P versus NP

If a solution can be checked quickly, can it always be found quickly?

The statement. P is the set of problems a computer can solve in time polynomial in the input size; NP is the set whose proposed solutions can be checked in polynomial time. Is P = NP?

In plain words. Checking a completed Sudoku is easy; solving one from scratch seems hard. Checking that a number factors as p × q is easy; finding p and q is the hardness cryptography relies on. P = NP would mean every such “easy to check” problem is secretly easy to solve. Almost everyone believes P ≠ NP; nobody can prove it.

What you need first. Stage 9 (logic, truth tables, counting; the satisfiability problem is a truth-table question), Stage 10 (factoring, why it is slow), Stage 1-2 for reading growth rates like n², 2ⁿ.

The parts. (1) What counts as fast: polynomial vs exponential time, and why the distinction survives changing computers. (2) NP-completeness (Cook-Levin, 1971): thousands of problems (satisfiability, travelling salesman, graph colouring) are each as hard as all of NP, so solving any one fast solves them all. (3) Why the obvious approaches fail: diagonalisation and circuit lower bounds each hit a proven barrier. (4) What a proof would need: a genuinely new way to show that no algorithm can exist.

What is known. Exponential lower bounds exist only for restricted models of computation; the three known barriers (relativisation, natural proofs, algebrisation) explain why current techniques cannot work. A Millennium Prize problem.

Tani ti. Asnjë kalkulator nuk e zgjidh këtë, por pjesët e saj janë të llogaritura. Provo një më poshtë, ose shkruaj tënde.

Mbaje punën tënde

Një llogari e lirë shtohet shënime në çdo mësim, një regjistrim të asaj që ju keni përfunduar, problemet tuaja të zgjidhura në një vend, dhe një mësues që ju mund të pyesni rreth kësaj faqeje. Matematika vetë është e hapur për të gjithë, të regjistruar apo jo.

Regjistrohu Hyr

Simbolet e përdorura këtu

Prek çdo simbol për përkufizimin e plotë, një fotografi dhe se çfarë do të thotë çdo shkronjë në të.

Pyetja që bëjnë njerëzit

Can I actually work on these?

You can understand them, which is the honest first step and what these pages are for. Working on them means the full path through the stages above and then the research literature, but every person who has made progress started by reading the statement.

Why are they unsolved if so many people have tried?

Usually because the existing tools provably cannot work (the barriers in P vs NP), or because the problem mixes two structures mathematics handles separately (additive and multiplicative in Goldbach and Collatz). A solution needs a genuinely new idea.

Më shumë në Frontiers

The Riemann Hypothesis Navier-Stokes existence and smoothness Goldbach's conjecture The Collatz conjecture The twin prime conjecture