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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.

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The answer to question 1 is yes; see Tristan Freiberg's Ph.D. thesis Strings of congruent primes in short intervals. 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.

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  • $\begingroup$ Thank you @Joni for your answer. I am impressed that you have extracted a result from page 72 which was not mentioned in the introduction. Just in case, could you tell me please about the phrase "However, we only show that these strings satisfy..." from that same page 72, does it mean that the result is technically stronger than the main result, or that it is only a partial result? (I think that the first is true, but would like to be sure). $\endgroup$ – Włodzimierz Holsztyński Jun 13 '14 at 5:40
  • $\begingroup$ The phrase refers to the author's result mentioned in the introduction that infinitely many intervals as short as $(x,x+\varepsilon \log x)$ ($\varepsilon>0$ arbitrary) contain two primes $p_n,p_{n+1}\equiv a \pmod q$. For the general case, the $k$ consecutive congruent primes were found on intervals of the form $(x,x+c(q)k\log x)$, so it's not really a generalization of the main result, but it complements it nicely. $\endgroup$ – Joni Teräväinen Jun 13 '14 at 18:50

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