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.



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 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.


