OPEN
This is open, and cannot be resolved with a finite computation.
Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
Originally asked to Erdős by Kalmár. Erdős believed the answer is yes. Romanoff
[Ro34] proved that the answer is yes if $C$ is an integer.
Ding
[Di25] has proved that this is true for almost all $C>1$.
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This page was last edited 28 October 2025. View history
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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 #244, https://www.erdosproblems.com/244, accessed 2026-04-11
