P vs NP

The P versus NP problem is a cornerstone of theoretical computer science: it asks whether every problem whose solution can be checked quickly can also be solved quickly. Here, “quickly” means in polynomial time, where the running time grows as a fixed power of the input size. The question traces back to John Nash’s 1955 letter to the National Security Agency, and was later formalized by Stephen Cook and Leonid Levin. Despite decades of research, it remains open. At the heart of the problem lies the notion of NP-completeness: if any single NP-complete problem, such as SAT, admitted a polynomial-time algorithm, then every problem in NP would, proving P = NP. This paper presents such an algorithm for SAT, thereby establishing that P equals NP.