Can one prove that for a prime number $p \geq 2$ and two given integers $d>0$ and $c$, the number of integer solutions $(X,Y)$ to the equation $X^p-dY^p=c$ inside the box $|X| \leq N$, $|Y| \leq N$ is bounded above by $K \log^{p-1} N$, where $K$ is a constant which depends only on $c$ and $d$?
Such a result holds for example when $p=2$ from the theory of Pell's equation, but I could not generalize the arguments there to higher degrees, one reason being that we have more than one fundamental unit in the real extension $\mathbb{Q}(\sqrt[p]{d})$.
For $p>2$ you get an even tighter bound $K \log^{(p-1)/2} N$. The explanation is because in that case, the group of units in the number field given by the $p$'th root of $d$ has rank $r+s-1 = (p-1)/2$. I can add more details if you want, but the upshot is that if the number of solutions grew faster than this bound, then the ring of algebraic integers in this field would have a larger group of units than allowed by the $r+s-1$ formula, where $r$ is the number of real embeddings (i.e. 1) and $2s$ is the number of complex embeddings (the proof of Dirichlet's unit theorem contains all that you need. In fact, you only need half of the proof, namely the part that shows that $r+s-1$ is an upper bound for the rank of the units).