Determine whether every problem whose solution can be quickly verified by a computer (in "P") can also be quickly solved by a computer (in "P"), or if some problems are inherently harder (in "NP").
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Every problem in the class P can indeed be quickly solved by a computer. The class P consists of decision problems that can be solved in polynomial time. This means that there exists an algorithm that can solve any problem in P within a reasonable amount of time, where the time required to solve the problem is a polynomial function of the problem's input size.
On the other hand, the class NP (nondeterministic polynomial time) consists of decision problems for which a solution can be quickly verified by a computer. This means that if someone claims to have a solution to an NP problem, it can be efficiently checked to verify its correctness.
It is still an open question in computer science whether P = NP or P ≠ NP. If P = NP, then it would imply that every problem in P is also in NP, and therefore every problem that can be quickly verified can also be quickly solved. This would mean that there is an efficient algorithm for each problem in NP.
However, if P ≠ NP, it would suggest that there are problems in NP that are inherently harder to solve than to verify. In other words, these problems would require exponential time or more to solve, even if the solution can be verified in polynomial time.
Until now, researchers have not been able to prove either P = NP or P ≠ NP. The question remains one of the biggest unsolved problems in computer science and mathematics.
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