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Let $f(n)$ be the minimal $m$ such that \[n! = a_1\cdots a_k\] with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that \[f(n)-2n \sim c\frac{n}{\log n}?\]
Erdős, Guy, and Selfridge [EGS82] have shown that \[f(n)-2n \asymp \frac{n}{\log n}.\]
Let $t(n)$ be the maximum $m$ such that \[n!=a_1\cdots a_n\] with $m=a_1\leq \cdots \leq a_n$. Obtain good upper bounds for $t(n)$. In particular does there exist some constant $c>0$ such that \[t(n) \leq \frac{n}{e}-c\frac{n}{\log n}\] for infinitely many $n$?
Erdős, Selfridge, and Straus have shown that \[\lim \frac{t(n)}{n}=\frac{1}{e}.\] Alladi and Grinstead [AlGr77] have obtained similar results when the $a_i$ are restricted to prime powers.
Let $A(n)$ denote the least value of $t$ such that \[n!=a_1\cdots a_t\] with $a_1\leq \cdots \leq a_t\leq n^2$. Is it true that \[A(n)=\frac{n}{2}-\frac{n}{2\log n}+o\left(\frac{n}{\log n}\right)?\]
If we change the condition to $a_t\leq n$ it can be shown that \[A(n)=n-\frac{n}{\log n}+o\left(\frac{n}{\log n}\right)\] via a greedy decomposition (use $n$ as often as possible, then $n-1$, and so on). Other questions can be asked for other restrictions on the sizes of the $a_t$.
Let $f(n)$ denote the minimal $m$ such that \[n! = a_1\cdots a_t\] with $a_1<\cdots <a_t=a_1+m$. What is the behaviour of $f(n)$?
Erdős and Graham write that they do not even know whether $f(n)=1$ infinitely often (i.e. whether a factorial is the product of two consecutive integers infinitely often).
Is it true that there are no solutions to \[n! = x^k\pm y^k\] when $k>2$?
Erdős and Obláth [ErOb37] proved this is true when $(x,y)=1$ and $k\neq 4$. Pollack and Shapiro [PoSh73] proved this is true when $(x,y)=1$ and $k=4$. The known methods break down without the condition $(x,y)=1$.
For any $k\geq 2$ let $g_k(n)$ denote the maximal value of \[n-(a_1+\cdots+a_k)\] where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$. Can one show that \[\sum_{n\leq x}g_k(n) \sim c_k x\log x\] for some constant $c_k$? Is it true that there is a constant $c_k$ such that for almost all $n<x$ we have \[g_k(n)=c_k\log x+o(\log x)?\]
Erdős and Graham write that it is easy to show that $g_k(n) \ll_k \log n$ always, but the best possible constant is unknown.

See also [401].

Is there some function $\omega(r)$ such that $\omega(r)\to \infty$ as $r\to\infty$, such that for all large $n$ there exist $a_1,a_2$ with \[a_1+a_2> n+\omega(r)\log n\] such that $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?
See also [400].
Does the equation \[2^m=a_1!+\cdots+a_k!\] with $a_1<a_2<\cdots <a_k$ have only finitely many solutions?
Asked by Burr and Erdős. Frankl and Lin [Li76] independently showed that the answer is yes, and the largest solution is \[2^7=2!+3!+5!.\] In fact Lin showed that the largest power of $2$ which can divide a sum of distinct factorials containing $2$ is $2^{254}$, and that there are only 5 solutions to $3^m=a_1!+\cdots+a_k!$ (when $m=0,1,2,3,6$).

See also [404].

Let $f(a,p)$ be the largest $k$ such that there are $a=a_1<\cdots<a_k$ such that \[p^k \mid (a_1!+\cdots+a_k!).\] Is $f(a,p)$ bounded by some absolute constant? What if this constant is allowed to depend on $a$ and $p?

Is there a prime $p$ and an infinite sequence $a_1<a_2<\cdots$ such that if $p^{m_k}$ is the highest power of $p$ dividing $\sum_{i\leq k}a_i!$ then $m_k\to \infty$?

See also [403]. Lin [Li76] has shown that $f(2,2) \leq 254$.
Let $p$ be a prime. Is it true that the equation \[(p-1)!+a^{p-1}=p^k\] has only finitely many solutions?
Erdős and Graham remark that it is probably true that in general $(p-1)!+a^{p-1}$ is rarely a power at all (although this can happen, for example $6!+2^6=28^2$).
If $\tau(n)$ counts the number of divisors of $n$, then what is the set of limit points of \[\frac{\tau((n+1)!)}{\tau(n!)}?\]
It can be shown that any number of the shape $1+1/k$ for $k\geq 1$ is a limit point, but no others are known.