Erdős Problem #306 — Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree.

Erdős Problem #306 asks whether every positive rational with squarefree denominator can be expressed as a finite sum of distinct reciprocals of semiprimes. It is the Erdős–Graham problem, also listed as problem D11 in Guy’s Unsolved Problems in Number Theory.

Known results

• Butler, Erdős and Graham (2015): the analogous statement with denominators having three distinct prime factors holds for positive integers.
• Johnson (1978): $1$ has a representation using $48$ semiprime reciprocals.
• A later computation found $17$ representations of $1$ using $47$ terms, but no shorter example was established.

June 2026 partial proof and conflicting claim

A 2026 paper proves the integer case and proves representability for every squarefree-denominator rational above an explicit threshold; the remaining $rac{a}{b}$ below that threshold is reduced to a gap-free-floor conjecture and remains open. A repository claims a theorem proving the full statement, but provides no independently verified proof and conflicts with the paper and a formalization retaining sorry.

Current status (as of June 2026): The integer case and all sufficiently large $rac{a}{b}$ are settled, while the small-rational $rac{a}{b}$ case remains open; a purported full proof is unverified.