Problem 316

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 $i=1,2$ ?

Source: erdosproblems.com/316

Number theory
Unit fractions

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

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 (1997).

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: Erdős and Graham 1980.
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.

Problem 316

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Proof claims

Discussion