Unsolved Problems

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?...

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 $n$?...

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?...

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?...

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?...

If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplici...

For how many prime numbers $p$ is a given integer $a$ (not $\pm 1$ or a perfect square) a primitive root modulo $p$?...

For coprime integers $a, b, c$ with $a + b = c$, is $c$ usually not much larger than the product of distinct primes dividing $abc$?...

Can the unit ball in $\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?...

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?...

What are the composition factors of finite groups appearing in genus-0 systems?...

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

Is every finite group the Galois group of some Galois extension of $\mathbb{Q}$?...

Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic?...

Are there infinitely many Leinster groups?...

Does generalized moonshine exist for all elements of the Monster group?...

Is every finitely presented periodic group finite?...

Is every group surjunctive?...

Is every discrete countable group sofic?...

What is the structure of the discrete spectrum of automorphic forms on reductive groups?...

Is there a relationship between the numbers of irreducible characters in blocks of a finite group and its local subgroups?...

Can representations of semisimple algebraic groups be characterized over the integers?...

What is the classification of complete minimal hypersurfaces in spheres of all dimensions?...