Let $M$ be a natural number $M>1$. For every prime $p_i$ not dividig $M$ take an arithmetic progression $A_i=k_i+np_i$ , $n\geq 0$ such that $k_i>p_i^2/M$. Is there any $M$ and some choice of the $k_i's$ such that $\cup A_i= \mathbb{N}$ ?

• MOTIVATION: it is not hard to see that if there is not such a choice for any $M$ Dirichlet's theorem in arithmetic progressions is an easy consequence. Of course this is much more general.

-Can we compute the function of the length of the intervals that we can cover using the primes not dividing $M$ below some $x$ and compare it to the growth function of the number of the primes below some $x$ ? We can notice that even for the primes below $M$ the arithmetic progressions can start from $1$ , the first term of every $A_i$ for $p_i < M$ can be less than $p_i$ this changes and the first term of the $A_i's$ becomes larger and larger comparing with the $p_i's$

related to Covering $\mathbb{N}$ with prime arithmetic progressions