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On some arithmetical properties of Stirling numbers. (English) Zbl 0752.11008

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.

MSC:

11B73 Bell and Stirling numbers
11D61 Exponential Diophantine equations
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