Solutions of $q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$ s.t. $q \geq 3$

Question

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

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

Answer 1

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 $z=3$.

Answer 2

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