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Claim. Let $f(z)=\sum_{k\ge1}a_kz^{n_k}$ be an entire function with $a_k\ne0$ for every $k$. G. Pólya, Untersuchungen über Lücken und Singularitäten von Potenzreihen, Math. Z. 29 (1929), no. 1, 549--640, doi:10.1007/BF01180553, proves that if $f$ has finite order and

$$\limsup_{k\to\infty}(n_{k+1}-n_k)=\infty,$$

then $f$ takes every complex value infinitely often. The statement follows the site's account of the paper, which is cited there as [Po29]; the paper is not held in this repository, so its own numbering of the result and its proof are not reconstructed here.

Covers. Every instance of Problem 517 in which $f$ has finite order. The question assumes $n_k/k\to\infty$, and this forces Pólya's gap condition: if the gaps were bounded, say $n_{k+1}-n_k\le G$ for all $k\ge k_0$, then $n_k\le n_{k_0}+G(k-k_0)$ and $n_k/k$ would stay bounded. So for an entire function of finite order the answer is yes. The case of infinite order is not covered and remains open beyond the subclass with $\sum1/n_k<\infty$, which Biernacki 1927 and Murai 1983 settle.

Depends on. Nothing in this wiki: the theorem is the paper's own, and the reduction of the question's hypothesis to the paper's is the elementary bound above.

Acceptance. Refereed: a journal publication, Mathematische Zeitschrift 29 (1929), no. 1, the DOI linked above; the publisher's record dates the issue December 1929, filled to the first of the month for this page's name. Reviewed is not listed: the site credits the theorem in its commentary on a problem it labels OPEN, which is commentary and not acceptance of a solution. Formalized is not listed: no Lean statement or proof of the theorem is recorded.