OPEN
This is open, and cannot be resolved with a finite computation.
Are there infinitely many $n$ (or any $n>105$) such that $n-2^k$ is prime for all $1<2^k<n$?
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.
The only known such $n$ are\[4,7,15,21,45,75,105.\]This is
A039669 in the OEIS.
Mientka and Weitzenkamp
[MiWe69] have proved there are no other such $n\leq 2^{44}$.
Vaughan
[Va73] has proved that the number of $n\leq N$ such that $n-2^k$ is prime for all $1<2^k<n$ is\[< \exp\left(-c\frac{\log \log \log N}{\log\log N}\log N\right)N\]for some constant $c>0$.
This is discussed in problem A19 of Guy's collection
[Gu04]. There is also further discussion on
the Prime Puzzles website.
Erdős made the stronger conjecture (see
[236]) that that number of $1<2^k<n$ for which $n-2^k$ is prime is $o(\log n)$.
View the LaTeX source
This page was last edited 05 March 2026. View history
External data from
the database - you can help update this
Formalised statement?
Yes
Related OEIS sequences:
A039669
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This problem looks difficult
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ebarschkis
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This problem looks tractable
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Additional thanks to: ebarschkis
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 #1142, https://www.erdosproblems.com/1142, accessed 2026-04-11
