Limits RegistryRegistry collection
Open and proven boundaries in combinatorics, number theory, geometry, and algorithms.
27 published records in this collectionAndrew 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 exact Euclidean kissing number in dimension 24 is 196560.
The exact Euclidean kissing number in dimension 8 is 240.
The kissing number problem asks how many equal, non-overlapping spheres can simultaneously touch a central sphere of the same size — a question that gets dramatically harder as dimension increases, and is only exactly solved in a handful of dimensions.
The exact Euclidean kissing number in dimension 1 is 2.
The exact Euclidean kissing number in dimension 3 is 12.
The exact Euclidean kissing number in dimension 2 is 6.
The hexagonal packing is the densest congruent-circle packing in the Euclidean plane.
The Leech lattice packing is optimal among all congruent-sphere packings in R^24.
The sphere packing problem asks what fraction of space can be filled by non-overlapping equal spheres. In eight dimensions, the extraordinarily symmetric E8 lattice achieves the provably optimal arrangement — a landmark 2016 result.
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.
Ramsey's theorem guarantees that large enough structures always contain order: however you two-color the edges of a large enough complete graph, a monochromatic triangle becomes unavoidable. R(3,3) is the smallest such threshold.
The current verified interval for the open Ramsey number R(4,6) is 36–40.
The current verified interval for the open Ramsey number R(5,5) is 43–46.
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.
Covering the plane with congruent circles and leaving no gaps forces the circles to overlap. The thinnest such covering — the arrangement with the least overlap, achieved by centering circles on a hexagonal lattice — has density 2π/√27 ≈ 1.209, meaning the circles' total area exceeds the plane's area by about 21%.
The exact three-color triangle Ramsey number is 17.