# Prime residua races and two views on primes

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.

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.
• 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. Jun 13, 2014 at 18:50