LUMINA
Explore the Unknown.
A space to question, experiment, and discover.
Some questions have resisted the best mathematicians for decades or centuries. Here you can study them, test ideas with real code, keep your evidence and share careful contributions with reviewers.
How to take part
- 1 Understand Read the precise statement, what is proved, what is only observed and what is still open.
- 2 Explore Run experiments in the IDE's sandbox and change the code.
- 3 Record Keep notes and save runs as evidence: the exact code, input, output and environment.
- 4 Contribute Submit an idea, an experiment, a proof draft, a counterexample or a literature review for review.
Scientific integrity. A program that runs successfully is not a proof, and checking many cases does not prove a statement about all cases. A problem's status here changes only with documented, authoritative recognition — never because of a run, an AI answer or an accepted contribution.
Research problems
6 of 6 problems
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Discrete dynamics Not a Millennium Prize Problem Status: Open
Collatz Conjecture
Start with any positive integer. Halve it if it is even, otherwise triple it and add one. Does every starting value eventually reach 1?
- Reference
- Jeffrey C. Lagarias
- Accepted contributions
- 0
- Status last verified
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Number theory Millennium Prize Problem Status: Open
Riemann Hypothesis
All non-trivial zeros of the Riemann zeta function have real part 1/2. If true, it pins down how regularly the prime numbers are distributed.
- Reference
- Clay Mathematics Institute
- Accepted contributions
- 0
- Status last verified
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Computational complexity Millennium Prize Problem Status: Open
P versus NP
If a proposed answer can be checked quickly, can it also be found quickly? Formally: is every problem in NP also in P?
- Reference
- Clay Mathematics Institute
- Accepted contributions
- 0
- Status last verified
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Analysis and PDE Millennium Prize Problem Status: Open
Navier–Stokes Existence and Smoothness
Do smooth, physically reasonable solutions of the 3D incompressible Navier–Stokes equations exist for all time, or can they break down in finite time?
- Reference
- Clay Mathematics Institute
- Accepted contributions
- 0
- Status last verified
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Mathematical physics Millennium Prize Problem Status: Open
Yang–Mills Existence and Mass Gap
Construct quantum Yang–Mills theory on four-dimensional space-time with mathematical rigour, and prove that its lightest particle has strictly positive mass.
- Reference
- Clay Mathematics Institute
- Accepted contributions
- 0
- Status last verified
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Arithmetic geometry Millennium Prize Problem Status: Open
Birch and Swinnerton-Dyer Conjecture
The number of independent rational points on an elliptic curve (its rank) equals the order of vanishing of its L-function at s = 1.
- Reference
- Clay Mathematics Institute
- Accepted contributions
- 0
- Status last verified
The Clay Mathematics Institute named seven Millennium Prize Problems in 2000. The Poincaré Conjecture was solved by Grigori Perelman; the Hodge Conjecture is not part of this workspace yet. The Collatz Conjecture is a famous open problem but not a Millennium Prize Problem. claymath.org