Erdős 730: mark solved and record the registered formal proof - #5091
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williamjblair wants to merge 1 commit into
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Erdős 730: mark solved and record the registered formal proof#5091williamjblair wants to merge 1 commit into
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A Lean 4 proof that `S.Infinite` holds for this file's `S` is registered
with the Palomar registry as PALOMAR-2026-08-22-000001: Palomar
recompiled the statement against Mathlib and checked the proof with the
Lean kernel and NanoDa under propext, Quot.sound and Classical.choice.
The proving theorem is `Erdos730.FullDensityTheorem.pairSet_infinite` in
williamjblair/lean-proofs at 03729c9c; its `PairSet` is this `S` written
as `{z | z.1 < z.2 ∧ …}`. It shows there are infinitely many consecutive
pairs (n, n+1), following the argument Liam Price posted on the
erdosproblems.com forum on 24 June 2026. The `sorry` stays, as for other
externally proved statements.
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Marks Erdős Problem 730 as solved and records the formal proof.
The proof
A Lean 4 proof that
{(n, m) : ℕ × ℕ | n < m ∧ n.centralBinom.primeFactors = m.centralBinom.primeFactors}is infinite is registered with the Palomar registry as PALOMAR-2026-08-22-000001. Palomar recompiled the statement on its own against Mathlib
db584cd6(Lean v4.33.0), checked the proof with both the Lean kernel and NanoDa under the axiomspropext,Quot.sound,Classical.choice, and its editorial review found no problems. The registered Challenge inlines this file'sSverbatim and statesS.Infinite.It shows that there are infinitely many consecutive pairs
(n, n+1), following the argument Liam Price posted on the erdosproblems.com forum on 24 June 2026; the analytic steps are reconstructed in the formalisation. The proof was produced with AI assistance, as the registry record's provenance states.For the
formal_proofchecklist inPROOFS.mdErdos730.FullDensityTheorem.pairSet_infiniteinwilliamjblair/lean-proofsat03729c9c.FullDensityCore.PairSet.Infinite, wherePairSetis{z | z.1 < z.2 ∧ z.1.centralBinom.primeFactors = z.2.centralBinom.primeFactors}, this file'sSwith the pair written asz.sorry, no custom axioms anywhere in the closure (scripts/check_axioms.shin that repository gates every headline theorem).#print axioms:[propext, Classical.choice, Quot.sound].lean4, commit-pinned link.Changes
erdos_730:research open→research solved,answer(sorry)→answer(True),@[formal_proof using lean4 at …]; thesorrystays.erdosproblems.com still lists
#730as open. If you would rather wait for the site's status to change, I am happy to hold this.