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Given a positive integer $n$, let $f_1(n)$ denote the number of pairs of consecutive prime numbers $\leq n$ which are incongruent modulo 3, and let $f_2(n)$ denote the number of pairs of consecutive prime numbers $\leq n$ which are congruent modulo 3. (Thus in particular $f_1(n) + f_2(n) = \pi(n) - 1$).

Numerically, one observes a strong bias towards $f_1(n)$ being larger than $f_2(n)$. In fact, the paper

Robert J. Lemke Oliver, Kannan Soundararajan: Unexpected biases in the distribution of consecutive primes

states a fairly general conjecture which would imply that $$ \lim_{n \rightarrow \infty} \frac{\log(f_1(n) - f_2(n))}{\log{n}} \ = \ 1. $$ The outcome of numerical computations seems to be in line with this claim.

Question: Which is the strongest result in this direction which is currently known, or which can be derived from known results by straightforward methods? -- In particular, can it at least be shown that there is no $n$ such that $f_2(n) > f_1(n)$?

The following plots of the ratio $q(n) := f_1(n)/f_2(n)$ and of the difference $d(n) := f_1(n) - f_2(n)$ range from $n = 2^{12}$ to $n = 2^{34}$ in logarithmic scale on the horizontal axis, and from $q(n) = 1$ to $q(n) = 2$ in linear scale, respectively, from $d(n) = 1$ to $d(n) = n$ in logarithmic scale, on the vertical axis:

plots of the ratio f_1(n)/f_2(n) and of the difference f_1(n)-f_2(n)

Observe that while the graph of $d$ almost looks like a straight line, it seems to have in fact a slightly, but steadily increasing slope.

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    $\begingroup$ I think even that very weak statement is not known and can't be derived from known results, but I'm not sure. $\endgroup$
    – Will Sawin
    Commented Sep 25, 2020 at 13:43
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    $\begingroup$ A more careful quantification of the Lemke Oliver-Soundararajan results should be able to derive these bounds from a suitably quantitative and uniform form of the Hardy-Littlewood conjectures, but unconditional results are likely out of reach right now (except possibly in the function field model, in which we do now have some reasonably good form of the HL conjectures in some cases). $\endgroup$
    – Terry Tao
    Commented Sep 25, 2020 at 15:50
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    $\begingroup$ Is it at the very least known, unconditionally, that $f_1(n)>f_2(n)$ infinitely often? $\endgroup$
    – Wojowu
    Commented Sep 25, 2020 at 21:57
  • $\begingroup$ @Wojowu I am not aware of such result. -- But maybe someone else is? $\endgroup$
    – Stefan Kohl ♦
    Commented Sep 26, 2020 at 12:15

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