Open-problem statements, with their sources attached.

$\pi^{\pi}$ 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.

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 are infinitely many Wall-Sun-Sun primes.

A Lucas–Wieferich prime associated with $(a,b)$ is an odd prime $p$, not dividing $a^2 - 4b$, such that $U_{p-\varepsilon}(a,b) \equiv 0 \pmod{p^2}$ where $U(a,b)$ is the Lucas sequence of the first kind and $\varepsilon$ is the Legendre symbol $\left({\tfrac {a^2-4b}{p}}\right)$. The discriminant of this number is the quantity $a^2 - 4b$. It is conjectured that there are infinitely many Lucas–Wieferich primes of any given non-one fundamental discriminant.

TODO: Source this conjecture

It is conjectured that there are infinitely many Wolstenholme primes. Reference:* Wikipedia

There are infinitely many prime numbers of the form k * 2 ^ k - 1 for k > 1.