Let $\ a>1\ \ r\ \ k\ $ be arbitrary natural numbers such that $\ a\ r\ $ are relatively prime. The natural conjecture below, is it known?, is probably true in full generality:

**Q1.** There exists a set $\ B\ $ of $\ k\ $ consecutive primes such that $\ q\equiv r \mod 2\!\cdot\! a\ $ for every $\ q\in B$.

(The connection of this conjecture to arithmetic progressions of consecutive primes is obvious but otherwise I don't feel that these two are too strongly related).

Primes, but $\ 2\ 3,\ $ fall into two residua classes $\mod 6,\ $ and also $\mod 4.\ $ This may be a start point of a discussion about primes with respect to either of these modules separately or especially to their comparison. The question below may be a step in this direction.

I assume that Conjecture Q1 is true (but even when it is not so then question Q2 may still make sense partially).

Let $\ p := \beta(a\ r; k)\ $ be the smallest natural number such that there exists a element set $\ B\ $ of consecutive primes such that:

- $\ p\in B$
- $\ |B| = k$
- $\ \forall_{q\in B}\ \ \ q\equiv r \mod 2\!\cdot a$

Furthermore, let $\ \beta_k := \min_{a\ r} \beta(a\ r;\ k)$.

**Q2.** Is it true that whenever $\ \beta(a\ r; k) = \beta_k,\ $ then $\ a = 2\ $ or $\ 3$?

Now let me risk:

**Q3.** Is $\ \min(\,\beta(2\ 1; k)\ \ \beta(2\ 3;k))\ <\ \min(\,\beta(3\ 1;\ k)\ \ \beta(3\ 5;\ k))\ \ $ for every $\ k \ge 9\,$? ...or for almost all $\ k\,$?

More questions come to mind, including comparisons of two residua for the same module, but the 3 questions above already outline the topic.