Open-problem statements, with their sources attached.

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

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.

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

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 $n$ 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.

The number $\pi$ is normal in base $10$, so every block of $k$ decimal digits has limiting frequency $10^{-k}$.

The sequence will eventually reach $1$.

If $n = ab$ is a crystal, then there are no other pairs of positive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and $B(c, d) ∈ ℕ$, i.e., the components of the crystals are unique.

Conjecture 1.1*: For any odd prime $k$, the sum associated with the classical theta function $θ_3$, $S(k)$ is positive.

Conjecture 4.1*: For any prime $k$ larger than $5$, $S(k) > k$.

Conjecture 4.2*: For any prime $k$ larger than $233$, $S(k) > 2k$.

Conjecture 4.3*: For any prime $k$ larger than $3119$, $S(k) > 3k$.

Conjecture 4.4*: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$), $nk < S(k)$.

Let D be the diagonal group of SL_n(ℝ) where n ≥ 3. Then any relatively compact D-orbit in SL_n(ℝ) / SL_n(ℤ) is closed.

Problem 10.1. Are there a transcendental number $\alpha$ and a positive real number $\xi$ such that $\lVert \xi \alpha^n \rVert$ tends to~$0$ as~$n$ tends to infinity? [Har19] (Trivial for $|\alpha| < 1$)