Unsolved · Math and physics
Thirteen locked doors.
Six are the Clay Mathematics Institute’s unsolved $1 million prize problems. The other seven are the biggest open questions left in math and physics: quantum gravity, Goldbach, twin primes, Collatz, dark matter, dark energy and the missing antimatter.
$0in unclaimed Clay prizes
13problems that would change everything
1solved so far: the Poincaré conjecture, 2003
The oldest door was asked in 1742 (Goldbach) and the newest in 1998 (dark energy); added together the thirteen have stood open for about 1,400 years. Eight are mathematics and computer science, five are physics. Six carry a $1 million prize from the Clay Mathematics Institute, which set seven problems in 2000; one of those seven, Poincaré, fell in 2003 and Grigoriy Perelman declined the money (Clay). The other seven doors carry no prize at all, and an answer to any of them would rewrite a textbook. In October 2026 the first door moved: an AI model proved the quasi-Riemann hypothesis, the zeros held inside a strip rather than on the line (Quanta); door one carries the details. Every number on this page carries its source.
What is ahead in this section
- The thirteen tilesNumber, year asked, the equation, the prize. Tap any tile for that door’s newest papers.
- Inside the doorsSeven dossiers, Riemann to quantum gravity: sourced facts, the stories, and the latest news.
- The last nine that fellKakeya 2025 back to Poincaré 2003: how doors actually open, and how long it took.
- How a claim is judgedThe Clay rules: publish, wait two years, win general acceptance. No AI can certify a proof.
- Build your own hero analogThirty-three strengths, eighteen agents, one door. Then Codex Obscura.
01Since 1859
ζ(s)=0 ⇒ Re(s)=½
Riemann HypothesisMathematicsDo all the nontrivial zeros of the Riemann zeta function lie on one line? The answer governs how prime numbers are spread out.
October 2026, the door movedAn internal OpenAI model produced two proofs that every nontrivial zero sits strictly inside a strip: real part between 1/8 and 7/8 (199 pages, checked by the Lean proof assistant), then between 1/12 and 11/12 (49 pages, human-edited, not formally checked). That is the quasi-Riemann hypothesis, not the line itself. Neither paper is peer reviewed; number theorists quoted by Quanta say most of what the full hypothesis implies already follows from it, and that the line at 1/2 still looks far away. Quanta, October 19, 2026 · the papers, openai/math
$1M Clay prize02Since 1971
P ≟ NP
P vs NPComputer scienceIf a solution is quick to check, is it always quick to find? A yes would upend cryptography, logistics and AI.
$1M Clay prize03Since 1845
∂u/∂t + (u·∇)u = −∇p + νΔu
Navier–StokesPhysics & mathDo the equations of fluid flow always have smooth solutions in 3D, or can they blow up?
$1M Clay prize04Since 1954
Δ > 0
Yang–Mills & Mass GapQuantum physicsProve the quantum theory behind the strong force exists, and that its lightest particle has positive mass.
$1M Clay prize05Since 1950
Hk,k ∩ H2k(X,ℚ)
Hodge ConjectureGeometryCan certain shapes in complex geometry always be built from simpler algebraic pieces?
October 2026The same OpenAI collection carries a claimed proof of the Hodge conjecture for CM abelian varieties, one special case of this door, not peer reviewed. openai/math
$1M Clay prize06Since the 1960s
rank E(ℚ) = ords=1 L(E,s)
Birch and Swinnerton-DyerNumber theoryDoes the behavior of one function tell you how many rational points an elliptic curve has?
$1M Clay prize07Since the 1930s
Gμν ↔ ℏ
Quantum GravityPhysicsUnite general relativity and quantum mechanics into one working theory of gravity.
No prize: the open frontier08Since 1742
2n = p + q
Goldbach ConjectureNumber theoryIs every even number greater than 2 the sum of two primes? Checked by computer into the quintillions, never proved.
No prize: the open frontier09Since 1849
p, p+2 ∈ ℙ ∞ly often
Twin Prime ConjectureNumber theoryAre there infinitely many pairs of primes just two apart, like 11 and 13?
