A387045 Positive numbers k with property that the largest circle on the xy-plane enclosing exactly k lattice points in its interior does not exist.
5, 6, 17, 18, 33, 34, 35, 36, 38, 50, 53, 54, 63, 70, 71, 72, 73, 89, 90, 97, 98, 102, 109, 110, 125, 126, 127, 128, 129, 150, 151, 165, 167, 168, 178, 188, 198, 209, 210, 217, 218, 219, 220, 221, 222, 242, 243, 257, 258, 259, 260, 277, 278, 285, 286, 294
Offset: 1
Examples
It can be proved that the largest circle enclosing exactly 5 or 6 lattice points in the interior on the xy-plane does not exist. Number 5 is the smallest nonnegative integer having this property and 6 is the next. Therefore, a(1)=5 and a(2)=6. Here is a brief argument. For details, please see my arxiv paper 2505.06234. First, let C be the circle going through (-1,0) centered at (1/2,1/2). It passes exactly 8 lattice points and encloses exactly 4. Now with (-1,0) fixed on the circle we can shrink C by an infinitesimal amount to circle C' so that C' only goes through one lattice point (-1,0). Then another infinitesimal perturbation will move C' to include exactly 5 lattice points in its interior. Another infinitesimal perturbation will move C' to include exactly 6 lattice points in its interior. Therefore, if the largest circle enclosing exactly 5 or 6 interior lattice points exists, then its radius is at least sqrt(10)/2. Second, a geometric argument shows that if the radius of a circle is at least sqrt(10)/2 then it encloses either exactly 4 interior lattice points or at least 7 interior lattice points.
Links
- Jianqiang Zhao, Table of n, a(n) for n = 1..189
- Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. This is an expanded version of the paper by Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, Geometry, Vol. 2 (2025), 12.
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