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Updated Oct 8

Problem 374

claims/: The 2 claim pages of Problem 374, one per claimant's result; the problem's standing derives from them.

Statement. For any $m\in \mathbb{N}$ , let $F(m)$ be the minimal $k\geq 2$ (if it exists) such that there are $a_1<\cdots <a_k=m$ with $a_1!\cdots a_k!$ a square. Let $D_k=\{ m : F(m)=k\}$ . What is the order of growth of $\lvert D_k\cap{1,\ldots,n}\rvert$ for $3\leq k\leq 6$ ? For example, is it true that $\lvert D_6\cap \{1,\ldots,n\}\rvert \gg n$ ?

Status. OPEN, the site's label (problem page, discussion thread and proof-claims tab read 2026-10-07). The site's proof-claims tab carries three proof claims. Zeraoulia's partial claim of 2026-07-27, the bound $D_6(x)=\omega(x/\log x)$ with certified computations, determines no order of growth and does not reach $D_6(x)\gg x$ , so it settles no instance and has no claim page. Hartley and Olson (claim page) and Yudin (claim page) each claim the full determination. Hartley and Olson's forum submission covered the positive lower density of $D_6$ , and their revised paper adds the orders of growth of $D_3$ through $D_5$ . The derived standing departs from the label: the two pending full claims agree, so the standing is claimed, answered; neither claim is accepted, and the site's curator has credited neither.

Source. erdosproblems.com/374, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #374, https://www.erdosproblems.com/374.

References.

[ErGr76] Erdős, P. and Graham, R. L., On products of factorials. Bull. Inst. Math. Acad. Sinica (1976), 337-355.
[LSS14] Luca, F. and Saradha, N. and Shorey, T. N., Squares and factorials in products of factorials. Monatsh. Math. (2014), 385-400.

Formalization. No formal-conjectures statement: the site's problem page showed none on 2026-10-06. The Lean development of Hartley and Olson, which this corpus has not built, is linked on its claim page.

Current assessment

The standing judges the site's formulation of 2026-09-04 above: the order of growth of $\lvert D_k\cap\{1,\ldots,n\}\rvert$ for $3\le k\le6$ , with the conjecture of Erdős and Graham [ErGr76] that $D_6$ has positive lower density as the example question. The classical facts, by the site's commentary and the sources: no $D_k$ contains a prime, $D_2$ is the set of squares above $1$ , $D_3$ is sparse against $D_4$ , the least element of $D_6$ is $527$ , and $D_k$ is empty for $k>6$ ([ErGr76], card); Luca, Saradha and Shorey [LSS14] bound $D_3$ by $X/\exp(c_0(\log X)^{1/4}(\log\log X)^{3/4})$ (card); and Tao's 2026 preprint, announced on the discussion thread on 2026-03-31, proves $(c+o(1))\sqrt n\le\lvert F_3\cap\{1,\ldots,n\}\rvert\le n^{1/2+o(1)}$ for $F_3=D_2\cup D_3$ , where $c=3.709751\ldots$ (OEIS A389117) is the asymptotic constant of the elementary subset $F_3^1$ , which includes the squares; since the squares contribute $\sqrt n+O(1)$ , this gives $(c-1+o(1))\sqrt n\le\lvert D_3\cap\{1,\ldots,n\}\rvert\le n^{1/2+o(1)}$ , and Yudin's constant for $D_3$ alone is $\kappa_3=c-1$ ; Tao calls this a weak answer to the case $k=3$ (card). The elementary identity $n!(n-1)!u!(u-1)!=\square$ for $n=v^2u$ puts every nonsquarefree nonsquare $n$ in $D_3\cup D_4$ , so $D_4$ has positive lower density. Three claims are pending and none is accepted; two have claim pages. Zeraoulia's preprint of 2026-07-27 derives $D_6(x)=\omega(x/\log x)$ from the fixed-cofactor theorem of Erdős and Graham and reports certified computations; it settles no instance of the question, so it has no claim page. Two independent preprints then assert the full determination: Hartley and Olson, posted on SSRN on 2026-09-29 with the positive lower density of $D_6$ and revised, in their repository, to the orders of growth of $D_3$ through $D_6$ , with a Lean development they report as sorry-free on the standard axioms (claim page), and Yudin, posted on arXiv on 2026-10-01, with $D_3(X)=\kappa_3\sqrt X+O_\varepsilon(X^{2/5+\varepsilon})$ , $\kappa_3=2.709751\ldots$ , and $D_5(X)\asymp D_6(X)\asymp X$ (claim page). Both are unreviewed manuscripts; neither has been refereed or credited by the curator, the corpus has not built Hartley and Olson's Lean development, and the two agree in their conclusions. Search scope: the site's problem page, discussion thread and proof-claims tab with its comments, read 2026-10-07, the arXiv record, the Zenodo record and the GitHub repository named on the Hartley–Olson claim page; the SSRN posting is outside that scope. Nothing on this page is independently reviewed.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (19)

Hartley and Olson's orders of growth of the factorial classes Yudin's determination of the orders of growth of the factorial classes problems/diophantine_problems/E0374/claims Diophantine Problems and Powers On products of factorials Conjecture and questions (pp. 345--346): D_3(x) = (c+o(1)) x^{1/2}, and three-factor squares with a_2 = a_1 - 3 Conjecture (p. 354): D_6(n) > cn Fact 1 (p. 342) with (9) and Fact 2: no prime lies in any D_k, every composite lies in F_6, D_k is empty for k > 6, and D_2 is the squares above 1 Fact 14 (p. 353): the least element of D_6 is 527 = 17 * 31 Fact 6 (p. 346): every n with a square factor above 1 lies in F_4, D_4 has positive density, and D_4(n)/D_3(n) tends to infinity Fact 7 (p. 346): if p in {2, 3, 5, 7, 11} is a proper divisor of n, then n lies in F_5 Theorem 2 (p. 342): D_3(n) = o(n)Theorem 3 (p. 347): for almost all primes p, 13p is not in F_5 diophantine_problems/luca_2014_squares_factorials_products_factorials Theorem 1 (p. 3): the set D_3 has at most X/exp(c_0 (log X)^{1/4} (log log X)^{3/4}) elements up to X diophantine_problems/tao_2026_products_consecutive_integers_unusual_anatomy Lemma 2.10 (p. 17): a hyperbola an^2 + h = bm^2 with coefficients of polynomial size has x^(o(1)) points with n at most x Theorem 1.10 (p. 9): the solutions of a_1! a_2! a_3! = m^2 with a_3 at most x number x^(1/2+o(1))Theorem 1.9 (p. 9): right endpoints of type F_3 intervals outside the elementary set number at most x^(1/2+o(1))

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