On the equation \(P(x)=n!\) and a question of Erdős. (English) Zbl 0762.11010

Berend, Daniel; Osgood, Charles F.

J. Number Theory 42, No. 2, 189-193 (1992).

Let \(P\) be a polynomial of degree \(\geq 2\) over \(\mathbb{Z}\) and \(s\) be a fixed non-zero integer. The authors prove that the set \(\{n\mid\) \(n\in\mathbb{N}\), \(P(x)=s\cdot n!\) has a solution \(x\in\mathbb{Z}\}\) is of density zero. This answers in particular a question of Erdős, which originated from a study of diophantine equations \(x^ p\pm y^ p=n!\) [P. Erdős and R. Obláth, Acta Litt. Sci. Szeged. 8, 241-255 (1937; Zbl 0017.00404)]. The proof of this theorem bases on results concerning \(G\)-functions, and is very enlightening.

Reviewer: Zhu Yaochen (Beijing)

MSC:

11D41

Higher degree equations; Fermat’s equation

Keywords:

\(G\)-functions; polynomial of higher degree; integer solution; density zero; diophantine equations

Citations:

Zbl 0017.00404

References:

[1]	Chudnovsky, G. V., The Thue-Siegel-Roth Theorem for values of algebraic functions, (Proc. Japan Acad. Ser. A Math. Sci., 59 (1983)), 281-284 · Zbl 0518.10037
[2]	Erdős, P.; Obláth, R., Über diophantische Gleichungen der Form \(n\)! = \(x^p\) ± \(y^p\) und \(n\)! ± \(m\)! = \(x^p\), Acta Litt. Sci. Szeged, 8, 241-255 (1937) · JFM 63.0113.02
[3]	Guy, R. K., Problems from western number theory conferences (1983)
[4]	Guy, R. K., (Unsolved Problems in Number Theory (1981), Springer-Verlag: Springer-Verlag New York) · Zbl 0474.10001
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