Refinement of the upper bound of the radius of a circular integral points set

Abstract

Let n be a product of the prime numbers whose positive integer powers are of the form a²+Db² where D> 4 is a square-free number and a, b are positive integers. For n≤ 3072, we obtained a refinement of the upper bound of the radius of a circular points set which was previously given in Tables 1 and 2 of Ganbileg's paper. In order to prove this, we showed that there are points on the circle with the radius R=n √D/2D such that mutual distances between these points are all integers. Consequently, if n is a product of the prime numbers whose squares are of the form a²+Db², then we showed that there are τ(n) points on the circle with the radius R=n √D/2D such that mutual distances between these points are all integer numbers, where τ(n) is the number of all positive divisors of n.