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Horst Alzer, Morsbacher Str. 10, 51545 Waldbröl, Germany, e-mail: h.alzer@gmx.de; Florian Luca, School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa, e-mail: florian.luca@wits.ac.za
Abstract: We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.
Keywords: Diophantine equation; factorial
Classification (MSC 2010): 11D61
DOI: 10.21136/MB.2016.0045-15
Full text available as PDF.
References:
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[6] J. Sándor: On some Diophantine equations involving the factorial of a number. Seminar Arghiriade. Univ. Timişoara 21 (1989), 4 pages. MR 1124179 | Zbl 0759.11011
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