On products of factorials

../

conjecture_p346: Erdős and Graham's unproved belief that D_3(x) = (c+o(1)) x^{1/2}, with their examples of square products a_1! a_2! a_3! with a_2 = a_1 - 3 and their question whether a_3 < a_2 < a_1 - 3 can occur.

conjecture_p354: Erdős and Graham's unproved belief that the integers needing six distinct factorials, the largest n!, to form a square product have positive lower density, D_6(n) > cn.

fact_1: Erdős and Graham's sets F_k and D_k = F_k - F_{k-1} for square products of at most k distinct factorials with largest n!, with their facts that 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 set of squares above 1.

fact_14: Erdős and Graham's Fact 14, that the least integer n needing six distinct factorials, the largest n!, to form a square product is 527 = 17 * 31.

fact_6: Erdős and Graham's observation that every n = m^2 r with m > 1 lies in F_4, so all multiples of 4 lie in F_4 and D_4 has positive density, with Fact 6, that D_4(n)/D_3(n) tends to infinity.

fact_7: Erdős and Graham's Fact 7, first observed by E. G. Straus: every n with a proper divisor p in {2, 3, 5, 7, 11} has a square product of at most five distinct factorials with largest n!.

question_p337: Erdős and Graham's suggestion, posed without proof, that a product of two or more disjoint blocks of consecutive integers, each of at least three integers, is a square in only finitely many cases.

theorem_1: Erdős and Graham's theorem that the products of a! over the subsets A of {1,...,n} take exp{(1+o(1)) n log log n / log n} distinct values.

theorem_2: Erdős and Graham's theorem that the integers n whose least square product of distinct factorials with largest n! has exactly three factors have density zero, D_3(n) = o(n).

theorem_3: Erdős and Graham's theorem that for almost all primes p the integer 13p has no square product of at most five distinct factorials with largest (13p)!.

---

P. Erdős, R. L. Graham: On products of factorials, Bull. Inst. Math. Acad. Sinica 4 (1976) no. 2, 337--355; MR 57 #256; Zentralblatt 346.10004.

Erdős and Graham study products of factorials in the spirit of the Erdős--Selfridge theorem that no product of two or more consecutive positive integers is a perfect power. Theorem 1 (p. 338) shows the possible values are sparse: the number $m(n)$ of distinct products $\prod_{a\in A}a!$ over the subsets $A$ of $[1,n]$ is $\exp\{(1+o(1))\,n\log\log n/\log n\}$. The paper then studies the equation $a_1!a_2!\cdots a_t!=y^2$ when the largest factor $n!$ is prescribed and the number $t$ of factorials is to be least; the abstract and introduction (pp. 337--338) say each increase in the allowed $t$ rather dramatically enlarges the set of $n$ for which the equation is solvable, until $t=6$, after which no increase occurs.

With $F_k$ the set of $n$ for which some set of at most $k$ distinct factorials with largest $n!$ has a square product, and $D_k=F_k-F_{k-1}$ (definitions, (9) and Facts 1--2, pp. 341--342): no prime lies in any $D_k$, every composite lies in $F_6$, so $D_k$ is empty for $k>6$, and $D_2$ is the set of squares above $1$. Theorem 2 (p. 342) proves $D_3(n)=o(n)$; the four-factor section (p. 346) shows every $n=m^2r$ with $m>1$ lies in $F_4$, so $D_4$ has positive density and $D_4(n)/D_3(n)\to\infty$ (Fact 6); Fact 7 (p. 346) puts every $n$ with a proper divisor in $\{2,3,5,7,11\}$ in $F_5$; Theorem 3 (p. 347) proves $13p\notin F_5$ for almost all primes $p$; and Fact 14 (p. 353) gives $527=17\cdot31$ as the least element of $D_6$, the check of $527$ being a computation the paper does not print. The authors state without proof that they are sure $D_3(x)=(c+o(1))x^{1/2}$ (p. 346) and reasonably certain that $D_6(n)>cn$ (p. 354). In the abstract and introduction they also suggest, without proof, that a product of disjoint blocks of consecutive integers, each of at least three integers, is a square only finitely often (p. 337). The paper closes with further unresolved questions.

Read status: claims checked for the statements on the result pages below, read clause by clause on the page images of the print; proofs were read for structure only. Table 1 on p. 353 has a misprint in the row for $323$, recorded on the Fact 14 page. Nothing here is independently reviewed.

Source: https://users.renyi.hu/~p_erdos/1976-25.pdf. No notice is printed on the scan, whose head reads "BULLETIN OF THE INSTITUTE OF MATHEMATICS ACADEMIA SINICA Volume 4, Number 2, December 1976"; the hosting archive's own notice "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only." (https://users.renyi.hu/~p_erdos/, read 2026-10-02) speaks for the site, not the paper; the publisher's site has no page for the 1976 article, and its site-wide footer reads "© Copyright 2026. Math Sinica All Rights Reserved." (https://www.math.sinica.edu.tw/bulletin/), every other right reserved.

Bears on. #374: the paper defines the sets $D_k$ the problem asks about and shows $D_k$ is empty for $k>6$ (Fact 1); for $k=3$ it proves the upper bound $D_3(n)=o(n)$ (Theorem 2) and conjectures $D_3(x)=(c+o(1))x^{1/2}$ (p. 346); for $k=4$ it states that $D_4$ has positive density (p. 346); for $k=6$ it finds the least element $527$ (Fact 14) and poses $D_6(n)>cn$, the problem's example question, as a conjecture (p. 354). It gives no order of growth for $D_5$ and proves none for $D_3$ or $D_6$. #363: the paper's suggestion on p. 337 (question) is the problem's finiteness question with intervals of at least three integers in place of four; the paper proves nothing about it.

Results.

• Theorem 1 (p. 338): the number of distinct products of distinct factorials up to $n!$ is $\exp\{(1+o(1))\,n\log\log n/\log n\}$.
• Fact 1 (p. 342), with (9) and the composite cases (p. 341) and Fact 2 (p. 342): the sets $F_k$, $D_k$; no prime in any $D_k$, every composite in $F_6$, $D_k$ empty for $k>6$, $D_2$ the squares above $1$.
• Theorem 2 (p. 342): $D_3(n)=o(n)$.
• Conjecture (pp. 345--346): $D_3(x)=(c+o(1))x^{1/2}$, with the three-factor examples and question.
• Fact 6 (p. 346): every $n=m^2r$ with $m>1$ lies in $F_4$, $D_4$ has positive density, and $D_4(n)/D_3(n)\to\infty$.
• Fact 7 (p. 346): a proper divisor in $\{2,3,5,7,11\}$ puts $n$ in $F_5$.
• Theorem 3 (p. 347): $13p\notin F_5$ for almost all primes $p$.
• Fact 14 (p. 353): the least element of $D_6$ is $527$.
• Conjecture (p. 354): $D_6(n)>cn$.
• Question (p. 337): products of disjoint blocks of at least three consecutive integers are perhaps squares only finitely often.

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