Open-problem statements, with their sources attached.

An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if n is not 4 or 5 mod 9.

There does not exist a $(2,5)$-perfect number

Taxicab number for $k=5$, $m=2$, and $n=2$ is not known. Whether such a number exists is also not known.

Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$. Whether such a number exists is also not known.

$e + \pi$ is transcendental.

$e\pi$ is transcendental.

$e^{\pi^2}$ is transcendental.

$e^e$ is transcendental.

$\pi^e$ is transcendental.

$\pi^{\sqrt{2}}$ is transcendental.

$\pi^{\pi}$ is transcendental.

$\pi^{\pi^{\pi}}$ is transcendental.

$\pi^{\pi^{\pi^\pi}}$ is transcendental.

$\pi^{\pi^{\pi^\pi}}$ is not an integer.

This would follow from $\pi^{\pi^{\pi^\pi}}$ being transcendental, but this formulation is of interest in its own right, as it could in principle be proven by direct computation. Reference:* YouTube

$\log(\pi)$ is transcendental.

$\log(\log(2))$ is transcendental.

$\sin(e)$ is transcendental.

The Catalan constant $G$ is transcendental.

Are there infinitely many primes p such that p + 2 is prime?

A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \equiv 1 \pmod{p^2}$, where $L_p$ is the $p$-th Lucas number. It is conjectured that there is at least one Wall–Sun–Sun prime.