perplexus.info :: Just Math : Rationality of circle

Rationality of circle (Posted on 2025-04-13)

Find the equation of a circle (x − h)² + (y − k)² = r² that has exactly one rational point on it. Also find the equation of a circle that exactly two rational points on it.

No Solution Yet Submitted by Danish Ahmed Khan

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Ideas but unproven

Comment 1 of 1

One point:

If you center the circle at (pi,0) with radius pi it will contain (0,0).

(x-pi)^2+y^2=pi^2

I don't see how it can possibly contain any other rational points.

Two points:

You can make the circle contain (1,0) and (0,1) with radius pi and it center on the line y=x.

Center = (a,a) where a is either of the solutions to

a^2 + (a+1)^2 = pi. (Approximately -1.649 or 0.649)

Again, I don't see how this circle could possibly contain any other rational points.

Posted by Jer on 2025-04-14 09:14:22

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