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


References:
[1] T. Andreescu, D. Andrica, I. Cucurezeanu: An Introduction to Diophantine Equations. A Problem-Based Approach. Birkhäuser, Basel (2010). DOI 10.1007/978-0-8176-4549-6 | MR 2723590 | Zbl 1226.11001
[2] I. G. Bashmakova: Diophantus and Diophantine Equations. The Dolciani Mathematical Expositions 20. The Mathematical Association of America, Washington (1997). MR 1483067 | Zbl 0883.11001
[3] H. Carnal: Aufgaben. Elem. Math. 67 (2012), 151-154. DOI 10.4171/EM/203 | Zbl 1247.97035
[4] F. Luca: The Diophantine equation $R(x)=n!$ and a result of M. Overholt. Glas. Mat. (3) 37 (2002), 269-273. MR 1951531 | Zbl 1085.11023
[5] F. Luca: On the Diophantine equation $f(n)=u!+v!$. Glas. Mat. (3) 48 (2013), 31-48. DOI 10.3336/gm.48.1.03 | MR 3064240 | Zbl 06201413
[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

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://mb.math.cas.cz/.

Affiliations:   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

 
PDF available at: