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It is well-known that the primes are uniformly distributed in residue classes modulo any fixed integer. More precisely, for each integer $q$ and each residue $a \in \mathbb{Z}/q\mathbb{Z}$ that is coprime to $q$, we have $$ \#\{ n < N : p_n \equiv a \bmod{q} \} \sim N/\varphi(q),$$ where, $p_n$ denotes the $n$-th prime and $\varphi(q)$ denotes the totient function. More precise bounds are also known, such as the Siegel–Walfisz theorem.

Is the analogous statement known for pairs of consecutive primes? Specifically, is it known that $$ \#\{ n < N : p_{n} \equiv a \bmod{q},\ p_{n+1} \equiv b \bmod{q}\} \sim N/\varphi(q)^2,$$ where $a,b$ are an arbitrary pair of residues coprime to $q$? I expect that the answer should be "no", since problems involving pairs of consecutive primes tend to be difficult.

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    $\begingroup$ Conjecturally the answer should be "yes", but I do also doubt this is known unconditionally. Certain biases are expected), but the ultimate asymptotics should be the same. See that paper also for known partial results. $\endgroup$
    – Wojowu
    Commented Feb 14, 2022 at 13:27
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    $\begingroup$ You might also have a look at "prime constellations." It's not exactly what you're asking about, but the concepts are somewhat related. $\endgroup$
    – Joshua Stucky
    Commented Feb 14, 2022 at 15:27
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    $\begingroup$ The paper that Wojowu linked to starts out by saying "if you look at the data, it will look like this is false" (and then goes on to explain where those biases come from). I mention this because my first instinct was to write some code to test this statement up to $10^6$ or so and from the data it looked false to me. $\endgroup$
    – Michael Lugo
    Commented Nov 1, 2023 at 14:43

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