Represent every a/b (b squarefree) as a sum of distinct 1/(pq) with p,q distinct primes (Erdős #306)
Statement
Let $a/b\in\mathbb{Q}_{>0}$ with $b$ squarefree. Do there always exist integers $1<n_1<\cdots<n_k$, each a product of two distinct primes, such that $$\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?$$ That is, can every such rational be written as a sum of distinct unit fractions whose denominators are squarefree semiprimes?
Acceptance. FULLY RESOLVES: a proof that every $a/b\in\mathbb{Q}_{>0}$ with $b$ squarefree admits such a squarefree-semiprime unit-fraction decomposition (a general construction, machine-checkable preferred), OR a proof/explicit counterexample exhibiting a specific $a/b$ with $b$ squarefree that provably cannot be so represented. ADVANCES (each independently checkable): (a) exhibit an explicit squarefree-semiprime decomposition for a target $a/b$ in the still-open below-threshold regime of Li (2026) — a case not already covered by the unconditional above-threshold theorem — giving the semiprime denominators and verifying the sum; (b) improve the shortest known decomposition of $1$ into distinct squarefree semiprimes below $47$ terms, with the verified term list; (c) prove the positive answer for a structured infinite family of $a/b$ in the below-threshold regime, or settle Li's residual conjecture. Deliver the general construction/proof, the counterexample, or the explicit verified decomposition(s).
Background
Posed by Erdős and Graham [ErGr80]; listed as open on erdosproblems.com/306 (fetched 2026-07-21, status 'open', last edited 21 June 2026). Write $\omega$ for the number of distinct prime factors of a denominator; the target here is $\omega=2$ (squarefree semiprimes). The $\omega=3$ analogue for $b=1$ was proved by Butler, Erdős and Graham [BEG15] — reputedly Erdős' last paper, appearing 19 years after his death. For the $\omega=2$ case with target $a/b=1$, explicit decompositions have long been known: the first by Barbeau [Ba77], the shortest known having $47$ terms (Watanabe [Wa20], whose introduction surveys the history). Shisheng Li (2026, arXiv:2606.15159) then broke the general case open: every natural number is a sum of distinct squarefree-semiprime unit fractions (the $\omega=2$ integer case, previously only conjectured), and for every squarefree $b$ every $a/b$ above an explicit threshold ($\min\{B_{N_b}/6,\,1/5\}$) is likewise $\omega=2$ representable, unconditionally; the same engine also gives the first complete proof of the rational $\omega\ge 3$ statement. What remains open — and keeps the site status 'open' after that preprint — is the $\omega=2$ regime for $a/b$ below this threshold, which Li reduces to a single explicit conjecture. (The site's comment-activity widget is marked 'claimed', but it records $0$ claimed proofs and the authoritative status is open.) Attacker's tool: a targeted search for squarefree-semiprime unit-fraction decompositions of small $a/b$ in the still-open below-threshold regime (greedy / branch-and-bound over admissible semiprime denominators with reciprocal-sum pruning), attacking Li's residual conjecture from the constructive side, or shortening Watanabe's $47$-term record for $1$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #306 (T. F. Bloom) | website |
| REF-02 | S. Li, Every natural number is a sum of distinct semiprime unit fractions (ω=2 case of Erdős–Graham; arXiv:2606.15159, 2026) | paper |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.