OPEN
This is open, and cannot be resolved with a finite computation.
Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all except finitely many integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$?
Even the case $k=3$ seems difficult. This may be true with the primes replaced by any set $A\subseteq \mathbb{N}$ such that\[\lvert A\cap [1,N]\rvert \gg N/\log N\]and\[\sum_{\substack{n\in A\\ n\leq N}}\frac{1}{n} -\log\log N\to \infty\]as $N\to \infty$.
For $k=1$ or $k=2$ any set $A$ such that $\sum_{n\in A}\frac{1}{n}=\infty$ has this property.
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This page was last edited 01 October 2025.
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