I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in Fermat's Last Theorem.
Problem. We consider for positive integers $x,y,z\geq 1$ and being pairwise coprime, $(x,y)=(x,z)=(y,z)=1$, and for a fixed integer $n\geq 3$ the diophantine equation that involves partial sums of exponentials $$\sum_{k=0}^n\frac{x^k}{k!}+\sum_{k=0}^n\frac{y^k}{k!}=\sum_{k=0}^n\frac{z^k}{k!}.\tag{1}$$ The problem asks if it is possible to determine if $(1)$ for our given integer $n>2$ has solutions.
Question. I don't know if previous problem is in the literature. Is it possible to determine for a given integer $n\geq 3$ if the diophantine equation (the problem) have solutions $(x,y,z)$ in coprime positive integers $x,y,z$? Many thanks.
If this problem is in the literature please refer it in your comments of answer.
Example (Case $n=2$). One can to get easily solutions for the equation with $n=2$ that isn't considered in our Problem. For example $(x,y,z)=(17,5,18)$.
Remark. I know that the polynomial $P(x)=1+x+\frac{x^2}{2!}+\ldots+\frac{x^n}{n!}$ is in the literature and enjoys of certain properties.
