Unsolved Problems

If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random with $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translat...

Let $r$ be fixed and let $H(r)$ be the Hamming ball of radius $r$ in $\mathbb{F}_2^n$. Let $f(r)$ be the smallest constant such that there exist infin...

How many rotated (about the origin) copies of the "pyjama set" $\{(x, y) \in \mathbb{R}^2 : \operatorname{dist}(x, \mathbb{Z}) \leq \varepsilon\}$ are...

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings?...

Let $N$ be large. For each prime $p$ with $N^{0.51} \leq p < 2N^{0.51}$, pick a residue $a(p) \in \mathbb{Z}/p\mathbb{Z}$. Is $\#\{n \in [N] : n \equi...

Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leq p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have s...

Can we pick residue classes $a_p \pmod p$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least 10 of them?...

What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \pmod p$, one for each prime $p \leq x$?...

Pick $x_1, \dots, x_k \in A_n$ at random. Is it true that, almost surely as $n \to \infty$, the random walk on this set of generators and their invers...

Find bounds in the classification theorem for approximate groups....

In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard?...

Let $d \geq 3$ be an odd integer. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ the following is true: given any homogeneous polynomial $F(\mathbf...

Finding a single solution to a polynomial equation $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of su...

Is every group well-approximated by finite groups?...

For every positive integer $k$, does there exist a Hadamard matrix of order $4k$?...

If a ring has no nil ideal other than $\{0\}$, does it follow that it has no nil one-sided ideal other than $\{0\}$?...

Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?...

For a left-and-right Noetherian ring $R$, is the intersection of all powers of the Jacobson radical $J(R)$ equal to zero?...

Do SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite dimensions?...

If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f...

Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugati...

For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2,5)$ finite?...

If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$?...

Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths?...

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?...

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?...

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?...

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

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=...