For which integers $a\ge 1$ 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 ${\sum }_{i\le k}{a}_i!$ then $m_k\to \infty$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is **Erdős problem #404** (Erdős–Graham, 1980). In particular, in general it is **not known** which pairs $(a,p)$ give a finite bound and which (if any) give an infinite bound. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.