If you travel around a circle and mark all the points on the circle where the distance you travelled is a prime number, where you would go through many rotations\*, do you end up marking the entire circumference of the circle as the number of points you pick goes to infinity? 

Or do you leave "holes" in the circle by virtue of only marking the prime points of the circumference?

\*I understand that you would have to define the circumference of your circle. Define it as a finite number, prime or not. How does the circumference you pick affect the answer/whether the circumference is prime or not?

Please let me know how I can explain this question further or whether this question does not make sense.