What is the oldest unsolved math problem? Currently how many math problems are currently unsolved?
Euclid proved that all even perfect numbers are of the form 2[sup]p-1[/sup](2[sup]p[/sup] - 1) where 2[sup]p[/sup] - 1 ( and hence p) is prime. It has been an unsolved problem since then to determine whether or not there are any odd perfect numbers. This is the oldest one that springs to mind.
As for how many unsolved problems there are: quite a few, I’d say. Here are seven of the most important ones ( under the link “Millenium Prize Problems”).
What about non-trival zeros in the Reimann Zeta Function. I don’t think it’s the oldest but it is very important. While in and of itself, it’s not a big deal, but in the long run it will open a whole new area of mathematics. (But then again, I could be mixing that up with something else from complex analysis)
Upon re-reading my reply I noticed something wrong. I didn’t mean “non-trivial zeros” I meant non-trivial answers/solutions to the Reimann Zeta Function. That makes much more sense.
Same thing, Joey. The second-oldest unsolved math problem (after the odd perfect numbers) is the Goldbach conjecture, which states that every even number is the sum of two primes.
There’s probably no way to say how many unsolved problems there are in mathematics. Even creating a complete list of unsolved problems would require going through every mathematical paper and book ever published and making note of every conjecture made. Then one would have to go back and figure out which of those conjectures are solved in some other paper. In addition, there are a lot of unsolved problems that have never been published. It would also be enormously difficult to go through those conjectures and winnow them down to just the important problems.
Great link, Jabba. (seven important unsolved problems)
Is there a betting pool going on which one will be solved first, and what year (decade?) it will be solved in?
The P vs NP twangs my pragmatics. I know that the illustration given isn’t the actual problem. The actual problem is strictly mathematical. But, If I were given the example problem used to illustrate the actual problem, I’d immediately whack all of the ‘must not room together’ pairs off of the list. That way they are, by definition, not on the final list.
If the list is too small with the pairs gone, I’d return single members of pairs until the list was complete (If A goes to the list then B is deleted from consideration). If the list is still too small, the problem is insoluble without someone rooming alone.
If the Dean forbade deleting the pairs for some reason, I’d guess how many ‘paired’ students ‘ought’ to be on the list and pick them first, then pick from the ‘unpaired’ list. The number that ‘ought’ to be on the list is more likely to be a political number than a mathematical one.
Either way, no name is going to get on the list if it has the least chance of messing up the list, so the final list will not need to be checked. And, yes, I know I’m leaving out the fun part. And missing the mathematical point. I’m probably on ly sharing this because I’m up too late.
I wonder how old the twin primes conjecture is (that there are infinitely many pairs of primes that differ by 2). Since Euclid proved there are infinitely many primes, it is conceivable he wondered about that too. There are probably lots of unsolved problems like that.
What disturbs me about the OP is the implication that math is a thing with a structure that can be determined.
Math is not a thing; it is a process, multifaceted and ongoing.
You can no more answer how many unsolved problems there are than you can answer the question how many unfinished roads exist in the world. Existing roads are constantly being extended, as well as repaved, reconfigured and rerouted, and new roads are created every time a subdivision goes up or a wilderness cleared.
Similarly, there are hundreds of subdisciplines in math, some finding new areas of interest in traditional subjects and some creating new areas to explore.
In a sense, an unsolved problem is presented - and presumably solved - every time a paper is written and presented. These create untold new problems of their own.
It would be an interesting philosophical argument whether the number of unsolved math problems is actually infinite. If the number is finite, however, then it is almost certainly large beyond our comprehension.
Not that interesting, cause it’s definitely infinite. There are a countable number of statements that might be true, and we can only ever determine the truth of a finite number of them.
I should note that when I talked (in my first post) of how great the number of unsolved problems in math is, I was speaking of just the conjectures that have been made so far. As Exapno Mapcase says, the notion of listing all possible unsolved problems is ridiculous. As mathematics grows (and it has been growing at an increasing rate), the number of unsolved conjectures increases. Solving one of them often means that other problems open up.
Yllaria, I don’t think you understand the example problem given in the Clay Mathematics Institute discussion of the P versus NP problem. It’s possible that every one of the 400 students has some other student that they are incompatible with and yet there will still be some set of 100 students none of whom are incompatible with each other. Yes, if the set of students with no incompatibilities is 100 or more, obviously there’s a trivial solution to the problem, but that proves nothing except that in a tiny set of cases your method can solve the problem. Notice that the problem says nothing about putting students into the same room. What it says is that if two students are incompatible, they cannot both be in the dormitory. Go back and read the problem carefully.
I gather from the Clay page that the incompatible-students problem is known to be NP-complete. What that means is that any NP problem (i.e., one with an answer that can be expressed in a polynomial amount of time) can be converted to an incompatible-students problem, in a polynomial amount of time. To extend that a bit, it means that if there are any exponential-time problems in NP, then the incompatible student problem is one of them.
In general, NP-complete problems often have easy solutions in particular special cases, or easy “almost optimal” solutions, but nobody has ever found an easy solution which always gives the best answer to such a problem.
For instance, another example of an NP-complete problem is the Travelling Salesman problem. A salesman wants to visit N different cities, and he knows the distance between any pair of cities. Find the route between cities which lets him travel the minimum distance.
Now, in some special cases, it’s easy to find the minimum distance: If the cities are all in a straight line, for instance. And no matter what arrangement of cities you have, it’s easy to find a pretty good route between them. But with a general arrangement of cities, the only known ways to find the perfect path involve checking an extremely large number of possibilities. Even if you can rule out some in advance, you’re still left with a very large number.