2021/05/12 by Yang Ji, Ji, Yang
Mathematics · Computer Science · Engineering · #Mathematics and Applications #Computational Geometry and Mesh Generation #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2105.05250
Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits on the diagonals, the midlines or the edges of the square, or the side-length of the square is n times the distance from the point to one side (both n and (n2+4) are prime numbers), the distances from this point to the four vertices can not be all rational. However, this paper does not prove a more general situation. The proof here can be extended to the whole plane, instead of being limited to the interior of the square. Key Words: discrete Geometry; rational distance; a square problem.