Open-problem statements, with their sources attached.

Every nontrivial zero $\rho$ of the Riemann zeta function satisfies $\operatorname{Re}(\rho)=\tfrac12$.

Determine whether $\mathbf{P}=\mathbf{NP}$: can every decision problem whose solutions are verifiable in polynomial time also be solved in polynomial time?

For three-dimensional incompressible flow, determine whether $\partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\Delta u$ with $\nabla\cdot u=0$ always has a global smooth solution, or exhibit finite-time breakdown.

Every even integer $n>2$ can be written as $n=p+q$ with $p$ and $q$ prime.

For every $n>0$, repeatedly apply $T(n)=n/2$ when $n$ is even and $T(n)=3n+1$ otherwise. Some iterate $T^m(n)$ equals $1$.

There are infinitely many primes $p$ for which $p+2$ is also prime.

On a smooth projective complex variety, every rational Hodge class of type $(p,p)$ is a rational linear combination of classes of algebraic cycles.

For an elliptic curve $E/\mathbb{Q}$, $\operatorname{rank}E(\mathbb{Q})=\operatorname{ord}_{s=1}L(E,s)$.

For every compact simple gauge group $G$, construct a nontrivial quantum Yang-Mills theory on $\mathbb{R}^4$ whose spectrum has a positive mass gap $\Delta>0$.

For every positive integer $n$, there is a prime $p$ satisfying $n^2<p<(n+1)^2$.

Every perfect number $n$, satisfying $n=\sum_{d\mid n,\ d<n}d$, is even.

If positive integers satisfy $A^x+B^y=C^z$ with $x,y,z>2$, then $A$, $B$, and $C$ share a prime factor.

Determine whether positive integers $a,b,c$ exist such that $a^2+b^2$, $a^2+c^2$, $b^2+c^2$, and $a^2+b^2+c^2$ are all perfect squares.

For every positive integer $n$ divisible by $4$, there is an $n\times n$ matrix $H$ with entries in $\{\pm1\}$ and $HH^{\mathsf T}=nI$.

Every finite nontrivial union-closed family $\mathcal{A}$ contains an element belonging to at least $|\mathcal{A}|/2$ members.

For $n$ runners with distinct constant speeds on a unit circle, each runner is at circular distance at least $1/n$ from every other runner at some time.

Every tree with $m$ edges has an injective labeling $f:V\to\{0,\ldots,m\}$ whose edge differences are exactly $\{1,\ldots,m\}$.

Every Jordan curve $\gamma:S^1\to\mathbb{R}^2$ contains four points that are the vertices of a nondegenerate square.

Every synchronizing deterministic finite automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$.

For every number of petals $k$, there is a constant $c_k$ such that every $n$-uniform family with more than $c_k^n$ sets contains a $k$-sunflower.