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Claim. Let $f(n)$ be the number of solutions of $n=2^k+p$ with $p$ prime; the paper does not state the range of $k$, and allowing or excluding $k=0$ changes $f(n)$ by at most one, which does not affect the theorem. Theorem 1 of P. Erdős, On integers of the form $2^k+p$ and some related problems (card), proved in answer to a question of Turán, states that $\limsup f(n)=\infty$, and in fact that $f(n)>c\log\log n$ for infinitely many $n$, with $c>0$ an absolute constant. The set $A=\{2^k:k\ge0\}$ has about $\log_2N$ elements up to $N$, so it satisfies the hypothesis $\lvert A\cap\{1,\ldots,N\}\rvert\gg\log N$ of Problem 237, and the theorem answers the problem's question yes for this $A$. The site's commentary credits this case to the paper. The general question is settled by Chen and Ding, the accepted full claim, whose Corollary 1.2 extends this theorem to every set of more than $\log x$ integers up to $x$ with the bound $\tfrac18\log\log x-1.6$.

Covers. The case $A=\{2^k:k\ge0\}$, answered yes. Not covered: any other set $A$ with $\lvert A\cap\{1,\ldots,N\}\rvert\gg\log N$; the general case is settled on Chen and Ding's page, not on this one.

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

Acceptance. Refereed: P. Erdős, On integers of the form $2^k+p$ and some related problems, Summa Brasil. Math. 2 (1950), fasc. 8, 113--123, the paper link (the Rényi Institute's archive of Erdős's papers). The site's commentary credits this case to the paper, but the site's label settles the problem through Chen and Ding and credits them, so no reviewed evidence is listed here. The page is dated by the fascicle's issue month, November 1950, printed on the fascicle's cover (Summa Brasil. Math. vol. 2, fasc. 8); the day in the page name is a placeholder.