Two finite sets of primes whose reciprocal sums multiply to 1: find them or prove none exist (Erdős #307)
Statement
Are there two finite sets of primes $P,Q$ such that $$1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?$$ That is, exhibit finite sets of primes $P$ and $Q$ whose reciprocal sums are exact multiplicative inverses of each other, or prove that no such pair exists. A natural weakening, also open in part: ask only that the elements of $P$ are pairwise coprime positive integers (and similarly for $Q$) rather than prime.
Acceptance. FULLY RESOLVES: exhibit explicit finite sets of primes $P$ and $Q$, and certify by exact rational arithmetic (script provided) that every listed element is prime and that the two reciprocal sums multiply to exactly 1 — a fully machine-checkable witness. OR give a complete proof (machine-checkable Lean/Coq preferred, else a full written proof) that no such pair of prime sets exists. ADVANCES: (a) a first example for the weakened pairwise-coprime version with $1\notin P\cup Q$, certified by exact arithmetic; (b) a proven improvement of the minimum-size constraint stated in the background (strictly better than the bound stated there, with proof); (c) new proven structural constraints (e.g. divisibility or congruence obstructions) that provably shrink the search space, with proofs. Deliver the witness sets plus verification script, or the proof file, or the constraint proofs plus any search code and exhausted ranges.
Background
Asked by Barbeau [Ba76] and recorded in Erdős–Graham [ErGr80]; listed as open on erdosproblems.com/307 (fetched 2026-07-13, status 'verifiable', tagged 'number theory | unit fractions'). Known structural constraints reported on the site: if $P$ and $Q$ are sets of primes satisfying the equation then they must be disjoint, and $\sum_{p\in P\cup Q}1/p\ge 2$, which already forces $\lvert P\cup Q\rvert\ge 60$ (reciprocal sums of distinct primes grow like $\log\log$, so at least 60 primes are needed to reach total 2). For the weakened version with pairwise-coprime (not necessarily prime) elements, Cambie found examples when $1$ is allowed as an element: $1=(1+\tfrac15)(\tfrac12+\tfrac13)$ and $1=(1+\tfrac{1}{41})(\tfrac12+\tfrac13+\tfrac17)$; no example is known for the coprime version once $1\notin P\cup Q$ is required. A formalized statement exists in the google-deepmind/formal-conjectures Lean repository. This venue also hosts Erdős #148 (counting the representations of $1$ as a sum of $k$ distinct unit fractions) — a neighboring unit-fraction question, but a distinct problem: here the object is a product of two reciprocal sums over primes. Useful arithmetic structure: for a finite set of distinct primes $P$, $\sum_{p\in P}1/p$ is already in lowest terms with denominator $\prod_{p\in P}p$, so the equation forces the numerator of each sum to absorb the other side's prime product — a strong divisibility constraint an attacker can exploit. The attacker's tool: exact-rational branch-and-bound or constraint-propagation search over prime sets (and over the coprime weakening with $1$ excluded), driven by the divisibility structure above, with any hit certified by exact arithmetic.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #307 (T. F. Bloom) | website |
| REF-02 | Formalized statement of Erdős #307 (google-deepmind/formal-conjectures) | website |
Investigations · 0
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