Cubic Roots of Irrationals

Carl Gustav Jacob Jacobi tried to do for cubic roots of natural numbers, what Lagrange had done for square roots of natural numbers; classify the regularities found in the case the cubic roots were irrational numbers.

He did not find regularities. To this day mathematicians are looking for any regularities in one cubic root, or sets of cubic roots of natural numbers.

This generalization from power two to power three, reminds one of other apparently simple generalizations that did not happen. Perhaps the most famous one concerns Diophantine Equations of natural numbers. For power two, we have an infinite set of triplets satisfying Pythagoras Theorem:

[2k(k+1) + 1]² = [2k(k+1)]² + [2k + 1]²

for any natural number k.

When Fermat looked into the generalization to power three, he claimed to have proved it was impossible for this case.

It was only many years later, in the early nineties of last century, that Andrew Wiles finally proved that statement to be true.

In Physics we have the exact solution for the two dimensional Ising Model, found in the middle of last century by Lars Onsager. To this day physicists are hard at work; with nothing to show.

"Jacobi's Last Theorem", as Professor Fritz Schweiger calls it, is patiently waiting for a modern day Wiles to conquer it.