Problem 316

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claims/: The 1 claim page of Problem 316, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that if $A\subset \mathbb{N}\backslash\{1\}$ is a finite set with $\sum_{n\in A}\frac{1}{n}<2$ then there is a partition $A=A_1\sqcup A_2$ such that

$$\sum_{n\in A_i}\frac{1}{n}<1$$

for $i=1,2$?

Formulation. The site's wording, accessed 2026-09-18 (page last edited 5 March 2026; source key [ErGr80]). $A$ is a finite set of integers at least $2$ with reciprocal sum below $2$, and the question asks for a partition of $A$ into two disjoint parts each of whose reciprocal sums is strictly below $1$. The strict inequalities matter: both counterexamples below split into two parts with sums at most $1$, because $\{2,3,6\}$ has reciprocal sum exactly $1$. The 1980 monograph (printed p. 41) asks the same question for $a_1<a_2<\cdots<a_t$ and adds that it fails when repetitions are allowed, with the multiset $2,3,3,5,5,5,5$; its wording does not exclude $a_1=1$, but with $1\in A$ no admissible split exists, so the site's $A\subset\mathbb N\setminus\{1\}$ makes the exclusion explicit.

Status. Disproved. The answer is no: the divisors of $120$ other than $1$ and $120$, the set the site attributes to Sándor's 1997 paper, have reciprocal sum $239/120<2$ and no admissible split, and the eleven-element set $\{2,3,4,5,6,7,10,11,13,14,15\}$ of the site's commentary (sum $120047/60060$) has none either; both are finite computations rechecked here with exact arithmetic. Sándor's paper (J. Number Theory 63 (1997), refereed) is not held, so his statements are recorded second-hand from the site. The site's label is DISPROVED (LEAN); its Lean qualifier is a catalog label whose referent is the formal-conjectures file, which proves the disproof by a decide +kernel step on the eleven-element set; nothing was built here, and no local kernel credit is claimed (see Formalization). The accepted claim is recorded on Sándor's claim page.

Source. erdosproblems.com/316, accessed 2026-09-18: the problem page (DISPROVED (LEAN), with the site's banner saying the question is answered in the negative and the proof verified in Lean; source key [ErGr80]; last edited 05 March 2026), its one-comment discussion thread (3 March 2026) and its empty proof-claim tab. The site cites [Sa97] in its commentary and thanks Tom Stobart. Cite as: T. F. Bloom, Erdős Problem #316, https://www.erdosproblems.com/316, accessed 2026-09-18.

References.

• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), printed p. 41 (the site gives no page; the thread's comment of 3 March 2026 does). Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Sa97] Sándor, Csaba, On a problem of Erdős. J. Number Theory 63 (1997), no. 2, 203--210, DOI 10.1006/jnth.1997.2113. Not held; the Crossref record places the article in the publisher's open archive (license dated 17 July 2013).
• [Ch06] Chen, Yong-Gao, On a conjecture of Erdős, Graham and Spencer. J. Number Theory 119 (2006), no. 2, 307--314, DOI 10.1016/j.jnt.2005.11.003. Named in the thread as an improvement of [Sa97]; not held, not read (bibliographic record from Crossref).
• [FaCh08] Fang, J.-H. and Chen, Y.-G., On a conjecture of Erdős, Graham and Spencer, II. Discrete Appl. Math. 156 (2008), no. 15, 2950--2958, DOI 10.1016/j.dam.2008.01.001. Named in the thread as a further improvement; not held, not read.

Formalization. Statement and disproof. The file ErdosProblems/316.lean of formal-conjectures at the pinned commit (main, 2026-09-18) declares erdos_316 : answer(False) ↔ ∀ A : Finset ℕ, 0 ∉ A → 1 ∉ A → ∑ n ∈ A, (1 / n : ℚ) < 2 → ∃ (A₁ A₂ : Finset ℕ), Disjoint A₁ A₂ ∧ A = A₁ ∪ A₂ ∧ ∑ n ∈ A₁, (1 / n : ℚ) < 1 ∧ ∑ n ∈ A₂, (1 / n : ℚ) < 1 under category research solved and proves it in the file: the witness is {2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 15}, and a decide +kernel step checks that every subset with reciprocal sum below $1$ has a complement with reciprocal sum at least $1$. The file also proves the multiset variant ($2,3,3,5,5,5,5$) and leaves Sándor's general statement for $n$ parts (erdos_316.variants.generalized) as sorry; its docstring attributes the disproof to Sándor's paper and the formalization to Mehta, so the file is carried as a formalization link on Sándor's claim page, where the change of witness is noted. The community database (2026-09-18) records the status "disproved (Lean)" and formal status Lean, with 2 September 2025 as that entry's last-update date, and the statement as formalized, with 31 August 2025 as that entry's last-update date; it does not say when either state was first set. Nothing was built or checked here.

Current assessment

The question (site formulation of 2026-09-18). The statement above; DISPROVED (LEAN); last edited 5 March 2026. The commentary makes three points: the multiset $2,3,3,5,5,5,5$ already fails the conclusion when repetitions are allowed; the answer is no for sets as well, which the site credits to Sándor [Sa97] with the proper divisors of $120$, the divisors other than $1$ and $120$, as his counterexample, together with his general theorem that for every $n\ge2$ a finite set $A\subseteq\mathbb N\setminus\{1\}$ exists with reciprocal sum below $n$ and no partition into $n$ parts of reciprocal sum below $1$; and the eleven-element set $\{2,3,4,5,6,7,10,11,13,14,15\}$, credited to Tom Stobart, is described as the minimal counterexample. The one comment (3 March 2026) says that the result of [Sa97] was improved by Chen [Ch06] and then by Fang and Chen [FaCh08], and locates the problem at [ErGr80, p. 41]. The proof-claim tab is empty. The community database record: disproved (Lean) as of its last update on 2 September 2025, statement formalized, no OEIS entry.

