Geodesic
Diophantine equations involving factorials
Mathematica Bohemica, Tome 142 (2017) no. 2, pp. 181-184
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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 $k=n$.
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 $k=n$.
DOI : 10.21136/MB.2016.0045-15
Classification : 11D61
Keywords: Diophantine equation; factorial 
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Alzer, Horst; Luca, Florian. Diophantine equations involving factorials. Mathematica Bohemica, Tome 142 (2017) no. 2, pp. 181-184. doi: 10.21136/MB.2016.0045-15

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