With the solving of Fermat’s Last Theorem in 1995, what has replaced it as the now-reigning champion of unsolved mathematical problems or theories?
My guess would be Goldbach’s Conjecture; if it’s not at the top, it’s very close.
The big ones that folks outside of the field may have heard of are the Clay Institute’s Millennium Prize Problems (except the Poincaré conjecture) and Goldbach’s conjecture. Some of Hilbert’s problems are still unresolved, although most of those are kinda vague. The Collatz conjecture is also still open, but it’s not clear that it qualifies as a big problem.
Edit: Wikipedia has a list of unsolved problems in mathematics.
Personally, I thought that the Goldbach conjecture was a bigger deal than Fermat’s, even before Fermat’s was proven.
There’s some question as to whether the Four Color Map theory has been proved, since it was proved by having a computer work through every possible permutation and not by pure mathematical means (also, no human mind is capable of knowing the proof, which rests on faith in the computer program).
Personally, I’d have to go with the Riemann hypothesis. With the exception of the Navier-Stokes problem, the Riemann hypothesis has gone unsolved the longest of all the “Millennium Problems”, and Navier-Stokes isn’t quite as “pure” mathematics as the Riemann hypothesis is.
I would assume that “does P = NP?” is not the problem to solve, though maybe that’s just because I have a computer science background.
Can a human be said to “know” a proof that takes hundreds of pages, like that of Fermat’s Last Theorem? Would it make the Four Color Theorem somehow “more proven” if, instead of using a computer, some mathematician had assigned hundreds of grad students to work through the possibilities? Can the operation of a computer be said to be anything other than mathematical?
I think he means that there is no actual mathematical proof for it. I don’t think working out all of the possible permutations constitute a mathematical proof, but I’m not a mathematician.
I don’t think the proof theorists would have a problem with it, but there definitely needs to be an associated proof that the program is correct for the whole thing to be airtight.
This is somewhat related to the classification “theorem” for finite simple groups, which is well over ten thousand pages long.
Goldbach’s conjecture has the advantage that it’s easy to understand and explain to people with relatively little math background. On the other hand, if I understand correctly, it’s not an “important” problem in the sense that other important math depends on whether or not it’s true, nor have any useful or important mathematical techniques been developed in the attempt to solve it.
Many (myself included) would say it’s the Riemann Hypothesis that’s the current Greatest Unsolved Math Problem. It’s just about the only one of Hilbert’s problems from 1900 that remains completely unresolved; it’s one of the seven Millennium problems; and it’s one that many mathematicians would dearly love to see a solution to.
It’s not at all uncommon for a math proof to break a situation down into several possible cases and then deal with each case separately.
Of course, there was a fair bit of “traditional” mathematics which went into proving that all maps could be reduced to a finite number of cases, and finding what those cases were.
I think the point is that the brute-force method of proof offers no illumination of the reasons that the theorem is true, and exposes no new thought paths into other areas of math. It’s a proof, yes, but an intellectually barren one. But a proof that is the result of reasoning instead can actually advance knowledge. As yet there is none for the Four Color Theorem.
The Riemann Hypothesis is definitely the king of math problems that mathematicians really care about, and the Goldbach Conjecture is definitely the king of math problems that mathematicians explain in one line to their non-mathematician friends. As it happens, they are both part of Hilbert’s Eighth Problem (though all this means is that Hilbert’s Eighth Problem is actually lots of different, somewhat related problems).
I’m not opposed to the thrust of your position, but, as Chronos points out, there was certainly a fair amount of reasoning involved in reducing the theorem to that large but finite number of cases. If you like, it was only that reduction which was “mathematically” proven, and the fact that all those cases work out was proven “non-mathematically”. But the reduction is unimpeachable.
I would assume you meant to say “now the problem to solve”, though maybe that’s just because I also have a computer science background. 
Well, it still won’t be airtight until we attach an associated proof that that proof is correct. Oh, wait, and another one that…
I don’t agree with this. There are certain proofs that don’t give any insight, but the four color theorem’s proof is not one of them–if nothing else, we know that every graph reduces to one of finitely many cases. And as Thudlow mentions, the basic pattern of the argument is very well established.
A relatively common pattern in the development of a proof goes something like this:
Conjecture: No number has such-and-such a property
Theorem 1: There are at most finitely many numbers that have such-and-such a property
Theorem 2: If there are any numbers which have such-and-such a property, they must all be below this given but unrealistically high upper bound (maybe something like 10^(10^100) ).
Theorem 3: The same as Theorem 2, but with a much more accessible upper bound (maybe something like 10^10 ).
Theorem 4: All numbers up to that upper bound are tested, and none of them had that property. Combined with Theorem 3, therefore, no number has that property, and the conjecture is now proven.
This is much the same way that the Four-Color theorem worked, except with graphs, not numbers.
Like I said on these boards at least once before, there’s no point trying to hard to engage in a binary classification of just “is a legitimate proof” vs. “is not a legitimate proof”; we have the ability to express more, and nothing’s stopping us from using it.
We can all agree that there was significant mathematical insight involved in reducing the four-color theorem to those finitely many cases, and we can all agree that there was little such mathematical insight involved in verifying those finitely many cases, and we can all hopefully agree that nonetheless we now have as convincing evidence for those finitely many cases being four-colorable as we have for anything and thus now know the four-color theorem to be true, and we can all agree that it would still be nice if we could find another way of establishing this fact that gave even more insight (perhaps by being of a “wholly mathematical” flavor, avoiding the exhaustive search, whatever), and so on. The only things we don’t all agree on are pointless debates over what to call the situation.