Result of solving an unsolved problem?

Question

I know that when some of the previously unsolved problems were solved they created new fields in mathematics. May someone explain to me what would be the result of a major problem like the Hodge Conjecture being solved vs a "smaller" problem like "Do quasiperfect numbers exist?" in today's society.

Thank you in advance.

Answer 1

Here's a theorem ($P=NP$) that would change the world if proven to be true:

http://en.wikipedia.org/wiki/P_versus_NP_problem

As it says on the wiki page:

"Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields."

I'm adding the following comment on May 4th:

Strictly speaking, to change the world we don't just need a proof that $P=NP$, we need a constructive proof that leads to efficient (i.e. practical) algorithms. So, without a doubt, if somebody proved $P=NP$ and they prove there is an efficient polynomial time algorithm, then that person will have dramatically changed the world.

So regardless of all the qualifications posted in response to this thread, the answer to the original question remains affirmative. There are open math problems whose solutions could change the world if they turn out to be true and somebody manages to prove them.

Answer 2

Most of the time, the actual result isn't important as the theory. The reason why problems are unsolved is because either the math doesn't exist yet, or some connection between current fields has not been established yet.

Either way, creating new math and connecting existing math are the real reasons why solving open problems is important.

For example, if the word of god came down and told us that yes, indeed the Hodge conjecture was true/false, it would be nice but not nearly as groundbreaking as a proof for it.

Answer 3

I agree with @avid19 that generally it isn't the results of the breakthrough solutions that matter as much as how they open up new ways of thinking and new classes of problems. Let's not forget that fields such as probability and combinatorics received great impetus by what today is a set of simple gambling problems addressed by Pascal and Fermat and others. Our entire way of thinking expanded, and the range of problems we could address exploded... most important to problems not thought soluble. I put the Four Color Theorem (which led to graph theory), Nash's Game Theory, chaos theory and many other domains in this class.

There are a whole host of problems for which we simply don't have mathematics yet to address. Since I work in computation, my favorite example of this is Collatz's conjecture: Determine whether the following algorithm for integers greater than 1 always terminates at 1:

• If $n$ is even, divide it my 2, otherwise compute $3n + 1$; iterate.

This has been verified for $1 < n < 2^{62}$ and my personal view is that indeed the function does always terminate at 1, but neither I nor anybody has a clue how to prove it. Paul Erdös said of this problem: "Mathematics may not be ready for such problems."

We don't yet have adequate mathematics to describe and predict a wide range of problems and are reduced to simulating the processes, such as the dynamics of clouds, specific neural activity in the brain, and so forth. Perhaps new concepts that arise in solving the Collatz Conjecture will ultimately lead to such mathematical ideas.