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.

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

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