| Title:
|
Diophantine equations involving factorials (English) |
| Author:
|
Alzer, Horst |
| Author:
|
Luca, Florian |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
142 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
181-184 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $k=n$. (English) |
| Keyword:
|
Diophantine equation |
| Keyword:
|
factorial |
| MSC:
|
11D61 |
| idZBL:
|
Zbl 06738578 |
| idMR:
|
MR3660174 |
| DOI:
|
10.21136/MB.2016.0045-15 |
| . |
| Date available:
|
2017-05-23T09:59:47Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146751 |
| . |
| Reference:
|
[1] Andreescu, T., Andrica, D., Cucurezeanu, I.: An Introduction to Diophantine Equations. A Problem-Based Approach.Birkhäuser, Basel (2010). Zbl 1226.11001, MR 2723590, 10.1007/978-0-8176-4549-6 |
| Reference:
|
[2] Bashmakova, I. G.: Diophantus and Diophantine Equations.The Dolciani Mathematical Expositions 20. The Mathematical Association of America, Washington (1997). Zbl 0883.11001, MR 1483067 |
| Reference:
|
[3] Carnal, H.: Aufgaben.Elem. Math. 67 (2012), 151-154. Zbl 1247.97035, 10.4171/EM/203 |
| Reference:
|
[4] Luca, F.: The Diophantine equation $R(x)=n!$ and a result of M. Overholt.Glas. Mat. (3) 37 (2002), 269-273. Zbl 1085.11023, MR 1951531 |
| Reference:
|
[5] Luca, F.: On the Diophantine equation $f(n)=u!+v!$.Glas. Mat. (3) 48 (2013), 31-48. Zbl 06201413, MR 3064240, 10.3336/gm.48.1.03 |
| Reference:
|
[6] Sándor, J.: On some Diophantine equations involving the factorial of a number.Seminar Arghiriade. Univ. Timişoara 21 (1989), 4 pages. Zbl 0759.11011, MR 1124179 |
| . |
