Problem 307

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claims/: The 2 claim pages of Problem 307, one per claimant's result; the problem's standing derives from them.

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

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

Formulation. The site's wording on 2026-09-18 (the page shows no last-edited date). Both sets must be nonempty for the product to be $1$. For a finite set $P$ of primes the sum $\sum_{p\in P}1/p$ equals $N(P)/\prod P$ with $N(P)=\sum_{p\in P}\prod P/p$, and $N(P)$ is coprime to $\prod P$ (modulo each $p\in P$ every term of $N(P)$ but $\prod P/p$ vanishes), so the fraction is in lowest terms and is never an integer; hence, if the product is $1$, then $N(P)N(Q)=\prod P\prod Q$ forces $N(P)=\prod Q$ and $N(Q)=\prod P$, and a common prime of $P$ and $Q$ would divide $N(P)$ and $\prod P$ at once, which is impossible: the sets are disjoint, as the site says and as Robert Israel observed in a comment on [MO19] on 14 January 2019 (the $p$-adic order of $\sum_{p\in P}1/p$ is $-1$ for each $p\in P$, so a shared prime would give the product order $-2$). For the two-cycles of the arithmetic derivative that solutions give (see the Current assessment), Ufnarovski and Åhlander (J. Integer Seq. 6 (2003)) proved that both members are squarefree with disjoint supports, as [Ko12] and [Bo26] report. The site's weakened version drops primality and asks only that the elements of $P$ be pairwise coprime and likewise for $Q$; it has solutions when $1$ is allowed as an element, and the site records none with $1\notin P\cup Q$. Expanding the product, a solution of the problem would give $1=\sum_{p\in P,q\in Q}1/(pq)$, a representation of $1$ by reciprocals of distinct semiprimes, which is the case $a/b=1$ of Problem 306 in a very special shape; that problem's representations do not conversely factor.

Status. 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.

Source. erdosproblems.com/307, accessed 2026-09-18: the problem page (VERIFIABLE, explained by the site as open but provable by a finite example; source key [ErGr80]; no last-edited date shown; additional thanks to Stijn Cambie), its discussion thread (6 comments, 9 August 2025 to 30 May 2026, one of them deleted) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #307, https://www.erdosproblems.com/307, accessed 2026-09-18.

References.

• [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: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [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.

Formalization. Statement only. The file ErdosProblems/307.lean of formal-conjectures at the linked commit (main) declares erdos_307 : answer(sorry) ↔ ∃ P Q : Finset ℕ, (∀ p ∈ P, p.Prime) ∧ (∀ q ∈ Q, q.Prime) ∧ 1 = (∑ p ∈ P, (p : ℚ)⁻¹) * (∑ q ∈ Q, (q : ℚ)⁻¹) under category research open with proof sorry. The same file carries the variants coprime (the weakened version with pairwise coprime elements, $0\notin P\cap Q$ and at least two elements in each set; category textbook, proved with $P=\{1,5\}$, $Q=\{2,3\}$), coprime_one_notMem (the same with $1\notin P\cup Q$; research open, sorry), and two barrier statements attributed in their docstrings to [Bo26]: barrier (for any solution with $Q$ nonempty, $59\le|P\cup Q|$ and $4\cdot10^{112}\le(\prod_{p\in P}p)^2$; category research solved, proof sorry, with a formal_proof using lean4 at attribute pointing to Closed.lean of ElVec1o/erdos307 at the linked commit, dated 16 August 2026 per the GitHub API) and barrier_sixty ($60\le|P\cup Q|$; the same category and attribute, pointing to Sixty.lean at the linked commit, dated 2 July 2026, whose docstring says the proof rests on a native_decide search). No build or review of the external files is recorded, and no kernel credit follows. The community database records the problem as verifiable (9 September 2025), the statement as formalized since 31 August 2025, no formal proof and no OEIS entry.

