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Statement

A sequence $A=\{a_1<a_2<\cdots\}$ of positive integers has Property P if $a_i\nmid a_j+a_k$ for $i<j<k$ (p. 1, Erdős and Sárközy's definition). Theorem (p. 1). The set $S\subset\mathbb N$ of displays (1)--(2) below has Property P, and its counting function satisfies

$$S(x)\gg\frac{\sqrt x}{\sqrt{\log x}\,(\log\log x)^2\,(\log\log\log x)^2}.$$

Here $S(x)$ counts the elements of $S$ below $x$, as the paper's $A(x)=\sum_{a_i<x}1$ (p. 1) does, and, by the paper's notation (Section 2, p. 2), $f\gg g$ means that there is a constant $c>0$ with $f(x)\ge c\,g(x)$ for all sufficiently large $x$; the bound therefore holds for every large $x$, not only along a sequence.

The construction (displays (1)--(2), p. 2): $S=\bigcup_{i\ge1}S_i$ with $S_i=\{n\in\mathbb N:n=q_i^4\nu^2\}$, where $\nu$ runs over the products of exactly $i$ distinct primes $p\equiv3\ (\mathrm{mod}\ 4)$ and $q_i$ is the $i$-th prime in the class $3$ modulo $4$; the factor $q_i^4$ is an indicator of the set $S_i$ that keeps the union in Property P.

Source. C. Elsholtz and S. Planitzer, On Erdős and Sárközy's sequences with Property P, Monatsh. Math. 182 (2017), no. 3, 565--575, DOI 10.1007/s00605-016-0995-9 (published online 18 October 2016; Crossref record read). The copy read for this page is arXiv:1609.07935v1 (26 September 2016, 8 pp.), whose pagination is used here; the journal text was not compared. The Theorem on p. 1 and the construction on p. 2, read in the text layer and on the page images.

Read depth. Claims checked: the definition, the Theorem, displays (1)--(2) and the statements of Lemmas 1 and 2 were read clause by clause in the text layer and on the page images. Section 3 (Property P of the union, Lemmas 1 and 2), Section 4 (products of $k$ distinct primes, Lemma A and Corollary 1) and Section 5 (the counting function, with the proof of the Theorem) were read for their structure and not checked step by step.

Proof pointer

Section 3: Lemma 1 (p. 2) shows that if a prime $p\equiv3\ (\mathrm{mod}\ 4)$ divides $n_1$ but not $\gcd(n_2,n_3)$, then $n_1^2\nmid n_2^2+n_3^2$, because $-1$ is a quadratic non-residue modulo $p$; Lemma 2 (p. 3) deduces that any union of the sets $S_i$ has Property P, through the indicator $q_i$ when the three elements do not all lie in one $S_i$ and through their equal number of prime factors when they do. Section 4 bounds from below the number of squarefree integers up to $x$ with exactly $k$ prime factors, all $\equiv3\ (\mathrm{mod}\ 4)$, for $k$ near $\tfrac12\log\log x$ (Corollary 1, p. 3, from Lemma A). Section 5 (pp. 5--7) gives $S_i(x)\gg\sqrt x/(\sqrt{\log x}\,(\log\log x)^{5/2}(\log\log\log x)^2)$ for every $i$ with $\lfloor\tfrac12\log\log\sqrt x\rfloor+2\le i\le\lfloor\tfrac12\log\log\sqrt x\rfloor+\lfloor\sqrt{\tfrac12\log\log\sqrt x}\rfloor$, and sums these bounds (p. 7). Not reconstructed here.

Dependencies

Lemma A (p. 3), the special case of X. Meng's Lemma 9 (cited as the arXiv preprint 1607.01882) for squarefree integers with $k$ prime factors, all $\equiv3\ (\mathrm{mod}\ 4)$, uniformly for $2\le k\le A\log\log x$; the computed bounds $0.0482<M(3,4)<0.0483$ of Languasco and Zaccagnini and bounds on the Gamma function and its derivatives obtained with Mathematica (p. 4); Mertens' and Stirling's formulas; the asymptotic $2i\log i$ for the $i$-th prime $\equiv3\ (\mathrm{mod}\ 4)$ (p. 5); and the fact that $-1$ is a quadratic non-residue modulo a prime $p\equiv3\ (\mathrm{mod}\ 4)$.

Bears on

• Problem 12: Property P with $i<j<k$ is the problem's condition (no element divides the sum of two distinct larger elements), so $S$ is a set of the kind the problem asks about, with counting function $\gg N^{1/2}/((\log N)^{1/2}(\log\log N)^2(\log\log\log N)^2)$ for all large $N$, the bound the site's commentary credits to Elsholtz and Planitzer. The bound is below $N^{1/2}$, so it gives neither the first question's positive liminf against $N^{1/2}$ nor a set with at least $N^{1-c}$ elements up to $N$ for every $c>0$, and the theorem says nothing about reciprocal sums; it answers none of the problem's three questions. The larger counting functions $N/(\log N)^{O(\log\log\log N)}$ of the 2026 constructions are pending claims recorded on the problem page.