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Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ?

$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$

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    Sorry $f$or delay in answer @J.D , Yes .2012-02-27

2 Answers 2

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The problem

$N=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$

with $N,x,y,z \in \mathbb{Z}$ was considered by Andrew Bremner and Richard Guy in "Two more representation problems" published in the Proceedings of the Edinburgh Mathematical Society, vol. 40 pp.1-17 in 1997. An online copy is available here. They showed solutions only occurred for those $N$ where the elliptic curve

$t^2=u^3+N^2u^2+8Nu+16$

has rank at least $1$.

For small $N>0$, the first solution is for $N=6$, with $x=18$, $y=4$ and $z=3$.

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    Wonderful! It is another reason for me to try to learn something about elliptic curves.2012-03-01
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Maple code that searches for solutions in specific range :

for q from 4 to 30 do for x from 1 to 200 do for y from 1 to 200 do for z from 1 to 200 do if x/y+y/z+z/x=q then print(q,x,y,z); end if; end do; end do; end do; end do; 

For $~q=9~$ ;$~(x,y,z)=(12,63,98)$