Current assessment

The question (site formulation of 2026-09-18). The statement above; VERIFIABLE. The commentary attributes the question to Barbeau [Ba76] and asks whether it can be answered once primality is dropped and the elements of $P$, and those of $Q$, are only required to be pairwise coprime; it reports that Cambie has found several examples of that weakened version, among them $1=(1+\frac15)(\frac12+\frac13)$ and $1=(1+\frac1{41})(\frac12+\frac13+\frac17)$, that no example of the weakened version is known with $1\notin P\cup Q$, and that for sets of primes it is easy to see that $P$ and $Q$ are disjoint with $\sum_{p\in P\cup Q}1/p\ge2$, from which it concludes that $P\cup Q$ has at least $60$ elements. Both examples are identities of fractions ($\frac65\cdot\frac56=1$ and $\frac{42}{41}\cdot\frac{41}{42}=1$). The thread (six comments): 9 August 2025, that every primary pseudoperfect number yields a solution once $1$ is allowed; 10 August 2025 (the account StijnC), that this concerns the weakened version, with $(1+\frac1{1805})(\frac12+\frac13+\frac17+\frac1{43})=1$ where $1805$ is not prime, the pair $P=\{1,3,11,331\}$, $Q=\{2,5,1559\}$, which does not arise from a primary pseudoperfect number (both products verified by exact arithmetic), two related versions with positive answers (all $\gcd(p,q)=1$ across the sets; or $\gcd$ of each set equal to $1$), and the remark that a solution's $P$ determines $Q$ through the factorization of large numbers; 28 November 2025, a deleted post and two replies rejecting an AI-generated proof attempt at its step 6, the poster then withdrawing (a thread post with no manuscript, so it has no claim page); 30 May 2026, a link to the MathOverflow thread [MO19]. The community database record (above) agrees with the label.

Origin. Printed p. 38 of the 1980 monograph. After recording Barbeau's 1977 example of $1$ as a sum of $101$ reciprocals of products of two distinct primes and Burshtein's earlier example in which no term divides a later one, the monograph says, citing Barbeau [Bar (76)], that "it is not known if $1$ can be expressed as the product of two sums of the form $\frac1{q_1}+\ldots+\frac1{q_k}$ where the $q_i$ are distinct primes. Perhaps this can be done if the $q_i$ are just assumed to be pairwise relatively prime." The site's weakened version is the monograph's suggestion; Cambie's examples answer it when $1$ is admitted, and the site records no example without $1$. Barbeau's own article of 1977 [Ba77] (p. 178) poses the question from the arithmetic derivative $D$: an integer $n$ with $D(D(n))=n\ne D(n)$ would give $1=(\sum_i1/p_i)(\sum_j1/q_j)$ for two sets of distinct primes, and the author writes that he does not know whether such a representation exists; his 101-term example of $1$ follows as the representation such a product would in particular give. The MathOverflow answer of 19 August 2019 points to the same monograph page, gives Barbeau's 1976 and 1977 references and lists Johnson's 48 two-prime denominators for $1$.

The elementary facts and their sources. Each is an author-recorded finite check on this page, with the public source that states it.

• Disjointness and the identities $N(P)=\prod Q$, $N(Q)=\prod P$, as in the Formulation; Robert Israel's comment on [MO19] of 14 January 2019 gives the disjointness by the $p$-adic order $-1$ of each sum, and [Bo26] credits him with the "disjointness core" (p. 3; p. 5 calls the disjointness lemma the Israel mechanism). Neither sum is $1$, since $\sum_{p\in P}1/p=N(P)/\prod P$ is in lowest terms with $\prod P>1$; so with $s=\sum_P1/p$ and $t=\sum_Q1/q$, $st=1$ and $s\ne1$ give $s+t>2$ strictly, that is, $\sum_{p\in P\cup Q}1/p>2$ (the site writes $\ge2$).
• At least $59$ primes. Julian Rosen's comment on [MO19] 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 $2$. Among sets of $k$ primes the reciprocal sum is largest for the first $k$ primes; the first $58$ primes have reciprocal sum $1.99874\ldots<2$ and the first $59$ (up to $277$) have $2.00235\ldots>2$ (exact rational arithmetic). So $|P\cup Q|\ge59$, which [Bo26] calls Rosen's bound (p. 2) and proves as its Lemma 4.2, titled after Rosen (p. 6). The site's conclusion that $|P\cup Q|\ge60$ does not follow from the displayed mass condition alone: $59$ primes can carry mass above $2$, so excluding $59$-element supports needs a further argument; [Bo26] says the same (p. 2) and supplies one by finite verification (below).
• The weakened version. $(1+1/5)(1/2+1/3)=1$ and $(1+1/41)(1/2+1/3+1/7)=1$; the thread's $(1+1/1805)(1/2+1/3+1/7+1/43)=1$ and $(1+1/3+1/11+1/331)(1/2+1/5+1/1559)=1$. Each is an identity of fractions; each uses $1$, which is coprime to everything.
• Relation to Problem 306. A solution would give $1=\sum_{P\times Q}1/(pq)$ with the $|P||Q|$ denominators distinct semiprimes (distinct because $P\cap Q=\emptyset$); Johnson's 48-term and Watanabe's 47-term representations of $1$ by semiprime reciprocals do not have this product shape, and Problem 306's statement, even if true, would not produce one.

