Cubic Number Fields and Jacobi's Problem

"C. G. J. Jacobi (1804-1851) was the first mathematician who studied multidimensional continued fractions. His main objective was to find a generalisation of Lagrange's famous theorem which characterises quadratic irrational numbers as periodic continued fractions. He proposed an algorithm for number pairs in the hope that this algorithm will become eventually periodic if and only if both numbers belong to a cubic number field. Jacobi found some examples, but he could not show that Lagrange's theorem is true for this algorithm. In fact, the problem is still open!"

That is Professor Fritz Schweiger quote.