The board / number theory / #730
Erdős Problem #730
SOLVEDdatabase updated 2025-08-31
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Are there infinitely many pairs of integers \(n < m\) such that \(\binom{2n}{n}\) and \(\binom{2m}{m}\) have the same set of prime divisors? Yes: there are infinitely many consecutive pairs \((n, n+1)\). The formal proof registered as [PALOMAR-2026-08-22-000001] proves `S.Infinite` for this `S`.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/730.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA129515
Original pageerdosproblems.com/730
Attempts
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The live page is OPEN but lists 1 claimed proof—a full proof claimed by Liam Price (using GPT Pro), submitted 2026-07-15—and lists “Currently working on this problem: None”; the claimed-proof stop con
SKIPPED