nawl

The three of us spent the better part of the evening stumbling our way through the beginnings of a proof. Induction requires a ridiculous number of cases, so we're pretty sure we know why nobody's bothered to do it before. It was going almost too peachily until we hit a snag. I thought I solved it while drifting in and out of sleep, but when I got back up to put it down on paper, it turned out I was just re-iterating the same argument that didn't quite work. I hate when that happens. So we need to come up with something stronger. Oh well, if it works I guess it won't be entirely trivial.

On a slightly different note, our "hard" Ramsey theory problem is still tempting to think about, even though we're officially supposed to have abandoned it.

An equation (or a set of equations) is called partition regular if you can't color the integers without ending up with a monochrome solution. Rado studied the heck out of linear equations in the 30s, but it looks like nothing's been done since. By "nothing", I mean that the names that keep popping up seem to have extended things to matrices and vectors, and one guy apparently came up with a solution to 1/x + 1/y = 1/z, but that's about it.

So non-linear equations are just crying out for a solution. Except that it's probably really hard. Finding a K4 in the graph of sums and products (our first small question) is equivalent to asking whether a system of Diophantine equations has a solution, but the general consensus seems to be that Diophantine equations are hard, and that it's not useful to say much more than that. (We tried playing with inequalities already. Didn't work. Could be that we overlooked something.) Diophantine equations (such as that nice one Fermat talked about) are supposed to have something to do with elliptic curves, so I looked up elliptic curves, but the information I found got very scary very quickly. As did the proof that Diophantine equations are officially hard.

Still, how cool is a problem that potentially involves a. graph theory, b. numbers whose decimal representations have more digits than the numbers of atoms in the universe, and c. plain old algebra?