On the Erdos-Turan conjecture

We give equivalent formulations of the Erdos Turan 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 is an element of 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 greater than or equal to 6. (C) 2003 Elsevier Inc. All rights reserved.