This is one of the seven Millennium Prize Problems
proposed by the Clay Mathematics Institute
in April 2000.
The Millennium Problem is that no-one can prove whether or not P is equivalent to NP. To show P is not equivalent to NP would require a complex mathematical proof. To show P is equivalent to NP would just require finding a polynomial algorithm for one particular NP problem. It is known that if one such algorithm existed then it could be adapted to solve any NP problem in polynomial time.
A worrying point is that RSA
decryption is an NP problem, and so if it was shown that P=NP, then the world would be thrown into chaos
as all RSA
privacy would go down. That may happen anyway with the development of quantum computers