---

Statement

Let $T_x$ be the set of numbers $1<n\le x$ with $pn>x$, where $p$ is the least prime divisor of $n$. The Schinzel–Szekeres set $S_x$ consists of the primitive elements of $T_x$: those elements of $T_x$ with no proper divisor in $T_x$ (p. 262). Every element of $T_x$ is divisible by an element of $S_x$, and every element of $S_x$ exceeds $\sqrt x$ (used on pp. 262 and 265).

Lemma 2.1 (printed p. 262). If $m,n\in S_x$ and $m\ne n$, then the least common multiple of $m$ and $n$ exceeds $x$.

The paper prints the least common multiple as $\{m,n\}$ and introduces the lemma as a property that $S_x$ "is easily seen to have"; no proof is printed.

Source. I. Z. Ruzsa, On the small sieve. II. Sifting by composite numbers, J. Number Theory 14 (1982), 260–268; Section 2, Lemma 2.1 on printed p. 262. The edition is identified in the source digest.

Read depth. Claims checked: the definitions and the statement were read on the page images.

Proof pointer

No proof is printed. One argument, written here: by primitivity neither of $m,n$ divides the other. Say the least prime factor $q$ of $m$ is at most the least prime factor of $n$. Then $n/\gcd(m,n)>1$ has only prime factors at least $q$, so $\operatorname{lcm}(m,n)=m\cdot n/\gcd(m,n)\ge qm>x$.

Dependencies

None.

Bears on

• Problem 542: the lemma says that $S_x$, a set of integers in $(1,x]$, satisfies the problem's hypothesis that pairwise least common multiples exceed $x$. With Lemma 2.5 it gives such a set leaving at most $x\log^{-c_3}x$ integers up to $x$ divisible by none of its elements.