Pintér, Á. On some arithmetical properties of Stirling numbers. (English) Zbl 0752.11008 Publ. Math. Debr. 40, No. 1-2, 91-95 (1992). Let \(S_k^n\) denote the number of partitions of a set of \(n\) elements into \(k\) non-empty subsets. These numbers are known as Stirling numbers of the second kind. Further, let \(p_1,\ldots,p_s\) (\(s\geq 0\)) be distinct primes and let \(S\) be the set of non-zero integers composed of these primes only. By combining some results of A. Baker [Proc. Camb. Philos. Soc. 65, 439–444 (1969; Zbl 0174.33803)] and A. Schinzel and R. Tijdeman [Acta Arith. 31, 199–204 (1976; Zbl 0303.10016)], the author obtains the following result. Let \(a\geq 1\) be a given integer.(1) If \(S_{n-a}^n\in S\) for some \(n>a\), then \(n<C_1\) for an effectively computable constant \(C_1\) depending on \(S\) and \(a\) only.(2) If \(S_{n-a}^n\) is a perfect \(m\)-th power for some \(m\geq 3\), then \(n<C_2\) for an effectively computable constant \(C_2\) depending on \(a\) only. Reviewer: Roelof J. Stroeker (Rotterdam) Cited in 1 Document MSC: 11B73 Bell and Stirling numbers 11D61 Exponential Diophantine equations Keywords:exponential diophantine equation; number of partitions; Stirling numbers; perfect \(m\)-th power Citations:Zbl 0174.33803; Zbl 0303.10016; Zbl 0339.10018 PDFBibTeX XMLCite \textit{Á. Pintér}, Publ. Math. Debr. 40, No. 1--2, 91--95 (1992; Zbl 0752.11008)