Problem 404

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claims/: The 3 claim pages of Problem 404, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $m_k\to \infty$?

Status. 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; 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.

Source. erdosproblems.com/404, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #404, https://www.erdosproblems.com/404.

References.

• [Li76] Lin, S., On two problems of Erdős concerning sums of distinct factorials. Bell Laboratories internal memorandum (1976); the site's citation prints 1960, a misprint (see Problem 403).

Formalization. None recorded: formal-conjectures has no statement file for the problem, and the community database lists it as not formalized. The Lean certificates of particular values of $f$ are linked from Kitamura's claim pages; this corpus has not built them.

Current assessment

The questions, as the site states them (page last edited 29 September 2025): for which $a\ge1$ and primes $p$ is the exponent of $p$ in a sum of distinct factorials beginning with $a!$ bounded; how does the largest exponent $f(a,p)$ behave; and is there a prime $p$ and an infinite increasing sequence whose partial factorial sums are divisible by ever higher powers of $p$? All three are open. The site's commentary records one result, Lin's $f(2,2)\le254$ from his 1976 memorandum, so the first question has the answer yes at $(2,2)$. Kitamura's two repositories of July 2026 compute $f(a,2)$ exactly for every $a\le256$, with $f(2,2)=254$ (so Lin's bound is sharp), $f(a,2)=v_2(a!)$ for odd $a$, and $f(34,2)=18444$ the largest value in the range, and compute $f(a,p)$ exactly at some pairs with $p=3,5,7,11,13$; each exact value answers the first question yes at its pair. Their lower-bound witnesses, such as $f(1,5)\ge20000$, decide nothing. Tao's thread posts of 29 September 2025 observe that $f(1,5)$ appears very large and possibly infinite, and give a lemma for lifting a lower bound on $f(a,p)$ from $N$ to $N+H$ when the subset sums of $H$ consecutive factorials cover the residues modulo the relevant power of $p$; the posts are comments without a manuscript and have no pages. No result on the behavior of $f$ beyond these values, and nothing on the third question, is recorded; no refereed result is known.