Open Problems

All non-trivial zeros of the Riemann zeta function have real part equal to 1/2.

Every positive integer eventually reaches 1 under the 3n+1 map.

Is every problem whose solution can be verified quickly also solvable quickly?

The rank of an elliptic curve equals the order of vanishing of its L-function at s=1.

Do smooth solutions to the 3D Navier-Stokes equations always exist globally in time?

There are infinitely many pairs of primes differing by 2.

Every even integer greater than 2 is the sum of two primes.

Certain cohomology classes on algebraic varieties are combinations of algebraic subvarieties.

Prove that quantum Yang-Mills theory exists and has a positive mass gap.

For coprime a+b=c, the radical rad(abc) is usually not much smaller than c.

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

Complete Lean 4 formalization of the four color theorem for planar graphs.

For every n >= 2, 4/n can be written as a sum of three unit fractions.

How many colors are needed to color the plane so no two points at distance 1 share a color?

A Besicovitch set in R^n has Hausdorff dimension n.

A polynomial map with constant nonzero Jacobian determinant is invertible.

Somos-k sequences produce integers for k <= 7 despite division in the recurrence.

In any union-closed family, some element appears in at least half the sets.

A Hadamard matrix of order n exists for every n divisible by 4.

A transcendence conjecture that would resolve many open problems in number theory.

Each Markov number determines a unique Markov triple up to permutation.

Can every bounded set in R^n be partitioned into n+1 sets of smaller diameter?

A set in R^d with Hausdorff dimension > d/2 has a distance set of positive measure.

Does there exist a rectangular box with integer edges, face diagonals, and space diagonal?

What is the largest area of a shape that can be moved around an L-shaped hallway?