# Is there atmost one rational point on the circumference of a circle with center $(\pi, e)$? [duplicate]

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 ...

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

• @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 ... – user495643 May 31 '18 at 10:02
• I believe the question of knowing whether $(e,\pi)$ are $\mathbb{Q}$-independent is still open – Max May 31 '18 at 10:43