Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Claim. The theorem erdos_289 of the linked file states, in Lean 4 over
Mathlib, that for all sufficiently large there are pairs
of natural numbers with , pairwise separated in the
sense that or for , whose interval
reciprocal sums add to exactly in .
This is the restricted statement of
Problem 289 as formal-conjectures
renders it: distinct, non-overlapping, non-adjacent intervals of at least two
integers. The file (about 9,500 lines, no sorry) builds the proof from
four anchor intervals , , , whose
reciprocals sum to , packets of four short intervals around
multiples of a parameter, and a thinning argument for a simultaneous
independent choice of intervals; by its docstring the proof follows a path
independent of the three solutions on the site's proof-claim tab.
Standing. Claimed. The proof was produced by the LEAP prover agent, the
system of Kung and coauthors (arXiv:2606.03303), which the docstring names as
its author; the file was committed to a fork of formal-conjectures on 1
October 2026 and proposed the same day in pull request 6781 of the main
repository, merged on 7 October 2026, which marks erdos_289 as research solved with answer(True) and links the fork's file as its formal_proof
while keeping a sorry body in the main file. The site's label is OPEN (page
last edited 22 September 2025; accessed 2026-10-07) and the proof is not on
its proof-claim tab; no curator, referee or named mathematician has accepted
it, and the repository's merge review is not an independent mathematical
review. The file is third-party Lean that this corpus has not built or
audited, so formalized is not listed and the claim is pending. The three
earlier claims of the same statement,
Land,
Tang and
Budden, are pending
as well.