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:


1. $\ p\in B$
2. $\ |B| = k$
3. $\ \forall\_{q\in B}\ \ \ q\equiv r \mod 2\!\cdot a$


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.



Prime residua races and two views on primes
The answer to question 1 is yes; see Tristan Freiberg's Ph.D. thesis [Strings of congruent primes in short intervals](https://papyrus.bib.umontreal.ca/xmlui/bitstream/handle/1866/4556/freiberg_tristan_m_2010_these.pdf;jsessionid=770A3B94C710637E4D59BA5A5D53DBD5?sequence=4). There it is mentioned that even the case $|B|=2$ was an old conjecture of Chowla, solved by Daniel Shiu in 2000. D. A. Goldston, C. Pintz and C. Y. Yildirim proved in 2006 that for any $\varepsilon>0$, the short interval $(x,x+\varepsilon \log x)$ contains two primes congruent to $a\pmod q$ for infinitey many integers $x$, but not that these two primes are consecutive. Starting from page 72 in the thesis, question 1 is settled. Given that question 1 was only solved recently, I suspect that questions 2 and 3 are open, if true.


