Institute of Mathematics

Abstract:  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 $n > k \geqslant3$. 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).