Erdős Problem #269
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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A positive integer \(n\) has all its prime factors in the set \(P\). By convention, \(1\) satisfies this for any \(P\) as it has no prime divisors.
The infinite, strictly increasing sequence \(\{a_0, a_1, \dots\}\) of integers whose prime factors all belong to \(P\).
The \(n\)-th partial least common multiple, \([a_0, \dots, a_{n-1}]\), which is the LCM of the first \(n\) integers in the sequence.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/269.Attempts
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Exact carry-factor reduction for all finite P, plus rigorous rational enclosures excluding denominators below 40–51 digits for all three-prime subsets of {2,3,5,7} (and an 18-digit bound for {2,3,5,7}
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