DISPROVED (LEAN)
This has been solved in the negative and the proof verified in Lean.
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 $A$.)
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.
View the LaTeX source
This page was last edited 08 April 2026. View history
|
Likes this problem
|
None
|
|
Interested in collaborating
|
None
|
|
Currently working on this problem
|
None
|
|
This problem looks difficult
|
None
|
|
This problem looks tractable
|
None
|
|
The results on this problem could be formalisable
|
None
|
|
I am working on formalising the results on this problem
|
None
|
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #429, https://www.erdosproblems.com/429, accessed 2026-04-11
