Problem Statement
Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$ for all large $N$. Let $f(n)$ count the number of solutions to $n=p+a$ for $p$ prime and $a\in A$. Is it true that $\limsup f(n)=\infty$?
Categories:
Number Theory Primes
Progress
Erdős [Er50] proved this when $A=\{2^k : k\geq 0\}$.The answer is yes, as proved by Chen and Ding [ChDi22] - in fact, the assumption that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$ can be replaced just with the assumption that $A$ is infinite.
See also [236].
Source: erdosproblems.com/237 | Last verified: January 14, 2026