On the other hand, Euler proved that irrational numbers require an infinite sequence to express them as continued fractions.
Sequences that attempt to solve Hermite's problem are often called multidimensional continued fractions.
Jacobi himself came up with an early example, finding a sequence corresponding to each pair of real numbers (x, y) that acted as a higher-dimensional analogue of continued fractions.
[4] He hoped to show that the sequence attached to (x, y) was eventually periodic if and only if both x and y belonged to a cubic number field, but was unable to do so and whether this is the case remains unsolved.
Various generalisations of this function to either the unit square [0, 1] × [0, 1] or the two-dimensional simplex have been made, though none has yet solved Hermite's problem.
[6][7] Two subtractive algorithms for finding a periodic representative of cubic vectors were proposed by Oleg Karpenkov.
The second (HAPD algorithm) is conjectured to work for all cases (including for complex cubic vectors) and all dimensions