No prize: the open frontier10Since 1937
n → n/2 or 3n+1
Collatz ConjectureMathematicsHalve even numbers, triple odd ones and add one. Does every start always fall to 1?
No prize: the open frontier11Since 1933
ΩDM ≈ 5Ωb
Dark MatterPhysicsMost matter in the universe is invisible and pulls by gravity alone. What is it made of?
No prize: the open frontier12Since 1998
Λ > 0
Dark EnergyCosmologySomething is speeding up the expansion of the universe. What is it, and why is it so small?
No prize: the open frontier13Since 1967
nB ≠ nB̄
Matter vs AntimatterParticle physicsThe Big Bang should have made equal amounts of both. Why is anything left at all?
No prize: the open frontier
Inside the doors: facts, stories and news
The Clay Mathematics Institute announced the seven Millennium Prize Problems on May 24, 2000, in Paris, $1 million each. Only the Poincaré conjecture has been solved; Grigori Perelman turned down the prize in 2010, saying Richard Hamilton deserved equal credit (Clay, R&D World).
01Riemann HypothesisBernhard Riemann, 1859
Every nontrivial zero of the zeta function sits on the line where the real part is one half.
Facts
- The first 10 trillion zeros have all been checked, and every one is on the line. Clay
- Platt and Trudgian rigorously verified the hypothesis up to height 3 trillion in 2020 using interval arithmetic. arXiv
- The trivial zeros are the negative even integers; zeta(2) equals pi squared over 6, about 1.64. Quanta
- Ingham's 1940 bound on zeros off the line stood for more than 80 years. Quanta
Stories
- Sept 24, 2018: Michael Atiyah presented a claimed proof at the Heidelberg Laureate Forum. Experts including John Baez and Steven Strogatz were openly doubtful. AMS Blog
- Larry Guth of MIT and James Maynard of Oxford won their result by deliberately skipping a standard simplification, a move they called a gambit. Quanta
Latest news
- 2024: Guth and Maynard set a new upper bound on zeta zeros with real part 3/4, beating Ingham and sharpening counts of primes in short intervals. Quanta
Why it matters: It controls how regularly the primes are spread out, and primes underpin number theory and modern cryptography.
02P vs NPStephen Cook and Leonid Levin, independently, 1971
If an answer is quick to check, is it always quick to find?
Facts
- Clay's example: choose 100 of 400 students while avoiding incompatible pairs; there are more possible groups than atoms in the known universe. Clay
- Gerhard Woeginger's list holds 116 claimed proofs: 61 say P equals NP, 49 say it does not, 1 says undecidable, 2 say unprovable. TU Eindhoven
- Razborov and Rudich showed in 1994 that a 'natural' proof that P differs from NP would also break cryptography. Quanta
- Richard Karp showed that almost every NP problem without a known fast algorithm is NP-complete. Quanta
Stories
- August 2010: Vinay Deolalikar of HP Labs claimed a proof. Scott Aaronson offered $200,000 of Aaronson’s own money if it held up; it did not. Phys.org
- Leonid Levin found the key result as a Soviet undergraduate, was denied a doctorate in 1972 for political reasons, and emigrated to the US in 1978. Quanta
Latest news
- Feb 25, 2025: Ryan Williams showed any computation taking time t can be simulated in about the square root of t log t space, beating a 1970s bound. arXiv
- Aug 2023: 'meta-complexity' research, including Hirahara's 2022 result on partial MCSP, opened a new line of attack. Quanta
Why it matters: It decides whether hard search problems in logistics, drug design and code-breaking can ever be solved efficiently.
03Navier–StokesNineteenth-century fluid equations; prize statement by Charles Fefferman
Do the equations of 3D fluid flow always have smooth solutions, or can they blow up?
