DISPROVED
This has been solved in the negative.
Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\in A$?
Weisenberg
[We24] has shown the answer is no: $A$ can be arbitrarily sparse and missing at least one residue class modulo every prime $p$, and yet $A+n$ is not contained in the primes for any $n\in \mathbb{Z}$. (Weisenberg gives several constructions of such an $A$.)
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T. F. Bloom, Erdős Problem #429, https://www.erdosproblems.com/429, accessed 2026-01-16
