Result of solving an unsolved problem? 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.

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.
