Unsolved Problems

Does $P = NP$? More formally: if the solution to a problem can be quickly verified (in polynomial time), can the solution also be quickly found (in po...

Do all non-trivial zeros of the Riemann zeta function $\zeta(s)$ have real part equal to $\frac{1}{2}$?...

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere....

Prove that Yang–Mills theory exists and has a mass gap on $\mathbb{R}^4$, meaning the quantum particles have positive masses....

Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?...

The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1...

On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles....

Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For...

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....

For any $\epsilon > 0$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $c > \text{rad}(abc...

What is the minimum number of colors needed to color the points of the plane such that no two points at distance 1 have the same color?...

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

Every finite simple graph on at least 3 vertices is uniquely determined by its vertex-deleted subgraphs....

If a graph is the union of $n$ cliques of size $n$, no two of which share more than one vertex, then the chromatic number is $n$....

No packing of congruent spheres in three dimensions has density greater than $\frac{\pi}{\sqrt{18}} \approx 0.74048$....

Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?...

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...

There is no set whose cardinality is strictly between that of the integers and the real numbers....

Does every bounded linear operator on a separable Hilbert space over the complex numbers have a non-trivial invariant subspace?...

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

For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$....

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

Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?...

Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1....

What is the exact value of $R(5,5)$, the smallest number $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic $K_5$?...

For any $n$ runners on a circular track with distinct constant speeds, each runner is "lonely" (distance at least $1/n$ from all others) at some time....

Every tree can be gracefully labeled: vertices can be assigned distinct labels from $\{0, 1, \ldots, |E|\}$ such that edge labels (absolute difference...

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

What is the largest area of a shape that can be maneuvered through an L-shaped corridor of unit width?...

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

Every topological manifold can be triangulated....

A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjec...

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

A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal....

If a function on $\mathbb{R}^n$ has zero integral over every congruent copy of a given domain, must the function be identically zero?...

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...

The diameter of the graph of a $d$-dimensional polytope with $n$ facets is bounded by a polynomial in $d$ and $n$....

Extend the Kronecker-Weber theorem on abelian extensions of the rationals to any base number field....

Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real a...

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 $n$?...