Erdős Problem #430
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The terms of the sequence of Erdős Problem 430 for a given \(n\): the integers \(m\) with \(1 < m < n\) all of whose prime factors are \(> n - m\). The decreasing sequence \(a_1 > a_2 > \cdots\) of the problem lists the elements of this set in decreasing order: \(a_1 = n - 1\) is its largest element and \(a_k\) is its largest element below \(a_{k-1}\). The integer \(1\) has no prime factors and is not a term of the sequence, which stops instead, as in the example on erdosproblems.com: for \(n = 8\) the sequence is
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exactly 100 exceptions are certified for 5 <= n <= 10^10 (last 267680), with an elementary reduction and standalone checker; the uniform semiprime-gap step remains open.
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