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Is it true that if is a set such that for all , where is the least common multiple, then
Is it true that there must be many which do not divide any ?
Is it true that if is a set such that for all , where is the least common multiple, then
Is it true that, if , there must be many which are not divisible by any ?
Source: erdosproblems.com/542
An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.
Solved, the site's label for two answers: yes to the first question and no to the second, both by Schinzel and Szekeres (Acta Sci. Math. (Szeged) 20 (1959), 221--229, a refereed journal); the label describes the corrected Statement. Their Theorem 1 gives with equality only for and . Their Theorem 3 and its construction (pp. 228--229) give admissible sets , with , whose reciprocal sums exceed for every and all large and which leave only integers divisible by no element, so no constant with such integers exists. Their Theorem 2 adds for large with , and Chen (1996) lowers that constant to , a partial result on the first question for large . Erdős's 1973 speculation that the sum is at most is recorded below. The claim pages Schinzel and Szekeres 1959, which records the acceptance and links the Lean file of 2026 that declares itself a formalization of their theorems, not built here, and Chen 1996 record the results, and the frontmatter standing, which judges the corrected Statement, derives from them.
The site's second question fails for every . For
the pairwise least common multiples exceed (a common
multiple of two distinct elements is at least twice the larger one), and
every divides some element, since for some multiple of
lies in ; so no divides no element, and the answer no holds
for a trivial reason (an observation of this page, confirmed by computation
for ). The failure covers every , so it is not a boundary
failure; it is a misprint that the poser's own words contradict. The site's
phrase copies Erdős's sentence in [Er73] (printed p. 135: "I thought that
(14.1) implies the existence of an absolute constant so that there are
integers which do not divide any of the 's. To my great
surprise this was disproved by Schinzel and Szekeres."). His other statement
of the same question, [Er80] p. 111, differs only in the failing element: it
counts, as , the integers up to divisible by none of the 's, and
records his 1940 conjecture for sequences
with pairwise least common multiples above .
Both texts report that Schinzel and Szekeres disproved the question, and
their construction ([ScSz59] p. 228) counts the integers divisible by no
element, so the report is true of that form and not of the printed one,
which no construction is needed to refute. The change replaces "which do not
divide any " by "which are not divisible by any ", the
counting of [Er80], and inserts "if ,": in the non-multiples form
the set meets the hypothesis vacuously and leaves no such integer,
because divides every . The element is the one value at which no
set can meet the conclusion: a set with in it is , while a
one-element set with leaves
non-multiples. That exclusion is this page's own correction; the
formal-conjectures statement of the second question
(erdos_542.parts.ii, under Formalization) adds the same hypothesis
for the same reason and counts with the site, and the sets of
[ScSz59] exclude . The defect is already in the poser's text: [Er73]
prints the divisor form, and the site copies it. The form rests on these
sources alone, not on which results settle it. The only result about the
site's wording is the observation above; it settles no instance of the
corrected Statement and counts for nothing. The first question is unchanged.
The problem's standing judges the corrected Statement.