Problem 405

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claims/: The 3 claim pages of Problem 405, one per claimant's result; the problem's standing derives from them.

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Statement. Let $p$ be an odd prime. Is it true that the equation

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

has only finitely many solutions?

Status. The site labels the problem PROVED (LEAN). The standing derived from the claim pages is solved, proved: Brindza and Erdős bounded every solution by an effective absolute constant (Brindza and Erdős 1991), and the three solutions were determined in two refereed papers of 1996, by Yu and Liu (Yu and Liu 1996) and by Le (Le 1996); the site's curator credits Brindza and Erdős and Yu and Liu. The Lean proof the site's label refers to is third-party work not built here.

Source. erdosproblems.com/405, accessed 2026-09-04. The site cites the problem from p. 80 of Erdős and Graham's 1980 problem book [ErGr80]. Cite as: T. F. Bloom, Erdős Problem #405, https://www.erdosproblems.com/405.

References.

• [BrEr91] Brindza, B. and Erdős, P., On some Diophantine problems involving powers and factorials. J. Austral. Math. Soc. Ser. A 51 (1991), no. 1, 1-7. Library home: brindza_1991_diophantine_problems_involving_powers_factorials.
• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 80. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Le96] Le, Maohua, On the Diophantine equation $x^{p-1}+(p-1)!=p^n$. Publ. Math. Debrecen 48 (1996), no. 1-2, 145-149. Not among the site's references.
• [YuLi96] Yu, Kunrui and Liu, Dehua, A complete resolution of a problem of Erdős and Graham. Rocky Mountain J. Math. 26 (1996), no. 3, 1235-1244.

Formalization. The formal-conjectures file FormalConjectures/ErdosProblems/405.lean (commit of 2026-09-27) states the finiteness for odd primes as erdos_405 and the three solutions as erdos_405.variants.yu_liu, both with sorry and tagged solved, proves that the case $p=2$ has infinitely many solutions, and names no formal proof. The file Erdos405.lean of Boris Alexeev's repository of Lean proofs proves the list of solutions; the three claim pages link it at a pinned commit. Nothing has been built here.

Current assessment

The question, as the site states it: for an odd prime $p$, does $(p-1)!+a^{p-1}=p^k$ have only finitely many solutions in positive integers $a,k$? The answer is yes, and the solutions are known.

The resolution. Brindza and Erdős [BrEr91], Theorem 2, prove that every solution satisfies $\max\{p,a,k\}<C$ for an effectively computable absolute constant $C$, by Baker's method and their bound for the Ramanujan--Nagell equation $x^2+D=p^k$; so the solutions are finite in number over all odd primes together, not only for each $p$. The solutions were then determined, in two refereed papers of 1996 whose order the publication records do not settle, by Yu and Liu [YuLi96] and by Le [Le96]: $(p,a,k)=(3,1,1)$, $(3,5,3)$ and $(5,1,2)$, that is $2!+1^2=3$, $2!+5^2=3^3$ and $4!+1^4=5^2$. The site credits Brindza and Erdős and Yu and Liu; Le's paper is not among its references, and the third-party Lean proof names the authors of all three papers as its informal authors. The three claim pages record each result and its acceptance.

Context. Erdős and Graham's book posed the question for every prime, but for $p=2$ the equation reads $1+a=2^k$ and has a solution for every $k$, so the restriction to odd primes is needed; the site makes the same correction. They also expected $(p-1)!+a^{p-1}$ to be a perfect power only rarely, and the site records $6!+2^6=28^2$ as a case where it is. Brindza and Erdős recall Liouville's theorem that for an odd prime $p$ the equation $(p-1)!+1=p^k$ holds only for $p=3$ and $p=5$, and observe that no composite $n$ satisfies $(n-1)!+a^{n-1}=n^k$.

Search scope, 2026-10-07: the site's problem page, discussion thread (no comments) and proof-claims page (none listed); the formal-conjectures statement file at the commit the Formalization field links; the Lean file in Alexeev's repository at the commit the claim pages link, neither built here; Le's paper through its publisher's record and the zbMATH reviews of it (Zbl 0867.11020) and of Yu and Liu's paper (Zbl 0886.11018).

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• brindza_1991_diophantine_problems_involving_powers_factorials
• brindza_1991_diophantine_problems_involving_powers_factorials / theorem_2
• brindza_1991_diophantine_problems_involving_powers_factorials / theorem_3