Erdos 1982 another property 239 related questions

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theorem_1: Erdős, Guy and Selfridge's theorem that n! = a_1 a_2 ... a_k has no solution with n < a_1 < a_2 < ... < a_k <= 2n once n > 239, with the number of solutions for each smaller n.

theorem_2: Erdős, Guy and Selfridge's theorem that n! = a_1 a_2 ... a_k has a solution with n < a_1 <= a_2 <= ... <= a_k <= 2n for every n > 13, for which the paper gives an outlined proof.

theorem_3: Erdős, Guy and Selfridge's theorem that the least possible largest factor f(n), over factorizations of n! into distinct integers greater than n, lies strictly between 2n + c_1 n/ln n and 2n + c_2 n/ln n for all large n, for some constants 0 < c_1 < c_2.

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P. Erdős, R. K. Guy, J. L. Selfridge: Another property of 239 and some related questions, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer. 34 (1982), 243--257; MR 84f:10023; Zentralblatt 536.10007. No notice is printed in the file (the only imprint is "CONGRESSUS NUMERANTIUM, VOL. 34 (1982), pp. 243-257" on p. 1, and pp. 1--2 and 14--15 carry no copyright or license line); the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, read 2026-10-02, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the publisher has no online page for Congressus Numerantium and the card gives no DOI, so the publisher's page was not consulted and no Crossref license is recorded; the term is unstated.

The paper considers writing n! = a_1 a_2 ... a_k under various constraints on the factors. Theorem 1 (p. 243) states there are no solutions with n < a_1 < a_2 < ... < a_k ≤ 2n for n > 239, all solutions being enumerated, which is the property of 239 in the title; Theorem 2 says solutions with the factors non-decreasing in (n,2n] exist for all n > 13, and the paper only outlines its proof. Theorem 3 (p. 244) treats f(n), the least possible value of the largest factor a_k when n! is written as a product of distinct integers greater than n, and proves there are constants 0 < c_1 < c_2 with 2n + c_1 n/ln n < f(n) < 2n + c_2 n/ln n for all sufficiently large n; the authors add that no doubt f(n) = 2n + cn/ln n + o(n/ln n) for some constant c, perhaps provable by a more careful application of their method. Further questions concern min(a_k - a_1), whether n! = a_1(a_1+1) has no solutions for n > 3 (never proved, the authors note), and the long-standing conjecture that n! = (x-1)(x+1) has no solution for n > 7 (pp. 244--245). The methods are prime-counting estimates on the interval (n,2n] together with explicit computation. This is the source for Problem 390: Theorem 3 gives the matching upper and lower bounds of order n/ln n for f(n) - 2n, while the existence of the constant c in f(n) - 2n ~ cn/log n is expected but left open.

The copy read for this card is the scan of the hosting archive, printed pages 243--257. Read status: claims checked; the introduction on pp. 243--245, which states Theorems 1--3, the expected asymptotic for f(n) and the further questions, was read clause by clause, with the remarks of p. 249 on max a_1; the proofs of Theorems 1--3 (pp. 249--256) were read for their structure and constants, and their estimates and computations were not rechecked.

Source: https://users.renyi.hu/~p_erdos/1982-01.pdf.

Bears on. #390: the paper's f(n) is the problem's f(n); Theorem 3 (p. 244) gives 2n + c_1 n/ln n < f(n) < 2n + c_2 n/ln n for all large n with constants 0 < c_1 < c_2 (c_1 arbitrarily close to 1/9 and c_2 = 1.7 in the proof), so f(n) - 2n has exact order n/ln n; the existence of the constant c in f(n) - 2n ~ cn/ln n, which is the problem's question, is expected (p. 244) and not proved (theorem_3), #391: reported, not proved here; p. 249 poses as an old problem of Erdős the largest a_1 when k = n in (0) under (5), which is the problem's t(n), says it is easy to see that max a_1 < n/e - cn/ln n, and reports that Erdős, Selfridge and Straus had recently proved max a_1 = n/e + o(n); the paper gives no proof of either, #398: background only; pp. 244--245 call the statement that n! = (x-1)(x+1) has no solution for n > 7 a long outstanding conjecture, and the paper proves nothing on it.

Results.

• Theorem 1 (p. 243): there are no solutions of n! = a_1...a_k with n < a_1 < a_2 < ... < a_k <= 2n for n > 239; Table 1 (p. 251) lists the n <= 242 with no solution and Table 2 (p. 252) gives the number of solutions for the rest.
• Theorem 2 (p. 244): solutions with n < a_1 <= a_2 <= ... <= a_k <= 2n (factors not necessarily distinct) exist for all n > 13; the proof (pp. 252--254) is outlined.
• Theorem 3 (p. 244): with f(n) the least possible largest factor in a representation of n! as a product of distinct integers greater than n, there are constants 0 < c_1 < c_2 with 2n + c_1 n/ln n < f(n) < 2n + c_2 n/ln n for all sufficiently large n; the page also records the unlabelled expectation of p. 244 that f(n) = 2n + cn/ln n + o(n/ln n) for some constant c.

Further questions, pp. 244--245 (unlabelled; no result pages): it has never been proved that n! = a_1(a_1+1) has no solutions for n > 3, and a long outstanding conjecture says that n! = (x-1)(x+1) has no solution for n > 7.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.