Abstract
This chapter discusses some problems on geometry and arithmetic. For 150 years, from the time of Gauss until now, lattice points have been the subject of various interesting mathematical inquiries. Different problems were posed on this theme. The chapter discusses a problem proposed by H. Steinhaus: “for every natural number (that is, positive integer) n, does there exist in the plane a circle having in its interior exactly n lattice points?” It is easy to show that there exist natural numbers n for which no circle with a lattice point as center has exactly n lattice points in its interior. If there is a circle with a lattice point as center and radius ≤ 1, then there is only one lattice point in its interior, but if the radius of the circle is > 1 and ≤ 2, then there are exactly five lattice points inside the circle. There is no circle with a lattice point as center in which there would lie exactly two, three, or four lattice points.
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