Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  070.04104
Autor:  Erdös, Pál; Fuchs, W.H.J.
Title:  On a problem of additive number theory. (In English)
Source:  J. London Math. Soc. 31, 67-73 (1956).
Review:  Let {ai} be a non decreasing infinite sequence of non negative integers, f(n) the number of solutions of ai+aj = n and r(n) the number of solution of ai+aj \leq n. Erdös and Turán [J. London Math. Soc. 16, 212-215 (1941; Zbl 061.07301)] conjectured that r(n)-cn = O(1) cannot hold. In the present paper the authors prove (1) If c > 0, then r(n) = cn+o(n1/4 log- ½ n) cannot hold. (2) If c > 0, or c = 0 and ak < Ak2 then

limsupn ––> oo 1/n sumk = 0n (f(k)-c)2 > 0.

Theorem 2. contains a theorem of Dirac-Newman (Zbl 043.04702) who proved that f(n) cannot be a constant for n \geq n0.
Reviewer:  S.Selberg
Classif.:  * 11B34 Representation functions
Index Words:  number theory

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