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Statement

A sequence $A$ of positive integers $a_1<a_2<\cdots$ has property P if "no term $a_i$ divides the sum of two larger terms" (p. 97). Theorem (p. 98). Every infinite set $A$ with property P has density $0$.

The paper reads the two larger terms as distinct members of the sequence: its conjecture (1) on p. 97, $\max A(x)=[\tfrac13x]+1$ for finite sets with property P, is attained by the $[\tfrac13x]+1$ largest integers not exceeding $x$ (p. 98), and that set violates the condition as soon as the two larger terms may coincide (for $x=3n$ it contains $2n$ and $3n$ with $2n\mid3n+3n$). The site's Problem 12 follows this reading ("no distinct $a,b,c\in A$"); the finite Problem 13 and Bedert's theorem use the reading in which the two larger terms may coincide.

Source. P. Erdős and A. Sárközi, On the divisibility properties of sequences of integers, Proc. London Math. Soc. (3) 21 (1970), 97--101; the definition and (1) on printed p. 97 (PDF p. 1 of the five-page Rényi scan), the Theorem and the remarks on printed p. 98 (PDF p. 2), read on the page images. Received 13 March 1970; DOI 10.1112/plms/s3-21.1.97 (Crossref record read).

Read depth. Claims checked: the definition, display (1), the Theorem and the best-possible remark were read clause by clause on the page images. The proof (Lemma 1, Lemma 2 and the deduction on pp. 98--101) was not read.

Proof pointer

Two lemmas (pp. 98--99). Lemma 1: for $l$ an integer, $x>x_0(l)$ and $a_1<\cdots<a_k\le x$ with $k>c_1x$, there is $d<l^{c_2}$ with $P(d)\le l$ such that more than $c_3x/(d\log l)$ of the $a_i$ have the form $dt$ with $p(t)>l$, where $p(n)$ and $P(n)$ are the least and greatest prime factors of $n$. A second lemma and a counting argument then show that a set of positive upper density contains a term dividing the sum of two larger terms. Not reconstructed here.

The remark after the Theorem (p. 98): the result is best possible in the sense that for any increasing $f(x)\to\infty$ there is a property-P sequence with $A(x_v)>x_v/f(x_v)$ along a sequence $x_v\to\infty$; the sequence consists of all integers $a$ with $y_i<a<\tfrac32y_i$ and $a\equiv1\ (\mathrm{mod}\ (2y_{i-1})!)$, $i=2,3,\ldots$, for a fast-growing $y_1<y_2<\cdots$, and $A(\tfrac32y_i)>y_i/(2\cdot(2y_{i-1})!)>y_i/f(y_i)$.

Dependencies

None outside the paper.

Bears on

• Problem 12: the first result on the infinite question, quoted by the site as "who proved that $A$ must have density $0$"; the best-possible remark is the site's "essentially best possible" construction. The paper's conjectures on the same page are on the conjecture page.