Verifiability. The site's label is a body note on this page, not a claim: one pair $(P,Q)$ would settle the question affirmatively by exact arithmetic, and no such pair is known; no finite computation can settle it negatively.

The pending partial claim and its formal pointers. The manuscript [Bo26] has the claim page Bonfioli's barrier of sixty primes, a pending partial claim covering the necessary conditions below and not the existence question. [Bo26] identifies the problem with the existence of a two-cycle of the arithmetic derivative $n\mapsto n'$ that is not a fixed point (abstract, pp. 1--2), which it credits to Ufnarovski and Åhlander's Conjecture 4 (2003) and to later independent restatements, and states: Lemma 4.2 (p. 6, titled after Rosen), any finite set of distinct primes with reciprocal sum above $2$ has at least $59$ elements (the computation above, with the same decimals); Theorem 4.3 (p. 6), for any solution $|P\cup Q|\ge59$, $\prod_{P\cup Q}p\ge\Pi_{59}\approx8.77\times10^{112}$ and $\min(\prod P,\prod Q)\ge2.09\times10^{56}$; Proposition 4.5 (p. 7), by an exact-integer verification over the $49{,}961$ admissible $59$-element supports, no solution has $|P\cup Q|=59$, hence $|P\cup Q|\ge60$ and $\min(\prod P,\prod Q)>3.50\times10^{57}$. It presents Kovič's 2012 conditions [Ko12] (Propositions 16, 17 and 19 and the search below $10^4$) as the prior literature, and shows that the proof of his Proposition 18 ($r+s\ge34$) is invalid (§14, p. 63). The manuscript's own contributions are the product bounds and the step from $59$ to $60$; it credits the disjointness to Israel and the $59$ to Rosen (pp. 1--3, 5 and 6). The paper says "The problem remains open" (p. 2) and that it claims no priority for the statement $|P\cup Q|\ge60$, which "predates this work on the problem page", but does claim the proof. Provenance declared by the source (acknowledgments, p. 62): the work was carried out "with substantial assistance" from Anthropic's Claude, which "contributed to the derivations, the exact and heuristic computations, the Lean 4 formalisation, and the drafting of this note" and was used "to stress-test and attempt to refute each claim"; all statements and the final text are said to have been reviewed by the author, who takes responsibility for them. The formal-conjectures file records the $59$ and $60$ barriers as variants with external formal-proof pointers into the paper's repository (Formalization); no build or review of the Lean files and no rerun of the finite verification is recorded, and no independent review or refereed publication of the manuscript was found. [Bo26] also reports (p. 3) an independent structural note by another author obtained through ResearchGate, which this page records only through that report.

Search scope. The status rests on these routes; none found an example, a proof of nonexistence or a proof claim.

• The site: problem page, discussion thread, empty proof-claim tab; the community database record; formal-conjectures 307.lean at the pinned commit, with the GitHub API for the repository and the two commits its attributes name.
• The primary sources read as stated: [ErGr80] (p. 38); [Bo26] (pp. 1--3, 5--7 and 62 of the PDF at the pinned commit); [MO19] (question, answer and comments through the Stack Exchange API); [Ba77] (pp. 178--181 of the Society's archive copy).
• The Zenodo API for record 22279869; an arXiv API metadata search for arithmetic-derivative cycles, products of prime reciprocals or "Erdős problem 307" in abstracts (three records, none on this problem). The API searches titles and abstracts only, so this zero is weak.
• The Journal of Integer Sequences article [Ko12] (§3.2), which [Bo26] presents as the prior literature.
• The finite checks above (exact rational arithmetic).

Not searched: MathSciNet, zbMATH, Google Scholar full text, X, ResearchGate. Not consulted: [Ba76] (J. Rec. Math.; no open archive found), Burshtein (1973), Ufnarovski and Åhlander (2003) (known through [Ko12] and [Bo26]), the Lean files of [Bo26].

Remaining gaps. (1) Barbeau's 1976 posing [Ba76] is known second-hand (the monograph and [MO19]); his 1977 article [Ba77] states the question first-hand. (2) The site's bound $60$ rests, beyond the elementary $59$, on an unrefereed finite verification; the commentary's inference overstates what the displayed inequality gives. (3) The manuscript's barrier and its formal pointers are recorded only at the statements listed and are unreviewed. (4) No source narrows the search beyond these bounds. There is no status-defining proof to compile.

Proof coverage. Nothing establishes a standing beyond open: the claim pages are partial claims, Kovič's accepted and Bonfioli's pending. The elementary structure (disjointness, the strict mass inequality, the count $59$, the examples of the weakened version) is an author-recorded finite check on this page, with Israel's and Rosen's comments as their sources; the manuscript's results are recorded at statement level and are not reviewed.

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