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Is the maximum size of a set such that is never squarefree (for all ) achieved by taking those ?
Source: erdosproblems.com/848
A full solution has been claimed but not yet accepted. The statement is true.
The site's label is DECIDABLE, meaning that the site counts the
problem as resolved except for a finite check (page last edited 6 December
2025; proof-claim tab accessed 2026-10-06); the label is recorded here, not
adopted as the standing. The standing derives from the
claim pages. The accepted partial claim
Sawhney,
a note the site's curator, Thomas Bloom, credits in the commentary and whose
Section 4 names ChatGPT 5 Pro as having assisted the proof, proves that for all
sufficiently large the class attains the maximum, with a
stability statement, and leaves the finite range of small unchecked and
unquantified. The pending partial claim
Sothanaphan 2026,
a note posted in the discussion thread on 23 March 2026 and produced, as it
declares, with GPT-5.2 Thinking and GPT-5.4 Thinking, makes the threshold
explicit: the class attains the maximum for every .
Two full claims assert the exact maximum for
every :
Pitchford 2026,
a candidate computer-assisted determination published on Zenodo on 28 July
2026 with OpenAI Codex as its reported author and Ian Pitchford as its
publisher, by certificates and exact-rational envelope arguments up to
Sothanaphan's threshold; and
Li 2026, released
on 30 July 2026 and submitted to the site's proof-claim tab on 16 August
2026 with a tools line naming OpenAI ChatGPT 5.5, OpenAI ChatGPT 5.6 and a
proof-engineering system of the author's own, with a Lean 4 project. The
site has acted on none of the three, no referee or named expert has
examined them, and nothing was built or audited here, so they are claimed
and the problem's standing is claimed, proved. The
site's commentary also records the bound
sent by van Doorn, from the fact that must be divisible by the
square of a prime , sharpened to about by
Weisenberg.