Problem 307

Are there two finite sets of primes $P,Q$ such that

1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?

Source: erdosproblems.com/307

Number theory
Unit fractions

Wiki page

Status

Open

No claim settles this problem.

Verifiable, the site's label for this open question: the site's explanation is that the problem is open but a finite example could prove it, since a positive answer would be witnessed by one pair $(P,Q)$ and checked by exact arithmetic. The label does not mean that such a pair has been found. The standing derived from the claim pages is open, claim none: the two claim pages are partial claims giving necessary conditions on a solution, Kovič's conditions on two-cycles (accepted, refereed) and Bonfioli's barrier of sixty primes (pending), and nothing settles or pends on the existence question. No example, no proof that none exists and no accepted claim on the existence question was found in the search whose scope the Current assessment records. The known bounds are three: the elementary ones recorded below (the sets are disjoint, as Robert Israel observed on [MO19] in 2019, and any solution uses at least $59$ primes, as Julian Rosen observed there); Kovič's refereed 2012 conditions [Ko12] (no solution with $|P|=|Q|=2$ , both products at least $10^4$ , and at least nine primes in an odd smaller product); and the manuscript [Bo26]'s finite verification pushing the count to $60$ with a large lower bound on the products. This is a bounded negative finding, not a certificate of openness.

Ba76: Barbeau, E. J., Computer challenge corner: Problem 477: A brute force program. J. Rec. Math. 9 (1976/77), p. 30, as the bibliography of [ErGr80] gives it under [Bar (76)]; the site gives the year 1976, and the unrefereed manuscript [Bo26] reports the same volume and page. No open archive copy was found, and the question is quoted second-hand from [ErGr80] and stated first-hand in [Ba77].
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), p. 38 (the site gives no page). Library home: Erdős and Graham 1980.
Ba77: Barbeau, E. J., Expressing one as a sum of distinct reciprocals: comments and a bibliography. Eureka (Ottawa) 3 (1977), no. 7 (August--September 1977), 178--181, in the Canadian Mathematical Society's archive, Crux_v3n07_Aug.pdf. It poses this problem's question first-hand (p. 178) and prints the 101-term example of $1$ with two-prime denominators that the monograph cites on the same page.
Bo26: Bonfioli, V., On the equation $n''=n$ and a problem of Erdős and Barbeau on products of prime--reciprocal sums. Version 1.74.0, 2 September 2026, 136 pp.; Zenodo record 22279869 (published 3 September 2026; software type; concept DOI 10.5281/zenodo.20684626); repository ElVec1o/erdos307 (created 13 June 2026; linked at its head commit of 6 September 2026 per the GitHub API on 2026-09-18). The PDF at that commit, paper/erdos307.pdf, is version 1.75.0, also dated 2 September 2026, 136 pp., and is the version cited on this page: the Rosen and Israel credits, pp. 1--3, 5 and 6; Lemma 4.2 and Theorem 4.3, p. 6; Proposition 4.5, p. 7; acknowledgments, p. 62. Unrefereed.
Ko12: Kovič, J., The arithmetic derivative and antiderivative. J. Integer Seq. 15 (2012), Article 12.3.8, published 25 March 2012, journal page; §3.2, pp. 7--9 (Propositions 16--19 and the computer search below $10^4$ ). Refereed.
MO19: MathOverflow question 320838, "Product of sum of reciprocals of prime numbers", asked 14 January 2019; one answer (19 August 2019) and four comments, among them Robert Israel's of 14 January 2019 (the two sets need not be assumed disjoint, since the $p$ -adic order of each reciprocal sum is $-1$ at its own primes) and Julian Rosen's of 15 January 2019 (two numbers with product $1$ sum to at least $2$ , and the reciprocals of the first $58$ primes sum to less than $2$ , so a solution needs at least $59$ primes), accessed 2026-10-07. A lead the discussion links, never a status source.

Problem 307

Status

Proof claims

Discussion