Fifty years of concerted effort by theoretical computer scientists has produced neither a proof that P equals NP nor a proof that it does not; most researchers believe P does not equal NP but this remains an unproven belief, not a result.
What would resolve this A proof either that some NP problem provably cannot be solved in polynomial time (P does not equal NP), or a genuine polynomial-time algorithm for an NP-complete problem (P equals NP); a Clay Mathematics Institute Millennium Prize of one million dollars is offered for a correct resolution either way.
OpenComputational complexity theoryClay Mathematics Institute