## Goldbach conjecture ★★★★

Author(s): Goldbach

\begin{conjecture} Every even integer greater than 2 is the sum of two primes. \end{conjecture}

## The Erdos-Turan conjecture on additive bases ★★★★

Author(s): Erdos; Turan

Let $B \subseteq {\mathbb N}$. The \emph{representation function} $r_B : {\mathbb N} \rightarrow {\mathbb N}$ for $B$ is given by the rule $r_B(k) = \#\{ (i,j) \in B \times B : i + j = k \}$. We call $B$ an \emph{additive basis} if $r_B$ is never $0$.

\begin{conjecture} If $B$ is an additive basis, then $r_B$ is unbounded. \end{conjecture}

## The Alon-Tarsi basis conjecture ★★

Author(s): Alon; Linial; Meshulam

\begin{conjecture} If $B_1,B_2,\ldots B_p$ are invertible $n \times n$ matrices with entries in ${\mathbb Z}_p$ for a prime $p$, then there is a $n \times (p-1)n$ submatrix $A$ of $[B_1 B_2 \ldots B_p]$ so that $A$ is an AT-base. \end{conjecture}

\begin{conjecture} For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$) so that the union (as multisets) of any $c(p)$ bases of the vector space $({\mathbb Z}_p)^n$ contains an additive basis. \end{conjecture}