The Erdos-Turan conjecture on additive bases
Let . The representation function for is given by the rule . We call an additive basis if is never .
This famous conjecture seems intuitively likely, but to date, there has been relatively little progress on it despite considerable attention. Two positive results are a theorem of Dirac [D] which shows that cannot be constant from some point on, and a theorem of Borwein, Choi, and Chu [BCC] which shows that cannot be bounded above by .
On the other hand, if we consider the related problem for subsets of integers instead of natural numbers, Nathanson [N] has shown that the conjecture does not hold.
[BCC] P. Borwein, S. Choi, and F. Chu, An old conjecture of Erdos-Turan on additive bases, Mathematics of Computation. Volume 75, Number 253, Pages 475–484.
[D] G. A. Dirac, Note on a problem in additive number theory, J. London Math. Soc. 26 (1951), 312–313.
[EG] P. Erdos and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden’s theorem and related topics, Enseign. Math. (2) 25 (1979), no. 3-4, 325–344 (1980). MathSciNet
*[ET] P. Erdos and P. Turan, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215. MathSciNet
* indicates original appearance(s) of problem.