Revision history of
1141. All highlighted changes are compared to the current version.
Current version
Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2<n$?
In
[Va99] it is asked whether $968$ is the largest integer with this property, but this is an error, since for example $968-9=7\cdot 137$.
The list of $n$ satisfying the given property is
A214583 in the OEIS. The largest known such $n$ is $1722$.
ChatGPT and Tang
have shown that the number of such $n$ in $[1,N]$ is at most $N^{1/2+o(1)}$.
An internal OpenAI model (see
[APSSV26b]) has resolved this, proving that for any integer $a\geq 1$ there are only finitely many $n$ such that $n-ak^2$ is prime for all $k$ with $(k,n)=1$ and $ak^2<n$, deducing this from a result of Pollack
[Po17].
See also
[1140] and
[1142].
2026-04-09 09:09:38
Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2<n$?
In
[Va99] it is asked whether $968$ is the largest integer with this property, but this is an error, since for example $968-9=7\cdot 137$.
The list of $n$ satisfying the given property is
A214583 in the OEIS. The largest known such $n$ is $1722$.
ChatGPT and Tang
have shown that the number of such $n$ in $[1,N]$ is at most $N^{1/2+o(1)}$.
See also
[1140] and
[1142].
