Unsolved Problems

For a matroid of rank $n$ with $n$ disjoint bases $B_1, \ldots, B_n$, can we always find an $n \times n$ matrix whose rows are the bases and whose col...

For a finite group $G$ and prime $p$, is the number of irreducible complex characters of $G$ whose degree is not divisible by $p$ equal to the corresp...

Is every group surjunctive? That is, for any group $G$, if $\phi: A^G \to A^G$ is a cellular automaton that is injective, must it also be surjective?...

Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$....

Are there infinitely many prime numbers of the form $2^p - 1$ where $p$ is prime?...

What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$?...

Among all centrally symmetric convex bodies in $\mathbb{R}^n$, does the cube (or cross-polytope) minimize the product of the body's volume and the vol...

Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources?...

What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees?...

What is the shortest path that guarantees escape from a forest of known shape and size, starting from an unknown location?...

If $n$ complete graphs, each with $n$ vertices, have the property that every pair of complete graphs shares at most one vertex, can the entire graph b...

For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?...

Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle?...

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

Is there a set whose cardinality is strictly between that of the integers and the real numbers?...

Are there infinitely many primes $p$ such that $2p + 1$ is also prime?...

On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles?...

For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$?...

Does every Reeb vector field on a closed contact manifold have at least one periodic orbit?...

In the $n$-body problem with $n \geq 4$, can non-collision singularities occur in finite time?...

For any $k$-chromatic graph, can its $k$-colorings be transformed into each other by recoloring one vertex at a time, staying within $k$ colors, in po...

Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\pi, 1)$ space?...

Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint?...

Can every tiling of $\mathbb{R}^n$ by unit hypercubes have two cubes that share a complete $(n-1)$-dimensional face?...

If $z_1, \ldots, z_n$ are complex numbers that are linearly independent over the rationals, then the transcendence degree of $\mathbb{Q}(z_1, \ldots, ...

For every even number $n$, are there infinitely many pairs of consecutive primes differing by $n$?...

For any linear hypergraph with $n$ edges, each of size $n$, can the vertices be colored with $n$ colors such that no edge is monochromatic?...

Can graph isomorphism be decided in quasi-polynomial time for all graphs?...

For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$?...

For every integer $n \geq 2$, can $\frac{4}{n}$ be expressed as the sum of three unit fractions $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$?...

What is the optimal error term in the formula for the number of lattice points inside a circle of radius $r$?...

Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=...

Can Schubert calculus be given a rigorous foundation?...

What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?...

Does every simple closed curve in the plane contain four points that form the vertices of a square?...

If a compact set in $\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?...

Can every graph be totally colored with at most $\Delta + 2$ colors, where $\Delta$ is the maximum degree?...

For every graph $G$, is the list chromatic number equal to the chromatic number?...

For a monotone graph property, is the threshold for a random graph to have this property at most a constant factor away from the expectation threshold...

For a univalent function $f(z) = z + \sum_{n=2}^\infty a_n z^n$ on the unit disk, is $|a_n| \leq n$ for all $n$?...

If all zeros of a polynomial lie in the closed unit disk, does each zero have at least one critical point within unit distance from it?...

Do there exist any odd perfect numbers? (A perfect number equals the sum of its proper divisors.)...

Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?...

For varieties over finite fields, are the $\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?...

If a dense linear order without endpoints is complete and has the countable chain condition, must it be isomorphic to the real numbers?...

Is the chromatic number of a graph at most its clique cover number times the maximum chromatic number of its neighborhoods?...

Is the number of sum-free subsets of $\{1, 2, \ldots, n\}$ equal to $O(2^{n/2})$?...

If polynomials satisfy certain necessary divisibility conditions, do they simultaneously produce infinitely many primes for integer inputs?...

Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?...

Is every piecewise-polynomial function $f: \mathbb{R}^n \to \mathbb{R}$ the maximum of finitely many minimums of finite collections of polynomials?...