What about $n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}}$ over positive integers?

Question

I've considered the following equation for positive integers $x,y,z\geq 1$, and for positive integers $n\geq 2$

$$n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}},\tag{1}$$ where the pattern of exponent is a cyclic combination of those fractions.

Question 1. Is it known from the literature? Alternatively, if isn't in the literature provide discussion for what work can be done to get the solutions of this equation. Many thanks.

If it is a well-known equation refer/comment it and I try to search the solution and read it from the literature.

Computational facts. The solutions $(x,y,z;n)$ that I can get using a Pari/GP program searching in the segment $1\leq x,y,z\leq 40$, and for $2\leq n\leq 40$, are $(x,y,z;n)=(1,2,1;4)$ $(2,6,2;8)$ and $(2,4,2;16)$. If I'm right the requirement/conditions $y-x\mid xy$ and $y-z\mid yz$ arise when I am under the assumption or case to find solutions of the form $\alpha:=\frac{1}{x}=\frac{1}{z}$ from the deduction $$\frac{1}{\alpha-\frac{1}{y}}=\log_2 n.\tag{2}$$

Question 2. In previous question I try to explore the equation $(1)$ that I've considered yesterday, but is it possible to state a more interesting equation than mine? Can you propose a similar and more interesting equation, or system of equations? Many thanks.

I evoke a more interesting equation than mine, I don't know if equations of the previous type are more or less interesting $$n^{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}}+n^{\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4}}+n^{\frac{1}{x_3}+\frac{1}{x_4}+\frac{1}{x_1}}=n^{\frac{1}{x_4}+\frac{1}{x_1}+\frac{1}{x_2}},$$ with Question 2 I am looking for tips to improve the mathematical content of my Question 1, asking about how to create a more interesing equation similar than $(1)$, you can add special requirements in your proposal.

Answer 1

Question 1: Inspired by the ones you found we can see that there are infinitely many solutions as follows: $$(x,y,z;n) = (k-1,\quad k(k-1),\quad k-1;\quad 2^k)$$ for any $k\ge 0$.

Edit re: Question 2: How about instead of $$n^{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}}+n^{\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4}}+n^{\frac{1}{x_3}+\frac{1}{x_4}+\frac{1}{x_1}}=n^{\frac{1}{x_4}+\frac{1}{x_1}+\frac{1}{x_2}}$$ you make it $$n^{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}}+n^{\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4}}=n^{\frac{1}{x_3}+\frac{1}{x_4}+\frac{1}{x_1}}+n^{\frac{1}{x_4}+\frac{1}{x_1}+\frac{1}{x_2}},$$ so that it has greater symmetry.