I don't know if either I didn't quite get the lemma or the problem has an elegant solution. So, I got this problem from a friend, and the hint was to use Hensel's lemma, which I didn't know and searched in Wikipedia, I couldn't see how to use it and solved it in other way.
What I understood of the lemma is that it says if $f(x)\in\mathbb{Z}[x]$ and $a$ is such that $f(a)\equiv_p0$ for some prime $p$ then $a$ is a unique root for the same equation modulo $p^k$.
The problem was to solve
$$2^x=3^y+17$$
over the integers. But I don't see any polynomial I could use
Since $2\equiv 3^{14}\pmod{17}$, we have $3^{14x}\equiv 3^y\pmod{17}$, hence $y\equiv 14x\pmod{16}$ from which $y\equiv 2x\pmod 4$. Reducing modulo $5$ gives $3^y\equiv 3^{2x}\equiv 4^x\pmod 5$, hence $2^x\equiv 4^x+2\pmod 5$ which is impossible for $x\in\{0,1,2,3\}$ hence impossible for all $x\in\Bbb Z$.