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 a1 < a2 < ···, A(x) = sumai \leq x 1 be an infinite sequence for which (1) ak = ai1+···+air, i < ··· < ir < k, is not solvable. I prove that A(x)/x ––> 0 and that sum {1 \over ai} < 103. Further I show that A(x) = o(x) is best possible, but there always exists a sequence xi ––> oo for which (2) A(x) < Cxi(\sqrt5-1)/2. On the other hand, there exists a sequence A for which (1) has no solutions, but A(x) > cx2/7 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') ar1+···+ar_{s1} = al1+···+al_{s2}, r1 < ··· < rs1; l1 < ··· < ls2; s1 \ne s2, is not solvable for every choice of s1 \ne s2. There exists such a sequence with A(x) > cx\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) < cx5/6 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

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