Facts
- Terence Tao proved finite-time blowup for an 'averaged' version of the equations in 2014, so any proof must use finer structure of the real ones. arXiv
- Tom Hou of Caltech showed numerical evidence of potential singular behavior, with maximum vorticity growing by a factor of 10 million. Berkeley Lab
- In Hou's generalized model, vorticity grew by a factor of 10 to the 21st power in a tornado-like blowup. Berkeley Lab
- The same equations describe weather, aircraft and blood flow. Clay
Stories
- Tao's paper also lays out a program for one day adapting Tao’s blowup construction to the true equations. arXiv
- DeepMind's first author, NYU postdoc Yongji Wang, pushed neural-network solutions to near machine precision, the accuracy computer-assisted proofs need. Google DeepMind
Latest news
- Sept 18, 2025: DeepMind with Brown, NYU and Stanford found new families of unstable singularities in three fluid equations using AI. They do not claim to solve the prize problem. Google DeepMind
- Apr 2, 2025: Hou presented potential-singularity evidence at Berkeley Lab. Berkeley Lab
Why it matters: Nobody has proved that the equations engineers rely on every day always behave sensibly.
04Yang–Mills & Mass GapChen-Ning Yang and Robert Mills, 1954; prize statement by Arthur Jaffe and Edward Witten
Prove quantum Yang–Mills theory exists in four dimensions and that its lightest particle has positive mass.
Facts
- The theory must hold for any compact simple gauge group with a mass gap greater than zero. Clay
- The strong force uses gauge group SU(3); the electroweak force uses SU(2)×U(1). Clay
- Wilson's lattice gauge theory (1977) and Balaban's lattice work (1987) are the main rigorous routes. Clay
- Experiments and simulations support the mass gap, but it has never been proved. Clay
Stories
- Gross, Wilczek and Politzer discovered asymptotic freedom in 1973 and won the 2004 Nobel Prize in Physics. Nobel Prize
- Yang and Mills's five-page 1954 paper launched the framework behind the entire Standard Model. Clay
Latest news
- Oct 15, 2025: Sourav Chatterjee of Stanford gave Harvard's Millennium Prize lecture on recent progress. Harvard CMSA
Why it matters: The Standard Model of particle physics rests on a theory that has never been put on rigorous mathematical footing.
05Hodge ConjectureW. V. D. Hodge, 1941; presented at the 1950 ICM
Certain features of nice algebraic shapes can always be built from pieces cut out by polynomial equations.
Facts
- It is known in special cases, such as dimension below four, and open in dimension four. Clay
- Pierre Deligne wrote Clay's official description. Clay
- DARPA listed it in 2008 among its 23 Mathematical Challenges. Britannica
- The degree-2 case is due to Kodaira and Spencer. Clay
Stories
- Hodge's idea got little notice for about nine years, until Hodge’s 1950 talk. Britannica
- Hodge first stated it with integer coefficients; Atiyah and Hirzebruch showed that version is false, so only the rational version survives. Clay
Latest news
- Apr 2025: Floccari and Fu gave a new proof for abelian fourfolds of Weil type with discriminant 1. arXiv
- Aug 2025: a preprint posted a conditional reduction for threefolds; it is not established. arXiv
Why it matters: It would join shape and equation at the deepest level, turning hard calculations into algebra.
06Birch and Swinnerton-DyerBryan Birch and Peter Swinnerton-Dyer, 1950s–1965
How an elliptic curve's L-function behaves at one point tells you how many rational solutions it has.
Facts
- Coates and Wiles (1977) proved a first case for curves with complex multiplication. Clay
- Gross–Zagier (1986) and Kolyvagin (1990) proved the conjecture for analytic rank 0 and 1. Clay
- Aug 2024: Elkies and Klagsbrun found a curve of rank at least 29, beating an 18-year-old record; its coefficients run past 60 digits. Quanta
- Hilbert's tenth problem shows no general method exists for whole-number solutions (Matiyasevich, 1970). Clay
Stories
- The conjecture was found experimentally on EDSAC, one of Cambridge's earliest computers. Clay
- Andrew Wiles, famous for Fermat's Last Theorem, wrote Clay's official description. Clay
Latest news
- Feb 4, 2026: Barry Mazur gave Harvard's Millennium Prize lecture on BSD. Harvard CMSA
- Sept 21–25, 2026: Clay held an Oxford workshop on recent BSD results. Clay
Why it matters: It would give a test for when equations have infinitely many rational solutions.
