Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
Diophantine equation; factorial
Diophantine equation; factorial
Summary:
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 $k=n$.
References:
[1] Andreescu, T., Andrica, D., Cucurezeanu, I.: 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] Bashmakova, I. G.: Diophantus and Diophantine Equations. The Dolciani Mathematical Expositions 20. The Mathematical Association of America, Washington (1997). MR 1483067 | Zbl 0883.11001
[3] Carnal, H.: Aufgaben. Elem. Math. 67 (2012), 151-154. DOI 10.4171/EM/203 | Zbl 1247.97035
[4] Luca, F.: 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] Luca, F.: 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] Sándor, J.: On some Diophantine equations involving the factorial of a number. Seminar Arghiriade. Univ. Timişoara 21 (1989), 4 pages. MR 1124179 | Zbl 0759.11011
Search
Partner of
