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!