Claims

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1992_10_01_berend_osgood: Berend and Osgood prove that for every integer polynomial P of degree at least 2 the n with P(x) = n! solvable have density zero, so for each fixed m the n with f(n) = m are o(N) up to N; a refereed partial result.

2002_01_01_luca: Luca proves that the abc conjecture implies that P(x) = n! has finitely many solutions for every integer polynomial P of degree at least 2; through the problem's reduction, f(n) tends to infinity under abc. Refereed, conditional.

2022_04_18_bui_pratt_zaharescu: Bui, Pratt and Zaharescu prove that the n in [N, 2N) with s n! = P(x) solvable number at most C N^(33/34) for fixed P of degree at least 2, so F_m(N) is O_m(N^(33/34)); a refereed partial result.

2026_05_03_turturean: A write-up claiming that, under the abc conjecture, n - O(log n) <= f(n) <= n - 2 for all large n, built from Tao's thread sketches by an audit-and-revise scaffold querying ChatGPT-5.5-Pro; conditional and unreviewed.

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