Origin. Printed p. 41 of the 1980 monograph: "Is it true that if $a_1<a_2<\cdots<a_t$ and $\sum_{k=1}^t\frac1{a_k}<2$ then there exist $\varepsilon_k=0$ or $1$ so that $\sum_{k=1}^t\frac{\varepsilon_k}{a_k}<1$ and $\sum_{k=1}^t\frac{1-\varepsilon_k}{a_k}<1$? This is not true if we just assume $a_1\le a_2\le\cdots\le a_t$, as, for example, the sequence 2, 3, 3, 5, 5, 5, 5, shows. It is conjectured by Spencer and the authors that in any case, if $\sum_{k=1}^t\frac1{a_k}\le N-\frac1{30}$ then the $a_k$ can be split into $N$ sequences $a_k^{(i)}$, $1\le i\le N$, so that $\sum_k\frac1{a_k^{(i)}}\le1$ for all $i$." The Spencer--Erdős--Graham conjecture of the last sentence is a different statement (sums at most $1$, a margin of $1/30$, $N$ parts); from their titles, [Ch06] and [FaCh08] concern it.

The disproof (finite checks, recomputed here). Write $\sigma(B)=\sum_{n\in B}1/n$.

• Sándor's set as the site describes it: $D=\{2,3,4,5,6,8,10,12,15,20,24,30,40,60\}$, the divisors of $120$ other than $1$ and $120$ (the site's term is proper divisors). Then $\sigma(D)=(360-1-120)/120=239/120<2$, and an exhaustive check of the $2^{14}$ subsets $B\subseteq D$ finds none with $\sigma(B)<1$ and $\sigma(D\setminus B)<1$. So $D$ answers the question in the negative. The check was run here with exact rational arithmetic.
• The site's eleven-element set $E=\{2,3,4,5,6,7,10,11,13,14,15\}$: $\sigma(E)=120047/60060<2$, and none of its $2^{11}$ subsets gives an admissible split (same check). The site describes it as the minimal counterexample.
• Both $D$ and $E$ split into two parts with sums at most $1$: $\{2,3,6\}$ has sum exactly $1$ and the rest has sum below $1$. The strict inequality in the question is therefore essential.
• The monograph's and the site's multiset $2,3,3,5,5,5,5$: total $59/30<2$, and no split of the seven terms gives two sums below $1$ (checked here).

These are author-recorded finite checks; the two implementations agree, no one else has reviewed them, and the decide +kernel step on $E$ in the formal-conjectures file was not rebuilt here.

What is second-hand. Sándor's paper is not held: its counterexample is taken from the site and confirmed by the computation above, and his general theorem (that for every $n\ge2$ some finite set $A$ has $\sigma(A)<n$ yet admits no partition into $n$ parts of sum below $1$) is recorded as the site's statement. Stobart's eleven-element set is recorded on Sándor's claim page and has no claim page of its own, because it appears only in the undated commentary. Chen [Ch06] and Fang--Chen [FaCh08] are the thread's leads; neither is held.

Search scope. The problem, discussion and proof-claim pages; the community database record; the formal-conjectures file at the pinned commit; Crossref records for [Sa97], [Ch06] and [FaCh08] (DOI lookup and bibliographic queries); an arXiv API search for abstracts naming reciprocals, partitions and Erdős with a sum below a bound (no record) and the arXiv API listing of the seventy-six most recent abstracts mentioning Egyptian or unit fractions (to 7 September 2026; none on this problem); printed p. 41 of [ErGr80]. Not searched: MathSciNet, zbMATH, Google Scholar, X. Nothing found changes the status; the disproof is a finite check.

Remaining gaps. (1) [Sa97], [Ch06] and [FaCh08] are not held; Sándor's general $n$-part theorem and the two improvements are second-hand. (2) [Ch06] and [FaCh08] are unread, so whether they settle the Spencer--Erdős--Graham splitting conjecture of p. 41 is not recorded. The disproof itself is complete as a finite check and has no proof to compile.

Progress and known results

• Erdős and Graham (1980, p. 41): the question; the multiset counterexample $2,3,3,5,5,5,5$ for repeated terms; the Spencer--Erdős--Graham splitting conjecture with the margin $1/30$.
• Sándor (1997, not held): the divisors of $120$ other than $1$ and $120$ answer the question in the negative (checked here: sum $239/120$, no admissible split); for every $n\ge2$ a finite set with reciprocal sum below $n$ and no partition into $n$ parts of sum below $1$ (site commentary).
• Stobart (site commentary, undated): the eleven-element counterexample $\{2,3,4,5,6,7,10,11,13,14,15\}$ (checked here; sum $120047/60060$).
• Chen (2006) and Fang--Chen (2008): improvements named in the thread; unread.
• Formalization: the formal-conjectures file proves the disproof by a decide +kernel step (not built here); Sándor's general statement is a sorry there.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1980_old_new_problems_results_combinatorial_number_theory