Problem 405

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

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

has only finitely many solutions?

Source: erdosproblems.com/405

Number theory
Factorials

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

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.

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 and Erdos 1991.
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: Erdős and Graham 1980.
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.

Problem 405

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