Problem 404

For which integers $a\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\cdots<a_n$ with

p^k \mid (a_1!+\cdots+a_n!)?

If $f(a,p)$ is the greatest such $k$ , how does this function behave?

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 $m_k\to \infty$ ?

Source: erdosproblems.com/404

Number theory
Factorials

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 29 September 2025) and credits Lin [Li76] with $f(2,2)\le254$ , a pending partial claim on its claim page (Lin, 1976); the exact values of $f(a,2)$ for $a\le256$ and of $f(a,p)$ at some pairs with $p=3,5,7,11,13$ posted in the thread in July 2026 are pending partial claims on their page for p = 2 and their page for odd primes; the September 2025 comments in the thread are posts without a manuscript and have none. The standing in the frontmatter derives from the claim pages.

Problem 404

Status

Proof claims

Discussion