## Goldbach conjecture ★★★★

Author(s): Goldbach

Conjecture   Every even integer greater than 2 is the sum of two primes.

Keywords: additive basis; prime

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

Author(s): Erdos; Turan

Let . The representation function for is given by the rule . We call an additive basis if is never .

Conjecture   If is an additive basis, then is unbounded.

Keywords: additive basis; representation function

## The Alon-Tarsi basis conjecture ★★

Author(s): Alon; Linial; Meshulam

Conjecture   If are invertible matrices with entries in for a prime , then there is a submatrix of so that is an AT-base.

Keywords: additive basis; matrix

## The additive basis conjecture ★★★

Author(s): Jaeger; Linial; Payan; Tarsi

Conjecture   For every prime , there is a constant (possibly ) so that the union (as multisets) of any bases of the vector space contains an additive basis.

Keywords: additive basis; matrix