On a diophantine problem concerning Stirling numbers. (English) Zbl 0811.11017

The Stirling number of the second kind \(S_ k^ n\) is defined to be the number of partitions of the set \(\{1,\dots, n\}\) into \(k\) non-empty subsets. Let \(b>a>1\) be rational integers. Consider the equation \(S_ a^ x= S_ b^ y\) to be solved in integers \(x\), \(y\) with \(x>a\) and \(y>b\). In the note under review it is proved, using the theory of linear forms in logarithms, that all the solutions of the above equation satisfy \[ \max \{x,y\}< C b(\log b)^ 3 \log (b!/a!) \log a, \] where \(C\) is an effectively computable absolute constant.


11B73 Bell and Stirling numbers
11D99 Diophantine equations
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