Open-problem statements, with their sources attached.

For any k ≥ 2, let a₁,...,aₖ and b₁,...,bₖ be integers with aᵢ > 0. Suppose that for every prime p there exists an integer n such that p ∤ ∏ i, (aᵢ n + bᵢ). Then there exist infinitely many n such that aᵢ n + bᵢ is prime for all i.

Zagier's conjecture*

The $\mathbb{Q}$-dimension of the vector space spanned by all multiple zeta values of weight $n$ equals $d_n$, where $d_n$ is the Zagier dimension sequence satisfying $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \geq 3$.