Let $k$ be an nonzero integer and $b>2$ a real. Is it true that there exist only finitely many positive integer pairs $(n,m)$ for which $$ 2n^2-\lfloor m^b\rfloor=k? $$
I don't know the answer, but I guess it is positive.. To be precise, I think the following may be true:
Conjecture: Let $\alpha,\beta$ be positive integers, $k$ a nonzero integer, and $a,b$ be distinct reals greater than $1$. Then there exist only finitely many positive integer pairs $(n,m)$ for which $$ \alpha \lfloor n^a\rfloor-\beta\lfloor m^b\rfloor=k. $$ [It includes, as a special case, this other question (which has a positive answer).]