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Claim. Satz II of Romanoff's paper: for each fixed integer $a\ge2$ there is $\beta>0$, depending only on $a$, such that every interval $(0,x)$ contains more than $\beta x$ integers of the form $p+a^n$ with $p$ prime. Since $\lfloor C^k\rfloor=C^k$ for integer $C$, the set of Problem 244 has positive lower density for every integer $C\ge2$, which is a yes in the reading of the problem page's Formulation. The proof is a second-moment argument: the Cauchy–Schwarz inequality bounds the number of represented integers below by the square of the number of pairs $(p,a^n)$ over the number of coincidences $p+a^n=p'+a^{n'}$, and Brun-type sieve bounds on the coincidences reduce the theorem to the convergence of an auxiliary series, $\sum_l\sigma(l)/l$ in the paper's notation, which the paper proves on its last two pages.

Covers. Every integer $C\ge2$, answered yes. Not covered: every non-integer $C>1$.

Depends on. Nothing on this wiki; the claim rests on the cited paper.

Acceptance. Refereed: N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 109 (1934), 668--678, the paper link. The site's commentary credits the theorem with the yes for integer $C$, but the site labels the problem OPEN, so that remark is not acceptance of the problem and the page lists no reviewed evidence.

Dating. The page is dated by the volume's publication month, December 1934, as the publisher's record gives it; the day in the page name is a placeholder.