| Title:
|
On some diophantine equations involving factorials (English) |
| Author:
|
Gnatowski, Maciej |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
151 |
| Issue:
|
3 |
| Year:
|
2026 |
| Pages:
|
423-426 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the Diophantine equations $(n!)^k - n^k = (k!)^n - k^n$ and $(n!)^k + n^k = (k!)^n + k^n$, where $k$ and $n$ are positive integers. According to H. Arzel, F. Luca (2017), only the first equation has nontrivial solutions. It is proved also that ${(n!)^k - n^k > (k!)^n - k^n}$ for $n > k \geqslant 3$. In the paper we prove that the stronger inequality $(n!)^k - n^k(n-1)^k > (k!)^n - k^n$ holds. We propose also an alternative proof of the results of H. Arzel, F. Luca (2017). (English) |
| Keyword:
|
Diophantine equations |
| Keyword:
|
factorial |
| MSC:
|
11D61 |
| DOI:
|
10.21136/MB.2025.0043-25 |
| . |
| Date available:
|
2026-08-24T07:51:44Z |
| Last updated:
|
2026-08-24 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153715 |
| . |
| Reference:
|
[1] Arzel, H., Luca, F.: Diophantine equations involving factorials.Math. Bohem. 142 (2017), 181-184. Zbl 1424.11080, MR 3660174, 10.21136/MB.2016.0045-15 |
| . |
