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Community consensus

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OP-0038Algorithms and data structuresOpen

The polynomial Hirsch conjecture

Santos disproved the linear bound in 2010. Whether the diameter is polynomially bounded is still unknown.

Community consensus· 1957

OP-0025AnalysisPartially solved

Sendov's conjecture

Tao proved it for polynomials of sufficiently large degree. The remaining degrees are open.

Community consensus· 1958

OP-0023ProbabilityOpen

The connective constant of self-avoiding walks on Z²

The hexagonal lattice constant is √(2+√2). Nobody knows the value for the square lattice.

Community consensus· 1953

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-0021Logic and foundationsOpen

Vaught's conjecture

Can a countable first-order theory have strictly between ℵ₀ and 2^ℵ₀ countable models?

Community consensus· 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-0016Geometry and topologyOpen

The kissing number in dimension 5

How many unit balls can touch a central unit ball in R^5? Somewhere between 40 and 44.

Community consensus· 1694

OP-0015Geometry and topologyOpen

Optimal sphere packing in dimension 4

Dimensions 1, 2, 3, 8 and 24 are settled. Dimension 4 is not, and the conjectured answer is the D4 lattice.

Community consensus· 1611

OP-0013CombinatoricsOpen

The Ramsey number R(5,5)

A single finite number nobody knows: the bounds are 43 and 46, and the answer is widely believed to be 43.

Community consensus· 1935

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