Claims

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1964_01_01_erdos_straus: Erdős and Straus's 1964 Theorem 3 proves that a reciprocal sum with limsup n_k^2/n_(k+1) at most 1 is rational exactly when the Sylvester recurrence holds eventually, given a limsup condition on the lcm of the terms.

2001_01_01_duverney: Duverney's 2001 Corollary 3.2 proves that a reciprocal sum whose ratios u_(n+1)/u_n^2 have summable deviation from 1 is rational exactly when the Sylvester recurrence holds eventually, with signs allowed.

2025_04_08_koizumi: Koizumi's Corollary 1 proves that a sequence of positive integers with a_n^2/a_(n+1) in [2/3, 4/3] and reciprocal sum 1 is Sylvester's sequence; Theorem 3 reduces the problem to a conjecture on pseudo-greedy expansions.

2026_09_11_cook: Cook, with a write-up by Astra and other AI agents, claims that the Sylvester recurrence holds eventually whenever the product-weighted error (P_n/a_n)(a_n^2/a_(n+1) - 1) is bounded above.

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