##
**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 123.25503

**Autor: ** Erdös, Pál

**Title: ** Remarks on number theory. III (In Hungarian)

**Source: ** Mat. Lapok 13, 28-37 (1962).

**Review: ** Let a_{1} < a_{2} < ···, A(x) = **sum**_{ai \leq x} 1 be an infinite sequence for which (1) a_{k} = a_{i1}+···+a_{ir}, i < ··· < i_{r} < k, is not solvable. I prove that A(x)/x ––> 0 and that **sum** {1 \over a_{i}} < 103. Further I show that A(x) = o(x) is best possible, but there always exists a sequence x_{i} ––> oo for which (2) A(x) < Cx_{i}^{(\sqrt5-1)/2}. On the other hand, there exists a sequence A for which (1) has no solutions, but A(x) > cx^{2/7} for every x. Perhaps (2) can be improved, but the exponent can certainly not be made smaller than 2/7. Consider now the sequences A for which the equation (1') a_{r1}+···+a_{r_{s1}} = a_{l1}+···+a_{l_{s2}}, r_{1} < ··· < r_{s1}; l_{1} < ··· < l_{s2}; s_{1} \ne s_{2}, is not solvable for every choice of s_{1} \ne s_{2}. There exists such a sequence with A(x) > c_{x}^{\alpha} for every x if \alpha is sufficiently small. On the other hand, I show by using Rényi's strengthening of the large sieve of Linnik that if A is such that (1') has no solutions, then A(x) < cx^{5/6} for every x if c is a sufficiently large absolute constant. Perhaps the exponent 5/6 can be improved, but I have not succeeded in doing this.

**Classif.: ** * 11B83 Special sequences of integers and polynomials

**Index Words: ** number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag