An equation demanding solutions in $\mathbb{Q}^3$

Question

Playing around the problem in the book of a library . I deduced a question to finding all $(a,b,c) \in \mathbb{Q}^3$ such that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=0$$ But now I don't know anything further?

Answer 1

By multiplying $a,b,c$ with a suitable rational number, you can reduce the problem to finding integer solutions with no factor in common.

Then, we want the nonzero integer solutions to $a^2c+b^2a+c^2b = 0$.

If $p$ is a prime number and $p^k \mid a$, then $p^k \mid c^2b$ so either $p^k \mid b$ (in which case $p^{2k} \mid c^2b$, so $p^{2k} \mid b$), either $p^k \mid c^2$ (in which case $k$ has to be even and $p^{k/2} \mid c$).

So by regrouping prime factors accordingly, you can write $a=u^2v, b = v^2w, c = w^2u$ with $u,v,w$ pairwise coprime.

You end up with the equation $u^2v^2w^2(u^3+v^3+w^3) = 0$. This has no nonzero integer solution as was proven by Fermat.