Alexander Nazaryan in The New Yorker:
On August 6, 2010, a computer scientist named Vinay Deolalikar published a paper with a name as concise as it was audacious: “P ≠ NP.” If Deolalikar was right, he had cut one of mathematics’ most tightly tied Gordian knots. In 2000, the P = NP problem was designated by the Clay Mathematics Institute as one of seven Millennium Problems—“important classic questions that have resisted solution for many years”—only one of which has been solved since. (The Poincaré Conjecture was vanquished in 2003 by the reclusive Russian mathematician Grigory Perelman, who refused the attached million-dollar prize.)
A few of the Clay problems are long-standing head-scratchers. The Reimann hypothesis, for example, made its debut in 1859. By contrast, P versus NP is relatively young, having been introduced by the University of Toronto mathematical theorist Stephen Cook in 1971, in a paper titled “The complexity of theorem-proving procedures,” though it had been touched upon two decades earlier in a letter by Kurt Gödel, whom David Foster Wallace branded “modern math’s absolute Prince of Darkness.” The question inherent in those three letters is a devilish one: Does P (problems that we can easily solve) equal NP (problems that we can easily check)?
Take your e-mail password as an analogy. Its veracity is checked within a nanosecond of your hitting the return key. But for someone to solve your password would probably be a fruitless pursuit, involving a near-infinite number of letter-number permutations—a trial and error lasting centuries upon centuries.