Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
Diophantine equations; factorial
Diophantine equations; factorial
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).
References:
[1] Arzel, H., Luca, F.: Diophantine equations involving factorials. Math. Bohem. 142 (2017), 181-184. DOI 10.21136/MB.2016.0045-15 | MR 3660174 | Zbl 1424.11080
Search
Partner of
