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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 $A$.)

In [Er80] Erdős further asks whether if $A$ does not cover all residue classes modulo $p^2$ for any prime $p$ then there are infinitely many $n$ such that all of $n+a_k$ are squarefree; a variant of Weisenberg's construction also shows the answer is no.

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This page was last edited 08 April 2026. View history

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• Aristotle formalized the second proof of [We24]. You used to be able to type-check it online here, but unfortunately that website no longer seems to support Lean version 4.24.0.

  (The site has been updated to address this comment.)
  Woett — 08:40 on 29 Jan 2026