All limits.

A VC-dimension-d class cannot realize more labelings than the binomial growth bound.

Capacity region of a stationary memoryless multiple-access channel.

Andrew Beal formulated this conjecture in 1993 while investigating generalizations of Fermat's Last Theorem. It asks whether every solution to A^x + B^y = C^z with all exponents greater than 2 forces A, B, and C to share a common prime factor — equivalently, that the equation has no solutions in pairwise-coprime positive integers under those exponents. It remains open and unproven.

Elliptic curves are central to number theory — they were key to Andrew Wiles' proof of Fermat's Last Theorem and underpin widely used cryptographic systems. This conjecture, formulated in the 1960s from numerical experiments, predicts a precise link between how many rational-number solutions an elliptic curve has and the behavior of a related complex analytic function at a single point. It remains one of the deepest open questions about when polynomial equations have rational solutions.

For certain well-behaved geometric spaces called projective algebraic varieties, this asks whether every 'Hodge cycle' — a topological feature identified through the space's structure — can always be built out of actual algebraic (equation-defined) pieces. It is a question about how far topology and algebra can be identified with each other, central to how mathematicians reason about the shape of solution sets to polynomial equations.

The Navier–Stokes equations, formulated in the 19th century, describe how fluids like air and water move and are the basis of weather prediction, aircraft design, and blood-flow modeling. Despite how thoroughly they're used in practice, nobody has proven that their solutions in three dimensions stay smooth (well-behaved) for all time rather than developing a singularity — or found a case where one does.

Formulated independently by Stephen Cook and Leonid Levin in 1971, this asks whether every problem whose solution is easy to check is also easy to solve. If P = NP, most modern cryptography would become breakable in principle; if P ≠ NP, it would confirm that whole classes of problems (from optimal scheduling to protein folding) are inherently intractable to solve exactly at scale. Almost all computer scientists believe P ≠ NP, but no proof exists.

Bernhard Riemann conjectured this in 1859 while studying the distribution of prime numbers. The zeta function's zeros away from the 'trivial' negative even integers all appear to sit exactly on the critical line Re(s) = 1/2 — a pattern verified for the first many trillion zeros but never proven in general. A proof would sharpen almost everything known about how primes are distributed.

The Vesuvius Challenge, a privately funded prize founded by Nat Friedman and Daniel Gross, offers a $1,000,000 Grand Prize (part of a $2.14M total open prize pool) for the first team to virtually unroll and read an entire carbonized Herculaneum scroll using CT scans and ML, without physically opening it. The deadline is June 25, 2027; the Grand Prize remains unclaimed as of August 2026, though smaller Progress and First Letters prizes have been awarded along the way.

XPRIZE Healthspan is a $101M competition (launched 2023, running through 2030) challenging teams to develop a real therapeutic — not just a way to slow aging, but to measurably reverse already-occurred functional decline. As of August 2026, 20 finalist teams are in clinical trials; 10 received $1M Milestone 2 awards, but no grand prize has been claimed.

XPRIZE Quantum Applications is a $5M competition, title-sponsored by Google Quantum AI and presented by GESDA, launched in 2024 to push quantum computing past toy benchmarks toward genuine, judged real-world advantage. Seven finalist teams were selected in December 2025 and are in Phase II as of August 2026; winners are due Spring 2027, with no prize yet claimed.

Yang–Mills theory underlies the Standard Model's description of the strong nuclear force, and its predictions have been repeatedly confirmed in particle accelerators. Physicists have observed empirically that the theory's particles behave as if they have positive mass (a 'mass gap'), even though the classical fields it's built from travel at the speed of light — but no one has constructed the theory with full mathematical rigor or proven the mass gap exists.

Maximum CHSH correlator in local hidden-variable theories.

Stable sampling density cannot be below occupied spectral measure when frequency is measured in hertz.

Classical Kerr spacetime has an event horizon only at or below unit dimensionless spin.

A historically proposed energy-limited information-processing rate whose universality depends on physical modeling assumptions.

Approximately 2BT eigenmodes are well concentrated in both a time interval of duration T and two-sided baseband bandwidth B.

Maximum in-interval energy fraction of a bandlimited signal is the largest prolate-spheroidal concentration eigenvalue.

Compactness bound for static spherical perfect-fluid stars satisfying Buchdahl's assumptions.

Minimum asymptotic lossy coding rate for a memoryless source and single-letter distortion.

Classical sphere-packing upper bound on the reliability exponent above critical rate.

Perceptron mistakes are bounded for linearly separable examples of norm at most R and margin gamma.

The Bellman optimality operator is a gamma contraction for a discounted finite MDP.

Value iteration converges geometrically under the discounted Bellman contraction.

Greedy lower bound on the size of a binary code with prescribed minimum Hamming distance.

Shannon capacity of a finite discrete memoryless channel.

Shannon–Hartley capacity of an ideal band-limited Gaussian channel.

Minimum asymptotic rate for lossless coding of a stationary memoryless source.

Using cyclic frequency, Gaussian windows uniquely attain the minimum uncertainty product up to symmetries.

Minimum orthogonalization time from energy uncertainty.

Maximum nonrotating neutron-star mass is an equation-of-state-dependent frontier, not a universal scalar.

Maximum mass in the original zero-temperature ideal neutron-gas general-relativistic model.

Maximum mass of a cold, nonrotating ideal electron-degenerate white dwarf as a function of composition.

Maximum zero-inter-symbol-interference symbol rate of an ideal real baseband channel of bandwidth B.

Areal radius of the event horizon in the Schwarzschild vacuum solution.

The hexagonal packing is the densest congruent-circle packing in the Euclidean plane.

Conventional Rayleigh angular/separation criterion for two incoherent point sources through an ideal circular aperture.

Conventional coherent periodic-structure resolution criterion requiring capture of diffraction orders.