Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33 (2019)

Do you know which is the lowest currently unsolved number?

Not unless there is proof that any number has one and only one set of three cubes. I don't think that is true.

What you want for asymmetric cryptography is "always hard to solve, easy to verify". Only a very small number of problems are believed to fall in this category, including factoring and discrete logarithm.

It's far more common to have problems that are "sometimes easy to solve, sometimes hard to solve, easy to verify". That describes this problem as well as most NP-hard problems that with good heuristic solutions. The problem is those "sometimes easy to solve" cases break your cryptography!

All problems in NP seem to be hard to solve and easy to verify. This is actually a really common property.

It's a fair question mostly because of how effective math has been at solving practical problems, but honestly working on problems like these is usually due to a genuine appreciation and enjoyment of mathematics. I'd even say the majority of mathematicians study math strictly out of a sense of wonder and curiosity rather than towards a commercial application and that the vast majority of math research is not at all practical.

The cost is justified due to a common shared interest in the subject, many other people are interested in math and pay people to work on it. It's not all that different than paying someone for music, or a painting, or to write a work of fiction. People get a profound sense of enjoyment and happiness from this work and are happy to pay the cost for the continued advancement of this field.

The article touches on this with:

> k = x³ + y³ + z³ is what number theorists call a Diophantine equation — a kind of algebraic structure whose properties have fascinated mathematicians for millennia. “They’re rich enough to encode [other mathematical] statements that have nothing to do with Diophantine equations,” said Browning. “They’re rich enough to simulate computers.”

More specifically, equations of such form come up in a proof of Godel's first incompleteness theorem. That being said, yeah, this effort is mostly a mathematical curiosity, mapping out the domain of knowledge of mathematics increasingly more completely. Sometimes an application comes about where either the solution or novel techniques used to arrive to the solution can become useful, but pure math research isn't typically driven by any existing applications as much as a "because it's there" attitude.

Sometimes these weird questions can lead to more practical applications down the road, e.g. once we know how to do something it gets added to the toolbox and eventually used, somewhat like how Euler's totient was a cute mathemtatical curiosity until it became one of the important steps in RSA key generation and a very useful tool almost two centuries after it was discovered. It is also the case that sometimes knowing where the 'holes' are in these sort of stepwise math series' can itself reveal something deeper or more important. Like the fabled exceptions that prove the rule, knowing where a general rule does not work is sometimes an incredibly useful discovery.

Often we don't know the "practical" benefits that will come out of some specific line of investigation. But some people find it interesting to think about. I guess better than spending time playing with facebook.

I'm sure a certain Joseph Fourier was asked that question too, quite some time ago. :)

One of the terms in the solution is a bit bigger than 2^51 so its square is around 2^102. Assuming the methods used need to actually compute these numbers, you’ll need something bigger than a 64-bit int. Not sure how good GPUs are going to be at bignum-ish stuff, but it’s not the most obvious fit

Added. Thanks!

42 has been solved. See the update at the bottom of the page.