Problem 393

../

claims/: The 4 claim pages of Problem 393, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $f(n)$ denote the minimal $m\geq 1$ such that

$$n! = a_1\cdots a_t$$

with $a_1<\cdots <a_t=a_1+m$. What is the behaviour of $f(n)$?

Status. Open. The site labels the problem OPEN and credits no solution; its remarks credit Berend and Osgood, and Bui, Pratt and Zaharescu, with density and counting bounds for the $n$ with $f(n)=m$, and Luca, under the abc conjecture, with $f(n)\to\infty$. The standing derives from the claim pages: the accepted partial claims Berend and Osgood 1992 and Bui, Pratt and Zaharescu 2023 are refereed bounds on how often $f(n)$ takes a given value; the accepted conditional claim Luca 2002 and the pending conditional claim Turturean 2026 rest on the unproved abc conjecture and derive nothing; so the problem is open with no settling or pending full claim.

Source. erdosproblems.com/393, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #393, https://www.erdosproblems.com/393.

References.

• [BPZ23] Bui, Hung M. and Pratt, Kyle and Zaharescu, Alexandru, Power savings for counting solutions to polynomial-factorial equations. Adv. Math. 422 (2023), Paper No. 109021, 32.
• [BeOs92] Berend, Daniel and Osgood, Charles F., On the equation $P(x)=n!$ and a question of Erdős. J. Number Theory (1992), 189-193.
• [Lu02] Luca, Florian, The Diophantine equation $P(x)=n!$ and a result of M. Overholt. Glas. Mat. Ser. III (2002), 269-273.

Formalization. None recorded.

Current assessment

The dated site formulation above asks for the behavior of $f(n)$, the least spread $m\ge1$ between the smallest and the largest factor when $n!$ is written as a product of distinct increasing integers $a_1<\cdots<a_t=a_1+m$. Erdős and Graham asked in particular whether $f(n)=1$ infinitely often, that is, whether a factorial is the product of two consecutive integers infinitely often; that remains open unconditionally. The factorization $n!=2\cdot3\cdots n$ gives $f(n)\le n-2$ for $n>2$.

Every result on the problem passes through one reduction. If $f(n)=m$, the factors form a set $S\subseteq\{0,\ldots,m\}$ of offsets from $a=a_1\ge1$ containing $0$ and $m$, and $n!=P_S(a)$ with $P_S(X)=\prod_{s\in S}(X+s)$, an integer polynomial of degree $|S|\ge2$; for fixed $m$ there are finitely many such $S$. So a theorem about the equation $P(x)=n!$ for a fixed polynomial of degree at least $2$ bounds $F_m(N)$, the number of $n\le N$ with $f(n)=m$. Berend and Osgood 1992 prove that the solvable $n$ have density zero for every such $P$ [BeOs92], so $F_m(N)=o(N)$ for each fixed $m$; Bui, Pratt and Zaharescu 2023 prove the power saving $F_m(N)\ll_m N^{33/34}$ [BPZ23]. Both are refereed and enter as accepted partial claims; they bound how often $f(n)$ is small without deciding whether $f(n)\to\infty$.

Under the abc conjecture more is known. Luca 2002 proves that abc implies finitely many solutions of $P(x)=n!$ for every integer $P$ of degree at least $2$ [Lu02], so through the reduction $f(n)\to\infty$ and $f(n)=1$ only finitely often; the page is refereed and accepted as a conditional claim, which derives nothing for the standing. Two thread sketches of 2025-09-16 and 2025-09-17 by Terence Tao outline an abc-conditional proof that $f(n)=n-O(\log n)$: a spread below $n-C\log n$ forces, through the power of $2$ in $n!$ and Stirling's formula, only $O(\log n)$ factors, all of size at least $n^{10}$, and then two nearby factors with radicals too small for abc. Turturean 2026 is a write-up of that argument, produced by an audit-and-revise scaffold querying ChatGPT-5.5-Pro, claiming $n-O(\log n)\le f(n)\le n-2$ for all large $n$ under abc; it is a pending conditional claim with no reviewer's acceptance.

The status search covered the site's problem page and discussion thread (accessed 2026-10-07), the journal records of the three cited papers and Luca's text; no formal-conjectures statement file exists for the problem, and no proof was checked here.

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.

• bui_2023_power_savings_counting_solutions_polynomial_factorial
• bui_2023_power_savings_counting_solutions_polynomial_factorial / proposition_3_2
• bui_2023_power_savings_counting_solutions_polynomial_factorial / theorem_1_1
• luca_2002_diophantine_equation_result_m
• luca_2002_diophantine_equation_result_m / proposition_1