Loading problem…
Status
On this page
Status
Topics
Problem 306
Let with squarefree. Are there integers , each the product of two distinct primes, such that
Source: erdosproblems.com/306
- Number theory
- Unit fractions
Status
Claimed, ProvedA full solution has been claimed but not yet accepted. The statement is true.
Open, in the site's label. The frontmatter standing is claimed
with claim proved, derived from two pending full claims, each on its own
page:
Tang's Lean proof
of June 2026, a Lean development asserting the whole statement with two
Rosser--Schoenfeld bounds taken as axioms, followed by a manuscript
published in October 2026, and
Li's elementary proof
of September 2026, a preprint with a Lean formalization and the one full
proof claim on the site's tab. A third page,
Li's earlier preprint
of June 2026, is a partial claim covering the case and every
above an explicit threshold. Three further partial pages record the
representations of by two-prime denominators that settle the instance
, Barbeau's of 1977, Johnson's of 1978 and Watanabe's of 2020, each
claimed. None of the six is refereed, independently reviewed or accepted
by the site, which labels the problem OPEN (page last edited 21 June 2026,
and the community database agreeing, as of 2026-10-07); this corpus has
built none of the Lean developments. No refereed source proves the statement
for any beyond finite examples for ; the refereed theorem on
record is Butler, Erdős and Graham's Theorem 1, the natural-number case with
three prime factors instead of two.