Problem 279

Erdős Problem #279: declared status 'open'. Formalized: yes. 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 $t\geq k$? Current best: 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 $N\to \infty$. Prize: no. Tags: covering systems, number theory, primes.