What are the unsolved math questions?

[Everten P.]

I would like to know a) what the questions are what they ask(simplified) c) why they are important

NOTE: I have not become a freshman yet so I have not learned advanced math so that is why I am asking

Edit: oh god what have I done I put a typo in sorry guys. D:

Edited July 12, 2014 by Everten P.
Misunderstanding

[cpast]

Are you sure you mean "unsolvable", not "open"? There are statements in math that are neither provable nor disprovable, and problems which are proven to not be solvable by an algorithm, and open problems which are not proven, but which people research. The first two are somewhat closed-off leaves; once you prove that a statement can neither be proven nor disproven, you have to either assume it (very very rare to do) or you're just done working out its implications. Same for unsolvable-by-algorithm problems.

[rpayne88]

I assume you mean the "millennial problems." IIRC correctly, there were seven of them and one has since been solved. That is all i know, though.

Edit: Who niinjaed whom?

[cpast]

I assume you mean the "millennial problems." IIRC correctly, there were seven of them and one has since been solved. That is all i know, though.

Edit: Who niinjaed whom?

I ninjaed you, because my reply was first Also, you do recall correctly - seven problems, Poincare has been solved.

@Everten: The Millennium Prize problems (the ones with a million dollar prize for each) are pretty much the most famous examples of "open" problems, which are distinct from "unsolvable" problems. (if they weren't solvable, people wouldn't be setting prizes for their solutions!) While they can't know for sure, it's suspected that each of them can be either proven or disproven; it would be a huge surprise if one of the conjectures turned out to be independent of standard mathematical framework (and that proof would itself constitute a solution to the problem - "neither" is just as much a solution as "yes" or "no").

[Aethon]

Here's a neat one. Do a little research on the unknowable number... Omega

http://www-2.dc.uba.ar/profesores/becher/ns.html

[ZetaX]

There are literally tens of thousends of open problems, and at least hundreds of them should be very difficult. If you want to understand a specific one, then I can try to elaborate (which will not always be possible: the Hodge conjecture(s) would alredy be pretty nasty, and stuff like the existence of motives would be completely over the top).

@Aethon: That article seems to exagerate the implication of the unsolvability of the haling problem (hence omega) a lot.

[Red Iron Crown]

More of a computability problem, but does P equal NP?

[SciMan]

Here's one, but I'm not sure if it's actually an unsolvable math question, or just a paradox.

Does a set of all sets include itself?

[cpast]

More of a computability problem, but does P equal NP?

This is probably the easiest to explain of the really famous open problems (and it's not inaccurate to call complexity theory a branch of math; theoretical computer science is either separated from math by a very very fuzzy line, or is in fact a subset of mathematics).

Here's the problem:

Computers can solve some problems more quickly than others. In this context, "more quickly" doesn't quite mean "in less time"; the exact amount of time depends on a lot of factors, such as (in real life) the processor speed, that we really don't care about -- we care about the sort of thing that depends only on an algorithm, not on how you implement that algorithm. To do this, what we do is look at how much longer it takes a computer to solve the problem if you plug in larger numbers. With some problems, like adding, adding 2 10-digit integers takes only twice as long as adding 2 5-digit integers. With other kinds of problems, like sorting, the time goes up by more - sorting a 10-item list takes more than twice as long as sorting a 5-item list.

With many problems, the time it takes to solve them goes up like a polynomial with the size of the input (i.e. if the input is n bits long, there's some function n^c for a constant c that increases more quickly than the time it takes to solve an n-bit case of the problem). We don't care exactly what the polynomial is; as far as we care, 2n and 100000n grow about the same rate, because n^1.1 grows faster than both of them. Adding is in this category - it takes around n steps to add n-bit numbers. Sorting is as well - it takes fewer than n^2 steps to sort an n-element list. This group of problems is called P, and it's essentially the group of problems that computers can solve quickly.

There's another group of problems called NP. These are problem for which a computer can quickly check that an answer is correct. Anything in P is also in NP - if a problem is easy to solve, it's easy to check the solution by solving it again. But there are other problems that are easy to check, but we don't know if they're easy to solve. Things like factoring numbers into their prime factors are easy to check (just multiply the factors together and see if they make the number in question; checking whether the factors are prime is, surprisingly, in P), but we have no idea if there's a fast way to actually solve the problem.

That's what P vs. NP is about. If P = NP, it means every question for which you can easily check the answer can also be easily solved. On the other hand, if P != NP, it means there are problems that are easy to check but not to solve. A CS professor described P = NP as being pretty much that there is no difference between appreciating a good symphony and being able to write one. Basically everyone would be shocked - utterly shocked - if P = NP; we're pretty much certain it doesn't, we just can't prove it yet (incidentally, PROOF - i.e. "is there a proof of this statement shorter than N characters long?" - is in NP; if P = NP, a computer can very quickly prove theorems.)

[cpast]

Here's one, but I'm not sure if it's actually an unsolvable math question, or just a paradox.

Does a set of all sets include itself?

There is no such thing as a set of all sets, for basically this reason. The real paradox is "does a set of all sets not containing themselves contain itself?", and that paradox led to very strict rules about what can be called a "set"; among other things, no set can contain itself, and anything which does is not a set but a class. All sets also form a proper class, and not a set.