Problem 710

Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$ . Obtain an asymptotic formula for $f(n)$ .

Source: erdosproblems.com/710

Number theory

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ClaimedAnswered

A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Labeled OPEN on the site at the access of 2026-10-06. The standing is claimed, derived from the pending full claim of 23 September 2026 on its claim page (Turturean, 2026), which asserts $f(n)\sim(\sqrt2/e)\,n\sqrt{\log n}$ , elicited from the AI system GPT-6-Astra Pro, with a manuscript not read, no curator action and no outside check. No asymptotic formula was found in the search whose scope the Current assessment records. What is known is the Erdős–Pomerance pair of bounds, $(2/\sqrt e+o(1))\,n\sqrt{\log n/\log\log n}\le f(n)\le(2+o(1))\,n\sqrt{\log n}$ (Theorems 2 and 3, 1980), with the sketched improvement of the upper constant to $c=\sqrt r/(1-r)=1.7398\ldots$ , $e^{-r}=r$ (display (11), p. 154), which is the constant the site prints; the two sides differ by a factor of order $\sqrt{\log\log n}$ . A forum attempt of February 2026, made with the AI system Opus 4.6 and heavy computation, conjectures that the lower bound is the truth and reports a computation to $n\le10^6$ ; it is a lead, not status. This is a bounded negative finding, not a certificate of openness. The site lists a prize for Erdős's offer of 1992 for an asymptotic formula (below).

ErPo80: Erdős, P. and Pomerance, C., Matching the natural numbers up to $n$ with distinct multiples in another interval. Indag. Math. (Proc.) 83 (1980), no. 2, 147--161, DOI 10.1016/1385-7258(80)90018-9; the definitions, p. 147; Theorem 2, p. 150; Theorem 3, p. 153; display (11), p. 154. Library home: Erdős 1980; result pages Theorem 2, Theorem 3 and inequality (11).
Er92c: Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. 15 (1992), 34--50, DOI 10.46298/hrj.1992.125; Section 1, display (3) and the offer, printed p. 36. Library home: Erdős 1992; result page Section 1.
vD26: van Doorn, W., On the length of an interval that contains distinct multiples of the first $n$ positive integers. Integers 26 (2026), #A7; Lemma 3 quotes the two Erdős–Pomerance bounds; Theorem 1 concerns $\max_mf(n,m)$ , not this problem's diagonal. Library home: Doorn 2026.
OEIS: Kalogeropoulos, G., Sequence A390246, The On-Line Encyclopedia of Integer Sequences (30 October 2025; entry last modified 12 August 2026, server time): the entry's name defines $a(n)$ as the least integer $k$ such that there exist $n$ distinct integers $b_1,\ldots,b_n$ with $n<b_i<n+k$ and $b_i$ divisible by $i$ for $1\le i\le n$ , the site's convention; $2,3,4,6,6,9,9,11,13,15,\ldots$

Problem 710

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