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I would like to ask the broad community what is known about the solutions of diophantine equation $$\frac{u}{v} +\frac{v}{w} +\frac{w}{u} =t$$ where $t,v,u,w\in \mathbb{N}.$

I read a book of W. Sierpinski, $\text{ 250 Problems in Elementary Number Theory}$ and there is that this is still an open problem. Is this true? Could anyone give me some references?

I know from MSE that that equation has a solutions of the form $(u,v,w,t) =(k,k,k,3)$ and $(u,v,w,t)=(k,2k,4k,5)$ and also I know that solutions of this equation need not be of the form $(u,v,w) =(a^2 b, b^2 c ,c^2 a).$

I would like to know what is known about solutions of this equation? Any references to this problem will be welcome.

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The values of $t$ for which a solution exists are tabulated at the Online Encyclopedia of Integer Sequences. There are also references there and links to further information. In particular, there is a reference to the paper, Andrew Bremner and Richard K. Guy, Two more representation problems, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 1, 1–17, MR1437807 (98d:11037). The review by Michel Olivier says solutions to $n=x/y+y/z+z/x\,(*)$ correspond to rational points on the elliptic curves $Y^2=X^3+(nX+4)^2$; the paper tabulates $n$ for which $(*)$ has a solution.

The OEIS entry also links to this page of Dave Rusin. That in turn links to these two discussions, which take up where Bremner & Guy left off.

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