The P ?= NP problem asks whether there’s a fast algorithm to find such a proof
(1) a finite but exponentially-large space of possible solutions; and
(2) a fast, mechanical way to check whether any claimed solution is “valid.” (For example, do the puzzle pieces now fit together in a rectangle? Does the proposed airline schedule achieve the desired profit? Does the neural network correctly classify the images in a test suite?)
We’re asking whether, under the above conditions, there’s a general method to find a valid solution whenever one exists, and which is enormously faster than just trying all the possibilities one by one, from now till the end of the universe, like in Jorge Luis Borges’ Library of Babel.
Why is it an important problem? when you can find a type of solution for NP problems that are not checking one by one possible solutions, you are crazy and genius.