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Statement

The paper's definitions (p. 71): "Let $A_x$ be a set of positive integers with the least common multiple of each pair of terms not exceeding $x$ and $|A_x|$ being the largest", and "let $B_x$ be the union of the set of positive integers not exceeding $\sqrt{x/2}$ and the set of even integers between $\sqrt{x/2}$ and $\sqrt{2x}$". Theorem (p. 71).

$$|A_x\setminus B_x|=o(\sqrt x).$$

In particular $|A_x|=|B_x|+o(\sqrt x)=\sqrt{\tfrac98x}+o(\sqrt x)$. The Note after the theorem says the $o(\sqrt x)$ can be given explicitly from the proof, and that $|A_x\cap B_x|=\sqrt{9x/8}+o(\sqrt x)$ and $|B_x\setminus A_x|=o(\sqrt x)$ (pp. 71--72).

The introduction records (p. 71) that Erdős proposed the problem in 1951 (the paper's [3], the Mat. Lapok problem), that $\sqrt{9x/8}+O(1)\le|A_x|\le\sqrt{4x}+O(1)$ with a proof in the paper's [4] (Erdős 1965), that Choi improved the upper bound to $1.638\sqrt x$ and then to $1.43\sqrt x$, and that the problem is E2 and part of B26 in Guy's book.

Source. Yong-Gao Chen, Sequences with bounded l.c.m. of each pair of terms, Acta Arith. 84 (1998), no. 1, 71--95, DOI 10.4064/aa-84-1-71-95 (the Crossref record was read); the retained file is the publisher's 25-page PDF; the Theorem is on printed p. 71 = PDF p. 1 and the Note runs over pp. 71--72, read in the text layer and on the page images.

Read depth. Claims checked: the definitions, the Theorem and the Note were read clause by clause on the page images of pp. 71--72. The proof (Sections 1--2, pp. 72--95) was not read beyond the statement of Lemma 1 (p. 72).

Proof pointer

Section 1 (pp. 72--76) proves sieve lemmas; Lemma 1 uses the Eratosthenes--Legendre sieve to find $k$ in $(c_1x^{1/2}+c_2,\,c_1(x^{1/2}+x^{1/4})+c_2)$ such that every prime factor of $\prod_i(a_ik+b_i)$ exceeds $\log x/(6\log M)$. Section 2 (pp. 76--95) proves the Theorem by a case analysis of the elements of $A_x$ against $B_x$. Not read here.

Dependencies

Standard sieve results (Halberstam--Richert, the paper's [6]).

Bears on

• Problem 441: answers the first question asymptotically: $g(N)=|A_N|\sim(9N/8)^{1/2}$, the value of Erdős's construction, which is the site's "Chen established the asymptotic".