About 100 years ago N. Anning [Discussions: Relating to a geometric representation of integral solutions of certain quadratic equations. Am. Math. Mon. 22, No. 9, 321 (1915; doi:10.2307/2972036)] found a constellation of 12 points on a circle such that each mutual distance is an integer and is among the solutions \((x,y)\) of the Diophantine equation \[x^2+xy+y^2=7^2 \cdot 13^2,\] i.e. for each solution \((x,y)\) the numbers \(|x|\) and \(|y|\) are equal to one of the distances or zero.
Moreover, Anning [loc. cit.] conjectured that there also exists an arrangement of 48 points on a circle, such that each mutual distance is an integer and is among the solutions of the Diophantine equation \[x^2+xy+y^2=7^2\cdot 13^2\cdot 19^2 \cdot 31^2.\]
In the paper under review the authors prove Anning’s conjecture. Moreover, they show that for each integer \(n\) there exist \(3\cdot 2^n\) points arranged on a circle such that each mutual distance is an integer and is among the solutions of the Diophantine equation \[x^2+xy+y^2=p_1^2p_2^2 \cdots p_n^2,\] with different prime numbers \(p_i\) of the form \(6k+1\), \(k\in \mathbb N\).