Millennium Prize Problems

Timeline

• 2026As of 2026, the only Millennium Prize problem to have been solved is the Poincaré conjecture.

Listen

• Let's Learn Everything!Maximum FunPodcast
• Dwarkesh PodcastDwarkesh PatelPodcast
• Hard ForkThe New York TimesPodcast
• The Peter McCormack ShowPeter McCormackPodcast

Debates

• Is P = NP?One view: Yes, some believe efficient algorithms exist for all problems whose solutions can be quickly verified. · Another: No, many believe P and NP are distinct classes, meaning some verifiable problems lack quick solutions.Open question
• Is the Riemann Hypothesis true?One view: Yes, extensive computational evidence and its implications for prime numbers strongly suggest its truth. · Another: No, a counterexample or an alternative theory describing prime distribution might eventually be found.Open question
• Do smooth solutions to the Navier-Stokes equations always exist?One view: Yes, for many physical scenarios, fluid flow appears well-behaved and predictable without singularities. · Another: No, singularities or breakdowns in smoothness might occur under extreme conditions, making prediction difficult.Open question

Glossary

• Millennium Prize ProblemsSeven challenging mathematical problems identified in 2000, each with a $1 million prize for its solution.
• P vs NP ProblemA core computer science question asking if every problem whose solution can be quickly verified can also be quickly solved.
• Riemann HypothesisA conjecture about the distribution of prime numbers, central to number theory.
• Navier-Stokes EquationsEquations that describe the motion of viscous fluid substances, like water or air.
• ConjectureA mathematical statement that is believed to be true but has not yet been proven or disproven.
• ProofA rigorous logical argument that establishes the truth of a mathematical statement.

Careers

Roles this can lead toward

Threads 5

Where this connects to other fields — and why it's worth knowing.

• Aesthetics Arts

  Math claims to run on cold logic, not feelings. Yet mathematicians constantly pick which proof to trust because it's 'elegant' or beautiful. So even the field that swears it needs no taste is secretly using beauty as a clue to what's true.

• Markets, Competition & Firms Economics

  Lots of things are valuable only because good answers are rare and hard to find. There's a famous math puzzle called 'P vs NP': if someone proved finding an answer is as easy as checking one, that rareness would vanish overnight. Whole industries would suddenly be worth almost nothing.

• The Cell & Molecular Biology Science

  Figuring out how a protein folds into its shape is a puzzle computer scientists call basically impossible to solve fast. Yet every cell in your body does it in a blink, using plain physics instead of math. Biology casually cracks a problem our best computers choke on.

• Epistemology Philosophy

  Computers have checked billions of cases and the Riemann Hypothesis holds every time, so mathematicians act sure it's true. But 'it worked a billion times' is guessing from evidence, the way science works, not airtight proof. Even math sneaks in a little faith.

• Matter, Energy & Forces Science

  There's a famous unsolved math problem, worth a million dollars, about how fluids swirl and churn. That same messy math controls the air over an airplane wing and tomorrow's weather forecast. We bet our lives on stuff we can't fully solve.