Reference request: a class of diophantine equations

Question

I am interested in some results on equations of the form $$\displaystyle a_0 x_0^{\alpha_0} + a_1 x_1^{\alpha_1} + a_2 x_2^{\alpha_2} = 0,$$ with $a_0, a_1, a_2 \in \mathbb{Z}$ and $\alpha_0, \alpha_1 | \alpha_2$. I was told that in the case $a_0 = a_1 = -a_2 = 1$ and $\alpha_0 = 2, \alpha_1 = 3, \alpha_2 = 6$ it is known that the number of solutions with $|x_i| \leq B$ is of order $\log B$. I am not aware of a reference for this result. I am also interested in any information on other cases.

Thanks in advance.

Answer 1

I just want to make a comment, that one can in general find a lot of information on the (primitive) integer solutions of the generalized Fermat equation $Ax^p+By^q=Cz^r$ (depending on the three cases $\delta<0$, $\delta=0$ and $\delta>0$, where $\delta=1-1/p-1/q-1/r$).
For example, in the survey "The ABC's of Number Theory" by Noam Elkies, there are many results and conjectures explained (and the implications of the abc-conjecture in this context). For the cases $(p,q,r)=(2,4,4), (2,3,6)$ one obtains elliptic curves, and even for general $A,B,C$ (where there may be infinitely many equivalence classes of solutions) the number of representatives in the range $\max(|A|, |B|, |C|)<N$ then grows as a multiple of $\log(N)^{\rho/2}$, where $\rho$ is the rank of the corresponding elliptic curve. If $\delta>0$, then there are only finitely many primitive solutions (Darmon-Granville), and for special cases, many partial results are known (some list is given in "The generalized Fermat equation : a progress report", by Michale Bennett; this includes cases with $p\mid q$).