Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.8 |

120 rue de Charonne

75011 Paris

France

Charles Helou

Department of Mathematics

Pennsylvania State University

25 Yearsley Mill Road

Media, PA 19063

USA

**Abstract:**

Following a statement of the well-known Erdos-Turán conjecture,
Erdos mentioned the following even stronger conjecture: if the
*n*-th term *a*_{n} of a sequence *A* of positive integers is bounded by
,
for some positive real constant ,
then
the number of representations of *n* as a sum of two terms
from *A* is an unbounded function of *n*. Here we show that if *a*_{n}
differs from
(or from a quadratic polynomial with rational
coefficients *q*(*n*)) by at most
,
then the number of
representations function is indeed unbounded.

Received July 19 2012.
revised version received October 14 2012.
Published in *Journal of Integer Sequences*, October 23 2012.

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