Question

The number of rational points on the circumference of a circle

Answer 1

infinity since there are many points on the circle which cant be predicted rational or irrational

Answer 2

Answer:

As per My knowledge

Step-by-step explanation:

Exactly two

Can you find a circle centered at P with 2 rational points on its circumference? It turns out you can iff π and e are linearly dependent over Q; that is, iff there exist rational numbers p, q, and r such that pπ+qe=r. To see this, first assume such rational numbers exist. Then, note that (π,e) is equidistant from the two points (q,p+rq) and (−q,−p+rq); indeed, the perpendicular bisector of these two points is simply the line px+qy=r, which P lies on. Conversely, assume two rational points Q and R lie on a circle centered at P. Then the perpendicular bisector of the segment QR passes through P. But this perpendicular bisector must have rational coefficients (since Q and R are both rational), which implies that some relation of the form pπ+qe=r exist

On the other hand, any circle with 3 (or more) rational points on its circumference must be centered at a rational point;

suppose e+π were rational. Then (1,π+e+1), (−1,π+e−1) are rational points and are equidistant from (π,e), which implies there is a circle with center (π,e) that contains two rational points.