The index

Explore the frontier

14 entries matching the current filters. Sorted by recently added.

Full hierarchy →

Notorious×Curated problem list×Clear all

OP-0039Computational geometryOpen

The Erdős unit distance problem

How often can the same distance occur among n points in the plane? The gap between the bounds is enormous.

The Open Problems Project (computational geometry)· 1946

OP-0037Algorithms and data structuresOpen

A strongly polynomial algorithm for linear programming

Smale's ninth problem: is there an LP algorithm whose arithmetic operation count does not depend on the bit size of the input?

Smale's problems for the 21st century· 1998

OP-0024AlgebraOpen

The Jacobian conjecture

If a polynomial map has constant non-zero Jacobian determinant, must it be invertible?

Smale's problems for the 21st century· 1939

OP-0022ProbabilityOpen

Conformal invariance of critical percolation on the square lattice

Smirnov proved it for the triangular lattice in 2001. The square lattice remains untouched.

Community consensus· 1992

OP-0020Geometry and topologyOpen

The smooth four-dimensional Poincaré conjecture

Is every smooth 4-manifold homeomorphic to S^4 also diffeomorphic to it?

Kirby's problem list in low-dimensional topology· 1961

OP-0019Number theoryOpen

The Littlewood conjecture

Can two real numbers be simultaneously badly approximable in the multiplicative sense?

Community consensus· 1930

OP-0018AnalysisDisputed

The invariant subspace problem for separable Hilbert spaces

Does every bounded operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?

Community consensus· 1935

OP-0017AnalysisPartially solved

The Kakeya set conjecture in dimensions at least 4

The three-dimensional case fell in 2025. Every dimension above it is still open.

Community consensus· 1971

OP-0012Graph theoryOpen

Hadwiger's conjecture

Is every graph with no K t minor properly colourable with t − 1 colours?

Open Problem Garden· 1943

OP-0010CombinatoricsOpen

Erdős's conjecture on arithmetic progressions

If the reciprocals of a set of integers diverge, must the set contain arbitrarily long arithmetic progressions?

Erdős Problems (Thomas Bloom)· Source reports OPEN· 1936$5,000 prize

OP-0008Number theoryDisputed

The abc conjecture

A claimed proof exists and is not accepted by the wider community — the conjecture's status is genuinely contested.

Community consensus· 1985

OP-0007Number theoryOpen

The binary Goldbach conjecture

Is every even integer greater than 2 a sum of two primes?

Community consensus· 1742

OP-0006Number theoryOpen

The twin prime conjecture

Are there infinitely many primes p such that p + 2 is also prime?

Community consensus· 1849

OP-0005Number theoryOpen

The Collatz conjecture

Does the 3n + 1 iteration reach 1 from every positive integer?

Community consensus· 1937