Unsolved Problems

Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?...

Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function?...

Is $\|f(A)\| \leq 2\sup_{z \in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?...

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

Do symmetric informationally complete POVMs exist in all dimensions?...

Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?...

Can a finite system of left cosets forming a partition of a group have distinct indices?...

Is there a minimum positive Mahler measure for non-cyclotomic polynomials?...

For which domains do non-zero functions exist with zero integrals over all congruent copies?...

Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time?...

Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?...

If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\geq 1/k$ from others) at some time?...

For a finite family of sets closed under unions, must some element appear in at least half the sets?...

What is the maximum number of points in an $n \times n$ grid with no three collinear?...

For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?...

Does every bridgeless graph have a collection of cycles covering each edge exactly twice?...

For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets?...

Does every finite connected vertex-transitive graph have a Hamiltonian path?...

What is the chromatic number of the plane with unit distance graph coloring?...

Can unknots be recognized in polynomial time?...

Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...

Do quantum invariants of knots relate asymptotically to hyperbolic volume?...

Are certain combinations of Pontryagin classes homotopy invariant?...

Must a Kakeya set in $\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?...

Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources?...

Is the Mandelbrot set locally connected?...

Does every regular compact contact-type level set carry a periodic orbit?...

If a billiard table is strictly convex and integrable, must its boundary be an ellipse?...

If the canonical bundle of a variety is nef, must it be semiample?...

Is the number of countable models of a complete first-order theory finite, $\aleph_0$, or $2^{\aleph_0}$?...

Is every simple group with $\aleph_0$-stable theory an algebraic group over an algebraically closed field?...

Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?...

Does the partition principle (PP) imply the axiom of choice (AC)?...

Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere?...

Does the generalized continuum hypothesis entail the diamond principle $\diamondsuit(E_{\text{cf}(\lambda)}^{\lambda^+})$ for every singular cardinal ...

Does the generalized continuum hypothesis imply the existence of an $\aleph_2$-Suslin tree?...

Does there exist an ultimate core model containing all large cardinals?...

If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem?...

Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?...

Does there exist a Jónsson algebra on $\aleph_\omega$?...

Is the open coloring axiom (OCA) consistent with $2^{\aleph_0} > \aleph_2$?...

Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?...

How many Sudoku puzzles have exactly one solution?...

How many Sudoku puzzles with exactly one solution are minimal (removing any clue creates multiple solutions)?...

What is the maximum number of givens for a minimal Sudoku puzzle?...

Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy?...

What is the outcome of a perfectly played game of chess?...

What is the perfect value of komi (compensation points) in Go?...

What is the largest possible cap set in $n$-dimensional affine space over the three-element field?...

Are the nim-sequences of all finite octal games eventually periodic?...