Jacobi´s Last Theorem

This distinguished German mathematician contributed to physics with the Hamilton-Jacobi equation, important in classical and quantum mechanics. Nevertheless my interest here is to highlight his contributions to number theory.

Last week Professor Fritz Schweiger from the University of Salzburg presented in the History and Pedagogy of Mathematics (HPM 2008) in Mexico City a paper on Jacobi´s Last Theorem, as he calls it. It is actually a conjecture, but paraphrasing Fermat´s Last Theorem, he chooses to call it this way. To this day nobody knows if Jacobi´s conjecture is true. He claimed that the trio:

(1, 4^1/3, 4^2/3) when expressed as infinite continued fractions, produce periodic values for the three fractions.

It is an interesting and apparently simple question, that so far has eluded mathematicians.

These problems appear in integer relation formulae:

Integer Relation Algorithm.

Thank you Professor Schweiger for teaching us these questions.