07Quantum GravityEinstein flagged it in 1916; Rosenfeld tried in 1930
Find one theory that covers both general relativity and quantum mechanics.
Facts
- String theory needs 10 dimensions, 6 curled up, with at least 10 to the 500th possible configurations. Quanta
- In 1996 Strominger and Vafa reproduced black-hole entropy by counting string-theory states. Quanta
- Loop quantum gravity treats space as discrete chunks linked into loops; Carlo Rovelli is among its developers. Quanta
- Feynman and DeWitt found quantized general relativity is non-renormalizable. Quanta
Stories
- String theory began in 1968 when Gabriele Veneziano found a formula for hadrons that turned out to describe vibrating strings. Quanta
- Rovelli was detained in 1977 for refusing military service and later wrote a physics book that sold more than 2 million copies. Quanta
Latest news
- Nov 17, 2025: Stelle's 1977 'quadratic gravity' is making a comeback after a 2024 result that it is asymptotically free. Quanta
- Mar 23, 2026: bootstrap papers argue string theory may be the only consistent option. Quanta
- Sept 25, 2026: progress on holographic models of de Sitter space, the kind our universe resembles. Quanta
Why it matters: Without it, physics cannot describe black-hole interiors or the Big Bang.
The last nine that were actually solved
S1Solved 2025
dim K = 3
Kakeya Conjecture (3D)GeometryPosed by Sōichi Kakeya in 1917: how little room do you need to turn a needle in every direction? Hong Wang and Joshua Zahl proved the 3D case in 2025. (Quanta)
Wang won the 2026 Fields MedalS2Solved 2024
D-mod ≃ QCoh
Geometric LanglandsMathematicsA key piece of the Langlands “grand unified theory of math.” Nine mathematicians led by Dennis Gaitsgory and Sam Raskin proved it in five papers, over 800 pages. (Quanta)
Settled after 30+ yearsS3Solved 2023
one tile, never repeating
The Einstein TileGeometryIs there a single shape that tiles the plane only without ever repeating? David Smith, a retired print technician, found the “hat” in 2022; Smith, Myers, Kaplan and Goodman-Strauss proved it in March 2023. (Quanta)
A 60-year questionS4Solved 2019
∑ψ(q)φ(q)/q = ∞
Duffin–Schaeffer ConjectureNumber theoryPosed in 1941: when can almost every real number be approximated by fractions in a given way? Dimitris Koukoulopoulos and James Maynard proved it in July 2019. (arXiv)
78 years openS5Solved 2019
s(f) ≥ √deg(f)
Sensitivity ConjectureComputer sciencePosed in 1992 by Nisan and Szegedy. Hao Huang proved it in July 2019 in two pages. (Quanta)
Two pages, 27 yearsS6Solved 2016
E8, Leech
Sphere Packing in 8 and 24 DimensionsGeometryWhat is the densest way to pack spheres? Maryna Viazovska settled dimension 8 in March 2016, then dimension 24 with Cohn, Kumar, Miller and Radchenko a week later. (Quanta)
Fields Medal 2022S7Detected 2015
h ∼ 10−21
Gravitational WavesPhysicsPredicted by Einstein in 1916. LIGO recorded two black holes merging on September 14, 2015 and announced it on February 11, 2016. (LIGO)
Nobel Prize 2017S8Found 2012
mH = 125.13 GeV
The Higgs BosonParticle physicsProposed in 1964 by Brout, Englert and Higgs. ATLAS and CMS at CERN announced the particle on 4 July 2012. (CERN)
Nobel Prize 2013S9Solved 2003
π1=0 ⇒ M≅S3
Poincaré ConjectureTopologyPosed by Henri Poincaré in 1904. Grigori Perelman proved it in 2002–2003, the only Millennium Prize Problem solved so far. (Clay)
$1M Clay prize, declined
Think you solved one? The Clay Mathematics Institute does not accept direct submissions. A solution must be published in a qualifying peer-reviewed outlet, wait at least two years, and win general acceptance from mathematicians (Clay rules). Publish first, then tell us at charles@digitalsuperintelligence.us and we will feature it here. No AI, this site included, can certify a proof.
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