P not equals NP

Definition

P not equals NP

The statement $\mathrm{P}\ne \mathrm{NP}$ asserts that some decision problem in NP does not belong to P:

$$\mathrm{P}\ne \mathrm{NP}⟺\exists \Pi \in \mathrm{NP}:\Pi \notin \mathrm{P}.$$

In other words, some decision problem has polynomial-length certificates for its yes-instances verifiable in deterministic polynomial time, but no deterministic polynomial-time algorithm solves it. Since $\mathrm{P}\subseteq \mathrm{NP}$ is known, this would make the inclusion strict.

This is an unproved conjecture, not a theorem. Its negation is P = NP.