On the Erd s–Turán conjecture

We give equivalent formulations of the Erd s–Turán conjecture on the unboundedness of the number of representations of the natural numbers by additive bases of order two of . These formulations allow for a quantitative exploration of the conjecture. They are expressed through some functions of 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.