Let $a$ be a fixed constant like 2 or 3 (for example). Let $P(x)$ be a single variable polynomial in x with integer coefficients.
What is known about the natural number solutions $(x,y)$ of
$$P(x) = a^y$$
can we find a bound $x < B$? Is this a known/named problem?
You should read about Hensel lifting. Start with the fact "$P(x) = a^y$" implies "$P(x) \cong 0 \pmod{a^y}$", although keep in mind that the implication does not work in the reverse direction.