Two families of periodic Jacobi algorithms with period lengths going to infinity (Q757467)

Let \(c\geq 2\) and \(m\geq 1\) be fixed integers. The authors establish that the Jacobi-Perron expansion of the vector \((x^ 2-(c^ m+c-1)x,x)\) is purely periodic of period length \(4m+1\), where x is selected to satisfy \(c^ m+c-1<x<c^ m+c.\) The key to their solution is to show that \(g(x)=x^ 3-(c^ m+c-1)x^ 2-(c^ m-1)x-c^ m\) has only one real root and that root x satisfies the previous inequalities. By a similar consideration it is also established that the Jacobi-Perron expansion of the vector \((x^ 2-c^ mx,x)\) is purely periodic, this time with period length \(3m+1\).