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Open problemChecked July 24, 2026

EditorialComplexity theory

P versus NP

The central question about the gap between efficiently checking and efficiently finding solutions.

Mathematical statement

Determine whether P=NP\mathbf{P}=\mathbf{NP}: can every decision problem whose solutions are verifiable in polynomial time also be solved in polynomial time?

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

Millennium.clay_prize_p_versus_np

Canonical source
Complete statement target, proof intentionally absentLean 4
def clay_prize_p_versus_np :    ClayPVersusNPResolution := by  sorry
Statement source
Lean Millennium Prize Problems
Lean version
v4.31.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References