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Every pair of twin primes $p$, $p+2$ will have either $p \equiv 1 \pmod{4}$ or $p \equiv 3 \pmod{4}$. Is there any work towards whether one of these is "more common", the way that Dirichlet's theorem on primes in arithmetic progressions came out of asking whether primes which are $1 \pmod 4$ are more common than primes which are $3 \pmod 4$?

I've made a few plots in python exploring this for $p \equiv \pm1 \pmod{4}$ using this list, as well as for whether $p$ is $5$ or $11 \pmod {12}$. The difference between the counts for each congruence class seems to fluctuate randomly, and the maximum difference (among the first 100,000 pairs of twin primes) was about 175 for both, achieved somewhere between the 15,000th pair and the 20,000th pair. (The plots are very rough, since I'm not that good with python, so I kind of eyeballed those stats.)

Obviously you can't just answer yes since we don't know if there are infinitely many twin primes at all. But I think the question is an interesting extension of the twin prime conjecture, and I would be interested to hear if you can answer no! Is it maybe possible to assume the twin prime conjecture and conclude that their must be infinitely many in each possible congruence class? ($1$ and $3 \pmod 4$; $5$ and $11 \pmod {12}$; $1, 2, 3, 4,$ and $6 \pmod 7$, etc.)

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    $\begingroup$ Or more generally the distribution of $p$ in invertible residue classes $a+m\mathbb{Z}$ such that $a+2$ is also invertible. $\endgroup$
    – Chris Wuthrich
    Commented Feb 23, 2023 at 9:53
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    $\begingroup$ This paper: arxiv.org/abs/1603.03720 and reference [9] within it, might suggest that there should be no differences in this case, but if you look at larger clusters of primes then numerical data might show differences among the allowed configurations. $\endgroup$
    – David Farmer
    Commented Feb 27, 2023 at 23:28

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