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Define the length of a set of arithmetic progressions of natural numbers $A=\lbrace A_1, A_2, \ldots \rbrace$ to be $\min_i | A_i |$: the length of the shortest sequence among all the progressions. Say that $A$ exactly covers a set $S$ if $\bigcup_i A_i = S$. Let $P'$ be the primes excluding 2.

What is the longest set of arithmetic progressions that exactly covers the primes $P'$?

In other words, I want to maximize the length of a set of such arithmetic progressions. Call this maximum $L_{\max}$.

$L_{\max} \ge 2$ because $$P' \;=\; \lbrace 3,5 \rbrace \cup \lbrace 7,11 \rbrace \cup \cdots \cup \lbrace 521,523 \rbrace \cup \cdots$$ Perhaps it is possible that $$P' \;=\; \lbrace 3, 11, 19 \rbrace \cup \lbrace 5, 17, 29,41,53 \rbrace \cup \lbrace 7,19,31,43 \rbrace \cup \cdots \;,$$ but I cannot get far with sequences of length $\ge 3$. (I know Green-Tao establishes that there are arbitrarily long arithmetic progressions in $P$, but I don't know if that helps with my question.)

I am number-theoretically naïve, and apologize if this question is nonsensical or trivial. In any case, I appreciate the tutoring!

Addendum. Although my question should be revised (as Idoneal suggests) in light of George Lowther's proof that 3 cannot be in a progression of length 4, George has shown that it is likely that $L_{\max}=3$ but certification requires resolving an open problem. So I've added the open-problem tag. Thanks for everyone's interest!

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# Covering the primes by arithmetic progressions

Define the length of a set of arithmetic progressions of natural numbers $A=\lbrace A_1, A_2, \ldots \rbrace$ to be $\min_i | A_i |$: the length of the shortest sequence among all the progressions. Say that $A$ exactly covers a set $S$ if $\bigcup_i A_i = S$. Let $P'$ be the primes excluding 2.

What is the longest set of arithmetic progressions that exactly covers the primes $P'$?

In other words, I want to maximize the length of a set of such arithmetic progressions. Call this maximum $L_{\max}$.

$L_{\max} \ge 2$ because $$P' \;=\; \lbrace 3,5 \rbrace \cup \lbrace 7,11 \rbrace \cup \cdots \cup \lbrace 521,523 \rbrace \cup \cdots$$ Perhaps it is possible that $$P' \;=\; \lbrace 3, 11, 19 \rbrace \cup \lbrace 5, 17, 29,41,53 \rbrace \cup \lbrace 7,19,31,43 \rbrace \cup \cdots \;,$$ but I cannot get far with sequences of length $\ge 3$. (I know Green-Tao establishes that there are arbitrarily long arithmetic progressions in $P$, but I don't know if that helps with my question.)

I am number-theoretically naïve, and apologize if this question is nonsensical or trivial. In any case, I appreciate the tutoring!