computer checking

yes thats true, but when we say diriclhets theorem in the elementary form we all know what we mean.

I give an answer with the hint for the proof

Do you think that i should write an answer and write down them?

I know that (C) implies (D).I have a very simple proof for that and for all the other :" to the previous question, if we take 2 arithmetic progressions for each pi>M, instead of 1 with the same condition then we have something stronger than the twin prime conjecture and polignac's conjecture in general, If we take 3 arithmetic progressions we have something stronger than the problem that infinitely many times exists every logical form with 3 primes ( for example p, p+2,p+6), etc. To these problems it is enough to take infinitely many M, not all..."

I think that if my conjecture is true then it means that something smaller than h(n) ( in the meaning that you dont use every prime ) is not bigger than p^2/M for every n .

Yes of course in this case we could have a counter example but if my first conjecture is true the Dirichlets theorem is a direct consequence.

thank you for your answer, the motivation to require p_i relatively prime to M has to do with the relation to diriclhlet's theorem and the related questions. I am sorry but i cant ubderstand why this version of my question is weaker, in my opinion it is stronger because it is more easy (maybe) to prove that the A_i's cant cover all the naturals if every k_i is greater than p_i^2/M than for all except finitely...

@S.Carnahan: i fixed that, is it clear now?

@S.Carnahan: if you write in the question: for each prime take 2 arithmetic progressions $A_i1, A_i2$ with the same condition ..., icant understand why this is unclear

@Gerhard Paseman: i am still waiting for an answer

Mainly i am looking for certain techniques and directions than arithmetical results

Thank you, for your advices too ,i appreciate any help , my email hasnt changed.