate, but such a tag is unavailable. Hence I put in both the number theory and geometry tags ... $\endgroup$ – user495643 May 31 '18 at 9:51
  • $\begingroup$ mmm but the thing is you've already reduced it to an algebra question (are 1,e,pi linearly independent?) (also, intuitively the answer is 'probably' yes) which isn't really geometry related anymore $\endgroup$ – Wen May 31 '18 at 9:53
  • $\begingroup$ @Wen: um, actually I have not fully reduced it ... I have shown that if two distinct rational points are there then $1,e ,\pi$ are linearly dependent over the rationals , but I don't whether the linear dependence of them does imply the exustence of two distinct rational points ... $\endgroup$ – user495643 May 31 '18 at 10:02
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    $\begingroup$ I believe the question of knowing whether $(e,\pi)$ are $\mathbb{Q}$-independent is still open $\endgroup$ – Max May 31 '18 at 10:43