Mathematica Bohemica, Vol. 142, No. 2, pp. 181-184, 2017
Diophantine equations involving factorials
Horst Alzer, Florian Luca
Received July 29, 2015. First published December 5, 2016.
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 $k=n$.
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Affiliations: Horst Alzer, Morsbacher Str. 10, 51545 Waldbröl, Germany, e-mail: email@example.com; Florian Luca, School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa, e-mail: firstname.lastname@example.org