Let $c_1, \ldots, c_k \in \mathbf N^+$ and $x_1,\ldots,x_k \in \mathbf Z \setminus \{0\}$. It is possible to prove by elementary means that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is a bounded sequence only if $|x_1|=\cdots=|x_k|$. (As usual, $\omega(x)$ is, for every non-zero $x \in \mathbf Z$, the number of distinct prime divisors of $x$, while $\omega(0) := \infty$.)

On the other hand, there seems to be more than a chance that the same conclusion may also come as an *easy* consequence of the $p$-adic Subspace Theorem. This sounds plausible to me, but I don't see how to proceed (and, to be honest, I'm not very familiar with the many uses of the Subspace Theorem and its variants and generalizations). So I thought to ask here, with the hope that the question is not completely trivial for the experts in the field and not totally uninteresting for the others.

> **Q.** Can the statement made in the first paragraph of this post be obtained, *in a more or less straightforward way*, from the $p$-adic Subspace Theorem? If so, is a proof along these lines written down anywhere?