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Claim. The answer to Problem 405 is yes, with the complete list. Le proves that for an odd prime $p$ and positive integers $a,k$ the equation

$$(p-1)!+a^{p-1}=p^k$$

holds only for $(p,a,k)=(3,1,1)$, $(3,5,3)$ and $(5,1,2)$, that is

$$2!+1^2=3,\qquad 2!+5^2=3^3,\qquad 4!+1^4=5^2 ;$$

the paper writes the equation as $x^{p-1}+(p-1)!=p^n$. The list is finite, so the question is answered. The finiteness, with an effective bound on every solution, was proved earlier by Brindza and Erdős on the claim page Brindza and Erdős 1991, and the same three solutions were determined independently, in the same year, by Yu and Liu on the claim page Yu and Liu 1996. The publication records (Publ. Math. Debrecen 48, no. 1-2, issued 1996; Rocky Mountain J. Math. 26, no. 3, issued September 1996) do not settle which determination came first, and this page claims no priority. The library holds no copy of the paper; the statement is taken from the publisher's record, from the zbMATH review of the paper (Zbl 0867.11020), which records that it solves the Erdős–Graham problem from p. 80 of their monograph with exactly these solutions, and from the Lean file below, whose header credits the three triples to Yu and Liu and to Le.

Depends on. No page of this wiki.

Formalization. The Lean file Erdos405.lean in Boris Alexeev's repository of Lean proofs declares itself a formalization of a solution to the problem, with Brindza, Erdős, Yu, Liu and Maohua Le as its informal authors and the AI systems Codex and GPT-5.6 Sol as its formal authors. Its theorem erdos_405 states that every solution with $p$ an odd prime is one of the three triples above, and one of its lemmas carries out the valuation step of Le's reduction, that for $a\neq1$ every odd prime divisor of $p-1$ divides $k$. The file was added on 2026-08-17 and the link pins the last commit that touched it at its path; the file's text at that commit contains no sorry. This corpus has not built or audited the file, so the page lists no formalized evidence.

Acceptance. The paper is M. Le, On the Diophantine equation $x^{p-1}+(p-1)!=p^n$, Publ. Math. Debrecen 48 (1996), no. 1-2, 145--149, a refereed journal, listed as refereed. The site's curator does not credit Le, so no reviewed evidence is listed. The page is dated by the publisher's record, which gives 1996 without a month; the first day of the year stands in for the issue date.