N vs NP. Existence or Constructive. I was discussing P vs NP problem with somebody who works in computer science. I work in mathematics and know very little about computer science. 
My opponent told me, if you solve P vs NP problem, keep the solution to yourself, so you can crack codes and stuff. I told him it does not matter because the theorem, if proven, is an existence statement and does not tell us how to implement the algorithms more efficiently. (Or even if it could, the constants can be so huge that it practically does not matter). 
My feeling is that since in mathematical theorems are usually of existence-type, it means that if you can solve P vs NP, it would not mean you can suddenly go on cracking codes easily. Thus, our disagreement can be summarized as saying: I say the solution to P vs NP problem is a theoretical curiosity, while he thinks it has practical consequences to algorithms. 
What are your thoughts?
 A: Well, I will try to answer the questions that I think you are asking. If the P=NP? question is solved positively, then the answer should be (quite possibly) in the form of an algorithm that solves an NP-hard problem in polynomial time. NP-hard problems are those problems that are at least as hard (computationally speaking) as any problem in the class NP, so that if you find an efficient (polynomial-time) algorithm to solve one of them, you are automatically finding an efficient algorithm for all problems in NP. Hence, in that case, the answer to the P=NP? question will have great practical value. However, as @Ittay Weiss points out, most people believe that P is a proper subset of NP. If that is the case, then you would not get any algorithm to crack codes, but on the other hand, you would get the one-million dollar prize offered by the Clay Institute :-). Moreover, a negative answer will justify all the work that is being done on heuristic algorithms. Finally, a third possibility is that the P vs. NP question lies beyond the axioms of complexity theory, so that either answer is unprovable with the current axioms. 
