Unsolved Problems

Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this ...

Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31)....

Every even integer greater than 2 can be expressed as the sum of two primes....

Every graph with chromatic number $k$ has a $K_k$ minor (where $K_k$ is the complete graph on $k$ vertices)....

What is the densest packing of congruent spheres in $n$ dimensions for $n \geq 4$?...

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

A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero diviso...

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

A Kakeya set (containing a unit line segment in every direction) in $\mathbb{R}^n$ must have Hausdorff dimension $n$....

For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement....

For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample....

Do solutions to the 3D Euler equations for incompressible fluid flow remain smooth for all time, given smooth initial data?...

If $\kappa$ is a singular strong limit cardinal, then $2^\kappa = \kappa^+$....

Is every abelian group $A$ such that $\text{Ext}^1(A, \mathbb{Z}) = 0$ a free abelian group?...

For certain constraint satisfaction problems (unique games), it is NP-hard to approximate the maximum fraction of satisfiable constraints beyond a cer...

Are there infinitely many primes of the form $n^2 + 1$?...

Find efficient algorithms for deciding whether a polynomial with integer coefficients has an integer root....

Find effective uniform bounds for the heights of rational points on algebraic curves....

For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?...

Find a strongly polynomial algorithm for linear programming....

Is the $C^r$ closing lemma true for dynamical systems?...

If $F: \mathbb{C}^n \to \mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible....

Does every simple closed curve in the plane contain all four vertices of some square?...

Every bridgeless graph has a cycle double cover: a collection of cycles that covers each edge exactly twice....

Prove that the general equation of the seventh degree cannot be solved using functions of only two variables....

Determine the structure of centralizers of generic diffeomorphisms....

Develop high-dimensional mathematics to model and predict behavior in large-scale distributed networks....

Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals....

Determine whether algebraic geometry can systematically replace linear algebra in optimization....

Create scalable mathematics for differential games, replacing traditional PDE approaches....

Develop asymptotics for systems with massive degrees of freedom....

Use mathematical duality and geometry as foundations for developing novel computational algorithms....

Find lower bounds for sensing complexity as data collection grows, addressing entropy maximization....

Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales....

Strengthen mathematical theory for isometric and rigid embedding relevant to protein folding....

Develop mathematics for creating optimal symmetric structures through nanoscale self-assembly....

Establish appropriate distance metrics on genome space incorporating biological utility....

If $\alpha$ is algebraic and irrational, and $\beta$ is algebraic and irrational, is $\alpha^\beta$ transcendental?...

Extend the theory of quadratic forms with algebraic numerical coefficients....

Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?...

Rigorously justify Schubert's enumerative geometry....

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 $\{0\}$?...

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?...

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?...

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 $C_{n+1} = 2^{C_n} - 1$....