Erdős Problem #295
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(k(N)\) denote the smallest \(k\) such that there exists \(N ≤ n_1 < ⋯ < n_k\) with \(\frac 1 {n_1} + ... + \frac 1 {n_k} = 1\). Such a \(k\) exists by `exists_k`.
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Problem #295 asks whether the shortest unit-fraction expansion above a denominator cutoff eventually exceeds (e−1)N by an unbounded amount. That asymptotic problem remains open. Round 1 closed k(17)=32. Round 2 proved the rigorous computational bounds 33≤k(18)≤35; on 11 October a reader, Kyunghan Bae, cut the upper bound to 34 with an explicit 34-term certificate we verified exactly, so now 33≤k(18)≤34. The harmonic maximum excludes at most 30 terms; exact GMP and Fraction recursions independently execute the same proved exhaustive search and exclude 31 and 32. A transparent four-for-one unit-fraction surgery on the k(17) identity supplied our 35 terms beginning at 18; the reader's five-for-four surgery on it gives 34. Full reproduction, sanitizer, serial, concurrency, and hostile source audits passed. On 13 October we added his certificate upper bounds k(19) ≤ 36, k(20) ≤ 38, k(21) ≤ 39 (re-verified three ways). Whether k(18) is 33 or 34 remains open, as does Erdős's asymptotic question.
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