On the Erdos-Turán conjecture

We give equivalent formulations of the Erdos-Turán conjecture on the unboundedness of the number of representations of the natural numbers by additive bases of order two of ℕ. These formulations allow for a quantitative exploration of the conjecture. They are expressed through some functions of x ∈ ℕ reflecting the behavior of bases up to x. We examine some properties of these functions and give numerical results showing that the maximum number of representations by any basis is ≥6.