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

Representation of Integers by Near Quadratic Sequences

Labib Haddad
120 rue de Charonne
75011 Paris

Charles Helou
Department of Mathematics
Pennsylvania State University
25 Yearsley Mill Road
Media, PA 19063


Following a statement of the well-known Erdos-Turán conjecture, Erdos mentioned the following even stronger conjecture: if the n-th term an of a sequence A of positive integers is bounded by $\alpha n^2$, for some positive real constant $\alpha$, 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 an differs from $\alpha n^2$ (or from a quadratic polynomial with rational coefficients q(n)) by at most $o (\sqrt {\log n})$, then the number of representations function is indeed unbounded.

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

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