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.



Reference request: a class of diophantine equations
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 $p\mid q$).


