MathDB

Define a ring in the plane to be the set of points at a distance of at least

r

and at most

R

from a specific point

O

, where

r<R

are positive real numbers. Rings are determined by the three parameters

(O, R, r)

. The area of a ring is labeled

S

. A point in the plane for which both its coordinates are integers is called an integer point.
a) For each positive integer

n

, show that there exists a ring not containing any integer point, for which

S>3n

and

R<2^{2^n}

.
b) Show that each ring satisfying

100\cdot R<S^2

contains an integer point.

Problem Statement