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This digest records statement-level content from arXiv:2603.28636v1, submitted 2026-03-30, the copy read for this digest. The edition is identified on the source card.

Definition and theorem statements

For each positive integer $m$, let $f(m)$ be the largest integer $r$ such that, for every set $A=\{a_1<\cdots<a_m\}$ of $m$ positive integers and every real number $x$, there are distinct $c_1,\ldots,c_r\in A$ and distinct integers $b_1,\ldots,b_r$ satisfying

$$x<b_i<x+2a_m,\qquad c_i\mid b_i\qquad(1\le i\le r).$$

The paper also writes $B=(x,x+2a_m)\cap\mathbb Z$ and lets $G(A,x)$ be the bipartite divisibility graph on $A$ and $B$.

Theorem 2.1 (E650; printed/physical p. 2). For every positive integer $m$,

$$f(m)=\min\{m,\lceil2\sqrt m\rceil\}.$$

Remark 2.2 (printed/physical p. 3). For every integer $m\ge4$,

$$f(m)=\lceil2\sqrt m\rceil.$$

Historical comparison (printed/physical p. 1). The paper recalls the Erdős–Surányi lower bound $f(m)\ge\sqrt m$ and the Erdős–Selfridge estimate $f(m^2)\le2m$, which gives the general upper bound $f(m)\le2\lceil\sqrt m\rceil$. Thus the earlier general bounds differed by a factor of two.

Theorem 3.1 (printed/physical p. 4). For all positive integers $s,t$,

$$f(st)\le s+t.$$

The paper's construction (printed/physical pp. 4–5) uses the Chinese Remainder Theorem to produce a set of $st$ positive integers and an interval of length $2\max A$ containing at most $s+t$ distinct multiples.

Theorem 4.1 (printed/physical p. 5). For every positive integer $m$,

$$f(m)\ge\min\{m,\lceil2\sqrt m\rceil\}.$$

The paper's lower-bound proof applies a generalization of Hall's theorem (Lemma 2.3, p. 3, the König–Ore formula) to $G(A,x)$. Together Theorems 3.1 and 4.1 give Theorem 2.1.

Interval-length range (printed/physical pp. 1--2 and 5). Although the displayed definition of $f(m)$ uses intervals of length $2a_m$, the introduction says that the same results remain valid when 2 is replaced by any real multiplier $c$ with $2\le c<3$. More precisely, Remark 3.3 says that, for every $\varepsilon\in(0,1)$, the upper-bound construction still works with an interval of length $(3-\varepsilon)\max A$ once $M$ is chosen larger than $\varepsilon^{-1}(3-\varepsilon)(s+tD)$. This records the source's parameter extension; it is not a new local proof.

Verification layers

Formal source. The paper cites Wouter van Doorn's ErdosProblem650.lean. Its Section 5 records Lean version leanprover/lean4:v4.28.0 and Mathlib commit 8f9d9cff6bd728b17a24e163c9402775d9e6a365. No local Lean environment or build was used.

Reported verification. The paper reports (Sections 1.1 and 5) that an initial draft produced by a large language model found the main strategy but contained a gap in the Case 2 injection, and that an automated theorem-proving system supplied a working variation and a complete Lean formalization. It says the final exposition and proofs are human-written. These are author-reported workflow claims, recorded without model or system names.

Local verification. The copy read for this digest is the arXiv v1 PDF, submitted 2026-03-30. All eight page images were read for the definition, historical comparison, interval-range and CRT statements, Theorems 2.1, 3.1 and 4.1, and the formalization account. This check records source statements and provenance only. Read status: claims checked for the definition of $f(m)$, Theorem 2.1, Remark 2.2, Lemma 2.3, Theorem 3.1, Remark 3.3 and Theorem 4.1, read clause by clause on the page images; the proofs of Sections 3 and 4 were read but not independently checked. Each result has its own page, linked from the source card.

Problem scope

Theorem 2.1 directly addresses E650. E860 is adjacent context only: the paper does not mention it, and E860's $h(n)$, the interval length needed to match the primes up to $n$ to distinct multiples, is a different function from $f(m)$.