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 $k=n$.
Keywords:  Diophantine equation; factorial
Classification MSC:  11D61
DOI:  10.21136/MB.2016.0045-15

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Affiliations:   Horst Alzer, Morsbacher Str. 10, 51545 Waldbröl, Germany, e-mail:; Florian Luca, School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa, e-mail:

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